Universidad de Sevilla
Escuela Superior de Ingenieros
Ingenier´ia de Telecomunicaciones
Quantum Dot Lasers
Candidato: Tutor en Espa˜na: Jaime P. Mora Pacheco Alejandro Carballar
Tutora en Italia: Gabriella Cincotti
A˜no Acad´emico: 2004/2005 Contents
Introducci´on 1
0 Resumen espa˜nol 4
Resumen Capitulo 1: Conceptos Fundamentales ...... 4
0.1 Pozoscu´anticos...... 5
0.2 PuntoCu´antico(QD) ...... 7
0.2.1 DensidaddeEstados...... 8
0.3 CristalesFot´onicos ...... 9
0.4 Definici´onyventajas...... 12
0.5 Fabricaci´ondeQDs...... 13
0.5.1 Elm´etodoStranski-Krastanow ...... 13
0.6 DOS:funci´ondedensidaddeestados ...... 15
0.7 EcuacionesdeestadodelQDLaser ...... 16
0.8 CorrienteUmbral...... 17
0.9 Q-Switching...... 17
0.10VCSELs...... 19
Resumen Capitulo 2: L´aseres de Puntos Cu´anticos ...... 11
I 0.11 EnsanchamientoHomog´eneo...... 20
0.12 EnsanchamientoInhomog´eneo...... 21
ResumenCapitulo3:EfectosNoLineales ...... 20
Resumen Capitulo 4: Aplicaci´on a Cristales Fot´onicos ...... 24
Conclusiones ...... 27
1 Fundamental Concepts 30
1.1 PrinciplesofLasers...... 31
1.1.1 Populationinversion ...... 32
1.1.2 Threshold Conditions & Optical Gain ...... 34
1.2 Quantumwells ...... 35
1.3 Quantumdot ...... 38
1.4 DensityofStates:DOS...... 39
1.5 ANewDevice:PhotonicCrystal ...... 40
2 Quantum Dot Lasers 43
2.1 QuantumDotsfabrication...... 47
2.1.1 Lithography Based QD Laser Fabrication ...... 49
ElectronBeamLithography ...... 49
DryEtching...... 50
2.1.2 Stranski-Krastanowmethod ...... 52
Temperature ...... 55
Verticallystacking ...... 56
Activated Alloy Phase Separation ...... 57
II Defectreductiontechnique ...... 58
2.2 DensityofStates:DOS...... 60
2.3 RateEquationsofQDLaser...... 64
2.4 Thresholdcurrent ...... 70
2.5 DifferentialGain ...... 77
2.6 Q-Switching...... 80
2.7 VCSELs...... 84
3 Non-linear Effects 88
3.1 Homogeneousbroadening ...... 90
3.2 Inhomogeneousbroadening ...... 95
3.2.1 GainandDifferentialGain ...... 100
3.2.2 ThresholdCurrent ...... 106
3.3 Cross-GainModulation ...... 108
4 ApplicationstoPhotonicCrystalStructures 111
Conclusions 125
Appendix 128
4.1 Appendix1:Fermi’sGoldenRule ...... 128
4.2 Appendix2:Smith-PurcellEffect ...... 131
4.3 Appendix3:MeasurementsTools ...... 132
4.3.1 AFM...... 132
4.3.2 TEM...... 132
III Acknowledgments 134
List of Figures 139
Bibliography 139
IV Introducci´on
Este Proyecto Fin de Carrera ha sido realizado en la Universidad de Roma Tre (Roma,
Italia) gracias a una beca Erasmus para el curso 2004-2005. Como proyecto realizado en el extranjero y acerca de una tecnolog´ia de ´ultima generaci´on, ha sido elaborado completamente en ingl´es, por lo que este documento pretende resumir en espa˜nol el contenido del mismo de forma breve pero explicativa.
La introducci´on de la mec´anica cu´antica aplicada a los dispositivos ´opticos es uno de los mayores avances en el campo de los l´aseres semiconductores. Entre sus principales ventajas est´an la fuerte reducci´on en la corriente umbral debida a un aumento de la densidad de estados y la estabilidad del l´aser con la temperatura. Estas y otras ventajas, provienen de la discretizaci´on en el n´umero de estados permitidos para los portadores en las distintas bandas.
El ´ultimo paso en lo que es la cuantizaci´on del tama˜no, que implica el pleno con-
finamiento de los portadores en tres dimensiones, es el punto cu´antico o Quantum Dot
(QD). La evoluci´on en cuanto al confinamiento de portadores en dispositivos semiconduc- tores puede apreciarse en la fig. .
1 Los QDs fueron enunciados por primera vez como ´ultimo caso de cuantizaci´on en l´aseres semiconductores a principios de los a˜nos 80, y desde entonces, se ha recorrido un largo camino con el fin de llegar a su implementaci´on f´isica, lo cual no fue posible hasta los ´ultimos a˜nos 90. Hoy d´ia, se puede decir que el desarrollo tecnol´ogico de los QDs ha permitido comenzar a cumplir las expectativas en su d´ia anunciadas, y se puede hablar de una tecnolog´ia madura, pero ´aun en fase de desarrollo.
Figure 1: Evolution of quantum confinement applied to semiconductor devices [4].
Por otra parte, hoy d´ia un nuevo tipo de dispositivos est´asiendo desarrollado y estudiado: los cristales fot´onicos. Los cristales fot´onicos o Photonic Crystals (PhC) son estructuras diel´ectricas peri´odicas caracterizadas por un ”ancho de banda fot´onico”. O lo
2 que es lo mismo, estas estructuras permiten la propagaci´on de la luz dentro de ellas s´olo para determinadas y seleccionadas frecuencias. Con esto, se consigue la propagaci´on de luz en diferentes direcciones con energ´ias espec´ificas [3]. QDs introducidos en estos disposi- tivos permiten explorar nuevas aplicaciones enfocadas a un mayor control de la luz emitida.
En este proyecto, se pretende estudiar el estado del arte de los Quantum Dots Lasers o l´aseres basados en puntos cu´anticos. Son estudiadas las caracter´isticas espectrales del l´aser, comportamientos din´amicos, proceso de fabricaci´on, etc. Por otra parte, se da un profundo estudio sobre la importancia de los efectos no lineales en estos l´aseres. Y para concluir, se estudia el uso de puntos cu´anticos en cristales fot´onicos en lo que es una aplicaci´on m´as pr´actica. Cabe decir que esta ´ultima parte se enmarca en un proyecto europeo de investigaci´on de la Universidad de Roma Tre, que en la fecha de partida del autor de este proyecto, estaba en espera de los par´ametros de simulaci´on que deb´ia de proveer el fabricante, con lo que los estudios realizados en el presente proyecto, se basan en simulaciones por ordenador previas.
El proyecto est´aestructurado en los siguientes cap´itulos o ”chapters”:
Capitulo 1: Conceptos Fundamentales. Se presentan algunos conceptos cl´asicos de • l´aseres basados en semiconductor y se introducen algunos conceptos relativos a la
cuantizaci´on en dichos l´aseres. Los cristales fot´onicos tambi´en son presentados.
Capitulo 2: L´aseres de Puntos Cu´anticos. El l´aser cu´antico es descrito con detalle. • Se caracteriza por su proceso de fabricaci´on, con una revisi´on de las diferentes tec-
nolog´ias existentes y las ventajas y desventajas de las mismas. Tambi´en se ofrece
3 un desarrollo anal´itico de distintas caracter´isticas como la ecuaciones de estado, la
densidad de estados, la corriente umbral, la ganancia ´optica, etc. La aplicaci´on fun-
damental del l´aser cu´antico en los VCSELs tambi´en es recogida por su importancia
en aplicaciones de telecomunicaciones.
Capitulo 3: Efectos No Lineales. Se da una segunda visi´on sobre algunos conceptos • ya presentados en el cap´itulo anterior, pues ahora se tienen en cuenta los efectos no
lineales. Se da un ´enfasis especial a la no-uniformidad del tama˜no del punto cu´antico
como principal lastre para el buen funcionamiento de los QD lasers.
Capitulo 4: Aplicaci´on a Cristales Fot´onicos. Se estudia la conveniencia de introducir • los puntos cu´anticos en PhC con el fin de controlar la emisi´on espont´anea de fotones.
Otras posibles ventajas son estudiadas.
De todos estos cap´itulos, se ofrecer´aa continuaci´on un breve resumen de los mismos, pudiendo encontrarse mayor informaci´on, ecuaciones y figuras, en el proyecto completo en ingl´es que se incluye tras este resumen. Las referencias bibliogr´aficas y a figuras son ´unicas en todo el documento completo (parte espa˜nola e inglesa) para facilitar el acceso a las mismas.
4 Capitulo 1:Conceptos
Fundamentales
En esta secci´on se da un breve repaso a algunos de los conceptos cl´asicos sobre el l´aser semiconductor, y en concreto siguiendo la estructura cl´asica de un l´aser Fabry-Perot para una mejor comprensi´on de los conceptos expuestos. El cap´itulo incluye tambi´en una breve descripci´on de la que se podr´ia llamar la tecnolog´ia previa al l´aser de puntos cu´anticos, como son los l´aseres basados en pozos cu´anticos o Quantum Well Lasers. Para concluir, se presentan los cristales fot´onicos como nuevas estructuras de gran inter´es en el campo de la electr´onica.
En este resumen, se tratar´an los conceptos de Quantum Wells y Photonic Crystals o lo que es lo mismo, Pozos Cu´antincos y Cristales Fot´onicos.
0.1 Pozos cu´anticos
Es conocido que en las bandas de conducci´on y de valencia de un material semiconductor, existen una serie de estados accesibles para electrones o huecos respectivamente. Esta serie
5 de estados disponible se da en un continuo, mientras que existe la posibilidad de conseguir niveles ´oestados discretos accesibles en dichas bandas mediante el confinamiento cu´antico de los portadores, usando para ellos los pozos cu´anticos.
Para conseguir este confinamiento, se puede pensar en la implementacion de los po- zos cu´anticos como en una fina capa de conductor con un determinado ancho de banda prohibida, emparedada entre dos tiras de otros semiconductores con mayor separaci´on entre las bandas de conducci´on y valencia [15]. Esta es la forma m´as sencilla, como puede verse en la figura 1.2. La anchura de banda del semiconductor de en medio (al que llamare- mos material estrecho), ha de ser al menos del orden del camino medio libre del portador entre colisiones en el medio. Esto quiere decir en valores concretos, en un rango oscilante entre el medio nanometro y los 10 ´o20 nm. A su vez, tambi´en ha de ser de una anchura menor que la longitud de onda de DeBroglie, es decir,
h λ = (1) p donde p es el momento del electron. Por ejemplo para el GaAs se tiene λ = 24nm.
La estructura del pozo cu´antico tiene la propiedad de confinar los portadores en una regi´on muy determinada y estrecha, en la dichos portadores se comportan de forma an´aloga al cl´asico problema de la mec´anica cu´antica de la part´icula encerrada en un pozo de potencial. Por tanto, en vez de tener una serie continua de posibles valores en la banda de conducci´on, los electrones en dicha banda en el material m´as estrecho estar´an confinados en estados discretos. En la banda de valencia ocurre exactamente lo mismo con los huecos.
Concretando sobre dichos niveles de energ´ia en el pozo cu´antico, se tiene que tener en cuenta el movimiento de los portadores en la direcci´on perpendicular a las ”heteroint-
6 erfaces”, puesto que este movimiento tambi´en est´acuantizado. Usando mec´anica cu´antica, la expresi´on para los niveles de energ´ia en un pozo cu´antico de estructura cuadrada (ideal), viene dada por: 2 2 π ¯h 2 Ej Ec = 2 j (2) − 2mnd
Donde j es el n´umero cu´antico usado para ir numerando los estados y d es el grosor del pozo cu´antico. Se puede demostrar 1.9, que la separaci´on entre niveles energ´eticos ha de ser a su vez mayor que la energ´ia t´ermica (producto kBT ), [15].
0.2 Punto Cu´antico (QD)
El punto cu´antico o Quantum Dot (QD) es el ´ultimo caso de cuantizaci´on posible en un s´olido, al existir una cuantizaci´on en las tres direcciones del espacio. Aplicado a semicon- ductores, se puede interpretar como un cristal de s´olo unos pocos nanometros de tama˜no, conveniente y coherentemente insertado en un entorno o matriz, con un ancho de banda prohibida mayor. Se suele decir para una mejor comprensi´on del QD, que su estructura y propiedades es muy similar a la de un ´atomo pero con una determinada forma geom´etrica.
Esta similitud permite aplicar al QD la f´isica at´omica pero con una mayor aplicaci´on pr´actica que en el caso de los ´atomos.
Cuando el movimiento de un portador de carga est´atan limitado a un peque˜no volumen, como ya se ha visto en el caso del pozo cu´antico, la energ´ia de dicho portador est´acuantizada. Si ahora quisi´eramos hallar una analog´ia a este comportamiento, como se hizo en el apartado previo con la part´icula en el pozo de potencial, se puede hablar de que la cuantizaci´on del electr´on es equivalente al caso del potencial de atracci´on de Coulomb
7 en un n´ucleo at´omico.
En el caso m´as simple e ideal, cuando la part´icula est´aencerrada en una caja rect- angular de potencial, y tomando como consideraciones que el potencial en el nivel m´as inferior es igual a cero, e igual a infinito en el l´imite, la cuantizaci´on de los niveles de energ´ia viene dada por:
2 2 2 2 2 π ¯h nx ny nz Enx,ny,nz = 2 + 2 + 2 (3) 2me Lx Ly Lz donde me es la masa efectiva del electr´on, y Lx,y,z las longitudes de la caja en cada una de las coordenadas espaciales. nx,y,z = 1, 2, 3, ... son los n´umeros cu´anticos.
0.2.1 Densidad de Estados
La funci´on de Densidad de Estados o Density of States (DOS), es la funci´on de energ´ia que mide el n´umero de niveles electr´onicos permitidos, dentro de un intervalo energ´etico y por unidad de volumen de cristal semiconductor. Esta funci´on se introduce debido a que el n´umero de niveles permitidos dentro de una banda de energ´ia, es a groso modo igual al n´umero de ´atomos dentro del cristal. Pero esto es cierto s´olo teniendo en cuenta que estos estados no est´an distribuidos uniformemente dentro de la banda energ´etica, cualquiera que sea la considerada. Una expresi´on muy general para el DOS es la dada por:
1 dN ρ(E) = ( ) (4) V dE
Donde se ve claramente que tiene dimensiones de energia−1 volumen−1. La forma de la · funci´on DOS depende significativamente de la naturaleza del cristal considerado, as´i como de su estructura interna. El c´alculo de esta funci´on, sobre todo a niveles cu´anticos, no es para nada sencillo. Para ver la importancia del DOS, se adjunta la figura 1.3, donde la
8 influencia de la cuantizaci´on en tama˜no se pone claramente de manifiesto en la forma del
DOS.
0.3 Cristales Fot´onicos
El cristal fot´onico, Photonic Crystal o PhC, es una estructura dielctrica y peri´odica caracterizado por la que se llama ”ancho de banda fot´onico”. Esto es, son estructuras que permiten la propagaci´on de la luz dentro de ellos pero s´olo para un rango de frecuencias seleccionadas. Al final, lo que tenemos es un dispositivo que previene la propapagaci´on de la luz en determinadas direcciones y con determinadas energ´ias [3]. Es evidente, que este grado de control sobre la luz puede ser de gran utilidad en multitud de aplicaciones.
En cierto modo, los PhC pueden ser considerados como una generalizaci´on de los reflectores dielctricos de Bragg, pero en 2 ´o3 direcciones en el espacio. Los cristales fot´onicos tridimensionales son, desde el punto de vista de la f´isica b´asica, el material ideal para el control de la propagaci´on de la luz, pero su fabricaci´on es a´un un reto.
Afortunadamente, los PhC en 2D ya son una realidad gracias a t´ecnicas est´andar de fabricaci´on como son por haz de electrones y litograf´ia modificada. El grado de control de la luz de estos dispositivos, es ya suficiente para su apliaci´on en ´optica integrada, incluyendo una posible aplicaci´on a QDs insertados dentro de estructura de 2D-PhCs.
Es m´as, hoy d´ia se sabe que en los cristales fot´onicos, la emisi´on de luz es modificada debido a que el ´indice de refraccion var´ia espacialmente en incrementos m´ultiplos de la longitud de onda. Por tanto, la interferencia de luz provocada por difracci´on de Bragg,
9 causa bandas de rechazo caracter´isticas y, en ´ultima instancia, un espectro en frecuencia dependiente de la direcci´on. Finalmente, la funci´on de densidad de estados es modificada localmente (LDOS), lo que a su vez modifica la tasa de radiaci´on espont´anea de los posibles puntos cu´anticos insertados en la estructura. Por tanto, es posible pensar que seleccionando los par´ametros adecuados en el PhC, ser´aposible modificar adecuadamente la funci´on
LDOS del QD emisor. Esta idea ser´aestudiada en el cap´itulo 4.
10 Capitulo 2: L´aseres de Puntos
Cu´anticos
En este cap´itulo se enuncian las principales ventajas del l´aser cu´antico. Posteriormente, se discuten los existentes procesos de fabricaci´on existentes en la actualidad y se vuelve a revisar el concepto de DOS (visto en el cap´itulo 1) aplicado a este tipo de l´aser. El l´aser cu´antico quedar´adefinido completamente mediante la presentaci´on de sus princi- pales caracter´isticas y propiedades como son la ecuaciones de estado, la corriente umbral y su ganancia ´optica. Por ´ultimo, en un aspecto m´as pr´actico, se ven aplicaciones tan importantes como el ”Q-switching” y los VCSELs.
En este resumen, resaltaremos el proceso de fabricaci´on que se ha impuesto en la actualidad: el m´etodo Stranski-Krastanow. Tambi´en se dar´auna breve explicaci´on de las principales caracter´isticas del l´aser y se incidir´aen los VCSELs como dispositivos muy
´utiles en comunicaciones ´opticas.
11 0.4 Definici´on y ventajas
La idea de ”explotar los efectos cu´anticos en l´aseres semiconductores de heteroestructuras para producir una modificaci´on en la longitud de onda” y conseguir ”menores umbrales de funcionamiento del l´aser” por medio de ”el cambio en la densidad de estados resultante de la reducci´on del numero de grados de libertad de movimiento de los portadores” fue introducida por Dingle y Henry en 1976. Posteriormente el l´aser de punto cu´antico como tal fue definido por Sakaki y Arakawa en los a˜nos 80. Con este l´aser, se tienen finalmente no s´olo reglas de dise˜no macrosc´opicas para el transporte de portadores de carga, sino tambi´en la posibilidad de hacer ingenier´ia de la funci´on de densidad de estados, mediante la realizaci´on de estructuras nanom´etricas (QDs) en la regi´on activa del l´aser.
Entre las principales ventajas te´oricas de este l´aser son destacables:
Disminuci´on del umbral de la corriente de transparencia y mayor independencia de • ´esta con la temperatura.
Mayor ganancia ´optica y diferencial. •
Alta estabilidad en la longitud de onda emitida en la salida. Menor chirp. •
Pero con los a˜nos, se fue comprobando que estas caracter´isticas eran demasiado ideales, sobre todo porque estaban basadas en asunciones tales como la existencia de QDs ideales e iguales, niveles ´unicos y discretos para los portadores, igual comportamiento
´oconfinamiento de electrones y huecos, etc. Estas asunciones han tenido que ir siendo reemplazadas por otras m´as realistas derivadas de los problemas de fabricaci´on de QDs ideales en tama˜no, volumen y forma, junto a otros efectos no lineales que ser´an discutidos
12 en el siguiente cap´itulo.
A continuaci´on se ir´adesgranando el estado del arte de esta tecnolog´ia, comenzando por el proceso de fabricaci´on de los puntos cu´anticos.
0.5 Fabricaci´on de QDs
A la hora de fabricar los QDs, el principal desaf´io es ser capaces de realizar estructuras de t´ipicament unos 10nm de tama˜no, de una forma eficiente y reproducible. Actualmente, existen una serie de procesos in situ y ex situ para su fabricaci´on. Los procesos in situ parecen haber tomado ventaja, y se basan sobre todo en la formaci´on de los QDs mediante un proceso de crecimiento, como es el m´etodo de Stranski-Krastanow [9]. Por contra, las t´ecnicas ex situ, aunque han sido mejoradas, adolecen del problema del uso de m´ascaras y litograf´ia, que impiden alcanzar la enorme precisi´on necesitada y hacen muy dif´icil evitar la dispersi´on lateral de los QDs creados. Algunos ejemplos de los QDs obtenidos mediante distintas t´ecnicas pueden apreciarse en la fig.2.1.
0.5.1 El m´etodo Stranski-Krastanow
Este m´etodo est´abasado en el crecimiento epitaxial. B´asicamente, desde mediados de los a˜nos 90, se ha venido usando y mejorando este m´etodo de fabricaci´on que permite la s´intesis directa de nanoestructuras, aplicando el fen´omeno de la formaci´on espont´anea de islas durante procesos de heteroepitaxia en capas ”tensionadas”. En esto consiste a grandes rasgos el proceso Stranski-Krastanow (proceso SK).
Originalmente, este proceso fue considerado perjudicial en la fabricaci´on de po- zos cu´anticos, por la indeseada aparici´on de islas en los pozos. Pero con una mayor in-
13 vestigaci´on, se advirti´oque estas islas pod´ian representar estructuras semiconductoras cero-dimensionales, y que para adecuadas condiciones de crecimiento, la distribuci´on del tama˜no (en altura y dimensi´on lateral) pod´ia ser muy estrecha, con un ensanchamiento inhomog´eneo reducido [21] (inhomogeneous line broadening; ver siguiente cap´itulo).
Este proceso se dice que es un ”crecimiento auto-organizado” de islas de escala nanom´etrica, y est´abasado en el conocimiento de que una capa de material semiconductor con una constante de tensor (lattice constant) diferente de la del sustrato, despu´es de un cierto grosor de deposici´on de dicho material sobre el sustrato, se producen una serie de
”racimos” de islas tridimensionales, se producen los deseados QDs. Para las primeras capas de deposici´on, los ´atomos se organizan en una capa conocida como ”wetting layer”. Esta capa act´ua como una zona de reserva de carga para los QDs. Por el hecho de que los QDs aparecen espont´aneamente durante el crecimiento, se dicen que son ”auto-generados” o
”auto-organizados”. Este proceso y otros pueden verse esquem´aticamente en fig.2.3 y cabe decir que permite la obtenci´on de un numero gigantesco de QDs mediante un simple paso de deposici´on
El tama˜no y la forma de las islas creadas pueden variar seg´un las condiciones del crecimiento epitaxial, siendo par´ametros determinantes la temperatura del sustrato y la composici´on del material de deposici´on, generalmente (Si)Ge o In(Ga)As sobre (Ga)As.
Con esta t´ecnica se consiguen f´acilmente islas con 20nm de longitud lateral y 2 nm de altura [9]. Evidentemente, a menor tama˜no del QD, menores longitudes de onda de los fotones emitidos, y la dispersi´on en el tama˜no se corresponde con un desplazamiento en onda de la luz emitida. De ah´i la importancia del tama˜no y la repetibilidad ”exacta” de los QDs.
14 Este proceso permite tambi´en que la primera capa de islas producidas, sea recubierta con otra fina capa del material del sustrato, de forma que se repita el proceso y se consiga una segunda capa de QDs sobre la primera, pudiendo conseguir un fuerte acoplamiento vertical entre ambas. Esta idea ser´afundamental en el estudio de los VCSELs, en este mismo cap´itulo.
Existen una serie de modificaciones a este proceso gen´erico, que permiten la ob- tenci´on de QDs redondeados ´opiramidales, o de menor o mayor longitudes de onda de emisi´on. Cada una de estas modificaciones se basan en distintas t´ecnicas como: control de la temperatura, ”vertically stacking”, ”activated alloy phase separation” y t´ecnicas cor- rectoras de defectos. Estas t´ecnicas se encuentran perfectamente ilustradas en las figuras
2.4-2.6.
0.6 DOS: funci´on de densidad de estados
En esta secci´on s´olo vamos a indicar la dificultad de los c´alculos necesarios para hallar el DOS en los QDs, principalmente debido al elevad´isimo n´umero de ´atomos en los QDs
(del orden de 105) y de las fuertes interacciones de Coulomb entre portadores de carga artificialmente confinados en un espacio menor que el radio de excitaci´on de Bohr [18].
El c´alculo de la funci´on puede seguirse en las ecuaciones 2.1-2.2, con las pertinentes consideraciones. Finalmente se llega a la conclusi´on de que debido a la naturaleza del proceso SK, cada QD es ´unico en forma y la diferencia con la forma media del conjunto de QDs se traduce en ligeras desviaciones en los niveles energ´eticos disponibles, lo cual se refleja en el DOS como un ensanchamiento de la funci´on, que idealmente, deber´ia tener
15 una forma similar a una delta, como puede verse en la fig.2.7.
0.7 Ecuaciones de estado del QD Laser
A partir de estas ecuaciones, se pueden estudiar muchas de las caracter´isticas y com- portamiento del l´aser. En el an´alisis realizado, se ha tenido en cuenta espec´ificamente la
”wetting layer” (WL), puesto que su papel como reserva de carga no puede ser olvidado.
Los efectos de escape y captura de los portadores en todas las capas tambien son tenidos en cuenta.
Para llegar a las ecuaciones, se parte de las figuras esquem´aticas figs. 2.8 y 2.9, con la estructura de bandas energ´eticas claramente definida [27]. Los portadores son inyectados en la SCL (separate confinement layer) y estos caen en la WL y luego en los estados de excitaci´on y reposo ´o”ground state” (GS). La recombinaci´on de estos portadores puede ser en una forma radiativa o no, con una constante temporal de recombinaci´on asociada en cada caso. La asociaci´on de estas variables a procesos tambi´en se aprecia en la fig. 2.9.
Usando la ecuaci´on 2.6 para describir la emisi´on de fotones en la direcci´on longitudinal del dispositivo, y teniendo en cuenta los procesos descritos con anterioridad, se llega finalmente a las ecuaciones siguientes para las capas WL, estados de excitaci´on y GS respectivamente
[28]: ∂N N L N (1 h ) N h N w = s s w − n + w n w (5) ∂t τdLw − τw2 τ2w − τwR ∂N h N (1 h ) N h N (1 f )h N (1 h )f Q n = w − n w n Q − n n + Q − n n (6) ∂t τw2 − τ2w − τ21 τ12
∂NQfn NQ(1 fn)hn NQfn(1 hn) NQ = − − fnfp vgg0(fn + fp 1)S (7) ∂t τ21 − τ12 − τQR − − donde Ni es la concentraci´on de electrones en las distintas capas, fn y fp son las funciones
16 de probabilidad de ocupaci´on de electrones y huecos en la GS (hn en la capa de excitaci´on), y Li el grosor efectivo de las capas en consideraci´on. Con S, la densidad de fotones a la entrada y salida de la zona de amplificaci´on puede hallarse, y por tanto, la ganancia ´optica.
Ver ecuaci´on y figura 2.10.
0.8 Corriente Umbral
Idealmente, deber´ia ser constante con la temperatura, y la temperatura caracter´istica infinitamente alta. Pero no es as´i porque todas las recombinaciones no se producen en los
QDs y la neutralidad de carga en ´estos no es cierta. En la realidad, la corriente umbral viene dada por la suma de las densidades de corriente procedentes de las recombinaciones radiativas en los QDs y en la OCL (Optical Confinement Layer). La OCL es una forma de denominar lo que son las regiones frontera (barreras de potencial) [37].
Para el caso de equilibrio, es decir, el tiempo de vida media de los portadores es mucho menor comparado con el tiempo de escape de portadores por excitaci´on t´ermica.
Se llega a una ecuaci´on 2.14, que recoje la dependencia ya hablada en sus dos t´erminos sumandos. La importancia de ambos sumandos var´ia con la temperatura, como puede verse en la fig. 2.12.
0.9 Q-Switching
Esta aplicaci´on es de gran importancia por sus posibilidades comerciales, al permitir pro- ducir pulsos muy cortos como salida del l´aser. El poder tener l´aseres pulsantes permite aplicaciones que van desde emisores de luz en sistemas de comunicaci´on a l´aseres usados
17 en cirug´ia.
B´asicamente, el m´etodo para obtener pulsos a la salida del l´aser consiste en cambiar la ganancia del medio mediante el cambio en la energ´ia suministrada al medio. El prob- lema es que los pulsos as´i producidos son bastante redondeados, por los retardos en la creaci´on de nuevas inversiones de poblaci´on. Esto limita tambi´en la longitud del pulso y la frecuencia de los mismos. Con el Q-switching, la salida del l´aser es cambiada controlando las p´erdidas en el medio [11], y los resultados obtenidos pueden verse en la fig. 2.16. Con el apoyo de esta figura explicamos brevemente el proceso. Con la corriente aplicada se van formando portadores que van incrementando la ganancia en el medio. Llega un punto en que la ganancia supera las p´erdidas y la densidad de fotones en la cavidad aumenta dr´asticamente. En este punto, este incremento de la densidad de fotones provoca un de- scenso en la ganancia de la secci´on, hasta que esta vuelve a ser inferior a las p´erdidas totales. La densidad de fotones vuelve a decaer dr´asticamente y a la salida del l´aser se ha producido un pulso de gran calidad.
El proceso optimizado se describe completamente en el apartado 2.6 del documento en ingl´es. Debido a la naturaleza discreta del QD: una vez que el estado es llenado por dos portadores de spin distinto, no es posible otro tipo de transici´on asociada a ese nivel de energ´ia, por lo que una vez ocupados todos los estados de los QDs, cualquier incremento en la ganancia del medio no ser´aposible, y nuevos portadores inyectados ser´an almacenados en otras capas. Por ello, la saturaci´on deber´ia ocurrir a la mayor ganancia posible para asegurar una ganancia diferencial a´un apreciable. Incrementando la longitud de la zona de absorci´on, tambi´en se provoca un incremento en el pico de potencia debido al hecho de que al formarse el pulso, hay un enorme n´umero de portadores en la cavidad disponibles
18 para producir fotones.
Cabe decir que por los motivos descritos y gracias a que la ganancia de saturaci´on en l´aseres QD es mayor que en otras estructuras (incluso que QW lasers) las caracter´isticas de Q-Switching de ´estos es excelente.
0.10 VCSELs
VCSELs es el acr´onimo de Vertical Surface Emitting Lasers, y son de gran utilidad en las actuales comunicaciones ´opticas, puesto que son dispositivos mucho m´as econ´omicos que los actuales l´aseres FP y dispositivos DFB (Distributed Feedback). Los VCSEls de GaAs han demostrado ser altamente fiables, de muy bajo coste, alta estabilidad de la longitud de onda emitida con la temperatura y baja divergenc´ia del flujo emitido. Adem´as, pueden emitir en todas las ventanas de inter´es en comunicaciones ´opticas, aunque en tercera ventana a´un tienen un amplio margen de mejora.
La estructura del VCSEL puede verse en la figura 2.17. B´asicamente, consiste en aunar el efecto de capas de QDs verticalmente acopladas, junto al uso de DBR (Distributed
Bragg Reflectors). Varios ejemplos del estado del arte actual de estos dispositivos pueden verse en la figura 2.18.
19 Capitulo 3: Efectos No Lineales
Los efectos no lineales que han ido apareciendo conforme los l´aseres de QDs fueron siendo desarrollados, han constitu´ido la principal r´emora para que ´estos lleguen a ser aqu´ellos de propiedades ideales descritas en su enunciaci´on. La inmensa mayor´ia de los efectos no lineales son derivados de la no-uniformidad de los puntos cu´anticos. En este cap´itulo se tratar´an el ensanchamiento homog´eneo y sobre todo el inhomog´eneo como las principales no-linealidades a tener en cuenta.
0.11 Ensanchamiento Homog´eneo
Este efecto no lineal, al igual que el ensanchamiento inhomog´eneo, no es exclusivo de los l´aseres QDs. En una forma general, se puede decir que consiste en que la respuesta de cada ´atomo u oscilador individual es ensanchada en la misma forma, al actuar osciladores en principio independientes en la misma forma. En nuestro caso, este efecto se manifiesta en que, a bajas temperaturas, QDs con diferentes energ´ias comienzan a lasear indepen- dientemente con ganancia en forma de delta, mientras que a temperatura ambiente, los
QDs contribuyen a una respuesta estrecha al lasear colectivamente debido al efecto que estamos tratando.
20 En los l´aseres QDs, este efecto es debido a desfases en la polarizaci´on de los distintos
QDs. Este ensanchamiento tiene la forma de una lorentziana, en la forma descrita por la ecuaci´on 3.4 y representado en la figura 3.1. Por lo anteriormente comentado, puede apreciarse que esta no-linealidad es en realidad beneficiosa en cierto rango de temperaturas, puesto que con la temperatura, QDs espacialmente aislados y energ´eticamente diferentes comienzan a lasear colectivamente, contribuyendo a una respuesta global en forma de delta. El problema ocurre a bajas temperaturas, como puede verse en la fig. 3.2.
0.12 Ensanchamiento Inhomog´eneo
En oposici´on al anterior efecto, el ensanchamiento inhomog´eneo, en general, provoca que en una colecci´on de ´atomos nominalmente id´enticos, estos tengan distintas frecuencias de resonancia distribuidas en torno a un valor central, con lo que una respuesta global ver´a una ensanchamiento final y una respuesta a la frecuencia central disminu´ida en amplitud.
En el l´aser ´atomos con diferentes posiciones en la estructura, tendr´an distintos alrede- dores, por defectos, dislocaciones o impurezas. Esto produce valores ligeramente distintos para niveles de energ´ia fijos, y por tanto, ligeras desviaciones en frecuencias de transici´on.
En los l´aseres de QDs, estos alrededores distintos, vienen dados fundamentalmente por la disporsi´on en el tama˜no que puede ser de hasta un 15%.
Para el estudio de este efecto, se parte de las ecuaci´on de ondas en 3D, junto a la asunci´on de un modelo matem´atico (ecuaciones 3.63.7) para expresar la dependencia exponencial de los niveles energ´eticos de ocupaci´on de portadores en relaci´on con el tama˜no del QD (ver fig. 3.3) [39]. Y por otra parte, se asocia a la fluctuaci´on en el tama˜no del QD
21 un proceso aleatorio gaussiano, con la consecuente funci´on de densidad de probabilidad ec. 3.8. Siendo por tanto el tama˜no una variable aleatoria, podemos calcular una serie de par´ametros referidos a los estados de ocupaci´on, puesto que ´estos, como hemos dicho, dependen del tama˜no.
Aplicando matem´aticas estad´isticas para trabajar con este proceso aleatorio, como puede verse a trav´es de las ecuaciones 3.8 a 3.12, se llega a que la desviaci´on est´andar de la variable energ´ia del estado de ocupaci´on, es el par´ametro que representar´ala fluctuaci´on en el tama˜no, y su equivalencia puede verse en la ecuaci´on 3.13. Una vez en este punto, se pueden hacer distintas deducciones [39].
Para ver la influencia de la desviaci´on en tama˜no en la probabilidad de ocupaci´on de estados, en la fig. 3.4 puede verse que con el aumento de la desviaci´on, la funci´on de probabilidad se ensancha, disminuyendo el pico de la misma y movi´endose a energ´ias menores. Esto significa que con una mayor desviaci´on, el estado m´as probable en ser ocu- pado es de menor energ´ia, con el consiguiente perjuicio en la respuesta final. La influencia del ensanchamiento inhomog´eneo aumenta casi linealmente con esta desviaci´on est´andar calculada.
En cuanto a la ganancia, siguiendo la ecuaci´on 3.19 para entender la influencia de la desviaci´on en tama˜no sobre la misma, puede verse que la dependencia es a trav´es de una integral, y que esta satura para una desviaci´on alta del tama˜no, puesto que esos QDs tienen menores niveles de energ´ia de transici´on y alcanzan la transparencia antes que otros
QDs menores. Puede verse en la figura 3.7. Con respecto a la ganancia diferencial, puede apreciarse en el detalle de la misma figura y en la fig. 3.8, donde la ganancia disminuye pr´acticamente inversamente proporcional al aumento de la desviaci´on.
22 Si por ´ultimo nos detenemos en c´omo se ve afectada la corriente umbral, en la ecuaci´on 3.18 se da la relaci´on de la funci´on de Fermi con la desviaci´on en el tama˜no.
Conforme ´esta ´ultima aumenta, la otra se reduce, y por tanto, se incrementa el valor de la corriente umbral. La corriente umbral depende de las recombinaciones radiativas en los
QDs y estas son proporcionales a las concentraciones de electrones y huecos (relacionadas con la funci´on de Fermi).
23 Capitulo 4: Aplicaci´on a Cristales
Fot´onicos
La idea de introducir puntos cu´anticos en el PhC (Photonic Crystal) viene de que ´estos, gracias a la modificaci´on local del DOS (LDOS), pueden permitir el control en la tasa de emisi´on espont´anea de los QDs. Para este fin, es necesario usar un PhC modificado, con una cavidad simple en su interior. Lo ideal ser´ia utilizar un PhC en 3D, pero actualmente es una utop´ia disponer de estos dispositivos, con lo que el experimento ser´arealizado con estructuras en 2D. Las caracter´isticas de la luz emitida espont´aneamente dependen fuertemente del entorno del emisor; tanto la tasa total de emisi´on y como el espectro son afectados. El LDOS cuenta el n´umero de modos electromagn´eticos en el que los fotones pueden ser emitidos teniendo en cuenta la situaci´on espec´ifica del emisor, y puede ser tambi´en interpretado como la densidad de fluctuaciones de carga. La recombinaci´on ra- diativa est´adescrita en la conocida como regla de oro de Fermi (ver Ap´endice), y permite iniciar el c´alculo del LDOS. As´i, se ve que tanto el LDOS como la radiaci´on dependen fuertemente de la frecuencia emitida y de la posici´on del emisor, pero no de la direcci´on de emisi´on. Por otro lado, en este resumen ya se dijo que en un PhC la emisi´on de luz es
24 modificada debido a que el ´indice de refracci´on var´ia espacialmente en regiones de longitud del orden o m´ultiplo de la longitud de onda. Por tanto, la interferencia por difracci´on de
Bragg provoca bandas de rechazo caracter´isticas, que s´i llevan a un espectro dependiente de la direcci´on. Finalmente, tenemos que el LDOS es modificado y por tanto tambi´en lo es la tasa de emisi´on espont´anea de los emisores QDs introducidos en la estructura fot´onica.
Para demostrar las posibilidades del uso conjunto de QDs y PhCs, se han estudiado varios experimentos, trabajando en FDTD (Finite Difference Time Domain) mediante varios programas software, pero sobre todo el ”RSoft Photonics CAD Suite”.
En el primero de estos experimentos, se realizaron medidas relativas a los ´angulos de emisi´on de la luz de los QDs. Seg´un el par´ametro ”a” o lattice, que caracteriza a todo PhC, se obtuvieron distintos resultados. En la figura 4.1 puede verse claramente que conforme el par´ametro ”a” es incrementado, los efectos del cristal son mayores sobre la luz emitida por los QDs, habiendo una mayor selectividad tanto en direcci´on como en la cantidad emitida.
En la figura fig. 4.2, puede apreciarse claramente en (b), la dependencia en frecuencia de la tasa total de recombinaci´on (radiativa y no radiativa). Se observan variaciones tan importantes como del 50% entre las muestras de un tipo de PhC y otro. Para el lattice de mayor tama˜no se aprecia una inhibici´on de hasta un 30%. Para un valor mayor del lattice se oberva una mejora de la emisi´on sobre un amplio rango de frecuencia. Luego el valor del lattice es decisivo para obtener una mejora o empeoramiento de la emisi´on.
Por otra parte se ha estudiado el experimento realizado por Arakawa [45] por ser el primero en demostrar f´isicamente la emisi´on de un ´unico fot´on por parte de QDs acoplados a un PhC modificado. Es importante hacer notar que en un r´egimen de acoplamiento d´ebil la tasa de decaimiento es a´un la que domina y siguiendo el desarrollo de las ecuaciones 4.1-
25 4.3, se obtiene el factor de Purcell (ver Ap´endice). La estructura PhC modifica usada es la inclu´ida en la figura 4.3 y el objetivo es conseguir el m´aximo acoplamiento entre el QD y la cavidad. Para poder realizar medidas, el sistema utilizado es el mostrado en el esquema de la figura 4.4. En ´el se usa un sistema basado en espectrocop´ia por fotoluminescencia.
La fracci´on de fotoluminescencia que se origina en la cavidad es mejorada en dos formas: a trav´es de una tasa de emisi´on mejorada, y el incremento en la eficiencia del acoplamiento.
Las medidas demostraron que el acoplamiento es dif´icil para elevados valores del factor de resonancia Q (del orden de 104). Sin embargo para cavidades con factores entre 102 y 103 se obtuvieron mejores resultados, recogidos en la figura 4.5.
Finalmente, en los QDs que no estaban en resonancia se obtuvo una mejora de esta 5 veces en la extensi´on de la vida del portador. El problema es que el desalineamiento entre los QDs y la cavidad hace que el acoplamiento sea dif´icil. Con un mejor ajuste espacial y espectral, las ecuaciones anteriores teorizan una mejora del orden de 400 veces, pero siempre en un r´egimen d´ebil de acoplamiento. Un buen gr´afico para apreciar la influencia de este desalineamiento en el acoplamiento viene recogido en la figura 4.6 (realizada en el programa Mathematica).
26 Conclusiones
En muchos casos, las ventajas de los QDs a´un no son del todo comprendidas. En la actualidad se trabaja con gran esfuerzo en este campo, y prueba de ello son la elevada cantidad de art´iculos que aparecen en las m´as importantes publicaciones, pero eso no impide que a´un hoy no se haya desarrollado una teor´ia completa que d´erespuesta a las cuestiones planteadas por esta nueve tecnolog´ia. Es por ello que este proyecto se hayan tenido en cuenta diferentes fuentes de informaci´on y puntos de vista para la interpretaci´on de los mismos problemas, con el objetivo de llegar a una mejor comprensi´on de la teor´ia y sus dificultades inherentes.
En los ´ultimos a˜nos se ha llevado a cabo un considerable avance en esta tecnolog´ia.
En el campo de la fabricaci´on, los nuevos tipos de m´etodos de crecimiento auto-ordenado de QDs, posibilitan la obtenci´on de puntos cu´anticos m´as uniformes en volumen y con menor dispersi´on lateral. Estos QDs al ser aplicados a los medios activos de l´aseres per- miten obtener mejores caracter´isticas de funcionamiento en los mismos, sobre todo en comparaci´on con aquellos desarrolados mediante t´ecnicas litogr´aficas, a pesar del consid- erable avance que tambi´en han experimentado estas t´ecnicas.
27 Pero a pesar de todo, el funcionamiento de estos l´aseres no ha llegado a´un a lo que se hab´ia predecido en los a˜nos 80 cuando fueron enunciados y establecido modelos t´eoricos sobre su funcionamiento. Esto es debido a que el aspecto clave sobre el que se basaban estas predicciones era el estrechamiento en la funci´on de densidad de estados. Pero a´un acerc´andonos hoy d´ia a la consecuci´on de esa estrecha funci´on, las bajas dimensiones de trabajo han hecho aparecer nuevos efectos no considerados (ni deseados) previamente, entre los cuales se revela el ensanchamiento inhomog´eneo como el principal escollo. De forma que en la actualidad, continuamente se est´an revisando y actualizando distintos modelos cada vez m´as fieles y realistas a los l´aseres fabricados.
En lo que al problema de la temperatura se refiere, se han realizado muchas con- sideraciones en este proyecto, pero lo que est´aclaro es que la predecida baja temperatura umbral a temperatura de trabajo, es s´olo alcanzable si las recombinaciones no radiati- vas son eliminadas en los QDs. Otro aspecto en el que hay que prestar atenci´on es en la necesidad de reducir las p´erdidas en la cavidad ´optica debidas al extremadamente peque˜no volumen efectivo de los QDs en la regi´on activa.
Pero a´un as´i, esta tecnolog´ia ya ha alcanzado algunas metas muy interesantes: ganancia ´optica mayor en al menos un orden de magnitud en comparaci´on con tec- nolog´ias usadas en la actualidad (l´aseres de pozos cu´anticos), mayor ganancia ´optica de saturaci´on, menor dependencia de la temperatura a bajas temperaturas y un chirp mucho menor. Otra importante ventaja alcanzada es el desarrollo de los VCSELs con capas de QDs en su interior, siendo dispositivos que ya se usan en la actualidad, emitiendo en 2a y 3a ventana de comunicaciones ´opticas, con alta fiabilidad y menor coste.
28 En cuanto a la aplicaci´on de introducir emisores de luz basados en puntos cu´anticos en estructuras consistentes en cristales fot´onicos, se ha demostrado que es posible con- trolar de forma independiente los estados de portadores y fotones mediante el dise˜no de apropiadas bandas de transici´on y de adecuadas distribuciones del ´indice de refracci´on.
Es m´as, se ha demostrado que dise˜nando una estructura de cristales fot´onicos adecuada, la tasa de recombinaciones espont´aneas de los QDs inmersos puede ser significantemente modificada.
En el caso de QDs acoplados a una cavidad inducida en un PhC, se ha observado una disminuci´on en dicha tasa de recombinaci´on de hasta 8 veces. Estos sistemas acopla- dos prometen incrementar la eficiencia externa de acoplamiento y la indiferenciaci´on de fotones emisores monofot´onicos. Por otra parte, los QDs no resonantes con dicha cavidad, presentan una disminuci´on del orden de 5 veces en la tasa de emisi´on en comparaci´on con los QDs en sistemas conductores cl´asicos, debido a que la densidad local de estados es disminu´ida en el cristal fot´onico. Esta extensi´on en la vida puede tener aplicaciones en otros dispositivos fot´onicos basados en QDs (como por ejemplo, conmutadores o interrup- tores), inclu´idas aplicaciones al procesado cu´antico de la informaci´on (por ej. memorias cu´anticas). Se puede concluir tambi´en de este estudio, que al existir una gran similitud obtenida en experimentos pr´acticos y simulaciones realizadas mediante software basado en FDTD, estas t´ecnicas de simulaci´on son perfectamente adecuadas para el estudio de las nuevas aplicaciones (nuevas estructuras) permitidas por esta tecnolog´ia.
29 Chapter 1
Fundamental Concepts
In this section we are going to give a brief overview of semiconductor fundamentals, mainly focusing on the principles of lasers. As these principles will be subsequently developed, different concepts from classic semiconductor device’s theory will be more clear, with a review of its meaning and properties. The explanations will be done considering a classical structure of Fabry-Perot laser, which is the most illustrative.
The chapter will end with the inclusion of the Quantum Well definition as a first step needed to understand the concept of Quantum Dot, which is later detailed. Finally, the chapter concludes with an introduction to Photonic Crystal and an explanation of why these new devices based on periodic structures are so interesting, and of course, why are so interesting for their use with Quantum Dots.
30 1.1 Principles of Lasers
It is well known than the word LASER comes from the acronym for light amplification by stimulated emission of radiation. A little word for one of the most important optical and electronic devices.
From its definition, the laser is recognized as a source of highly directional, monochromatic, coherent light, and as such it has solved some longstanding optical prob- lems. The light from a laser, depending on the type, can be a continuous beam of low or medium power, or it can be a short burst of intense light delivering millions of watts.
Focusing on the words of the acronym we are going to explain the general properties of a laser.
From the last letters of the word laser, amplification of stimulated emission, we have it principle of operation. The randomly process of the emission of radiation when excited electrons fall to lower energy states on the valence band is called spontaneous emission. If only this effect exist, the number of electrons remaining in the upper level of energy E2, that is called the population of E2, would be expected to decay in an exponential way, with an average decay time describing how long in average, an electron stays in the upper level. However, we want that emission to happens when we decide, without waiting the spontaneous emission to occur. If conditions are right, it can be stimulated to fall to the lower level and emit its photon in a much shorter time than the corresponding spontaneous decay. To realize this, it must be applicated an stimulus, which is a radiation of photons of the proper wavelength and same phase. With this stimulus, the electron is induced to
31 drop in energy from E2 to E1, the lower level, contributing a photon whose wave function is in phase with the radiation field. It is clear that if this process continues and other electrons are stimulated to emit photons in the same fashion, a large radiation field can build up; the light intensity (density of photons) grows avalanche-like in the system. And this radiation will be monochromatic since each photon will have exactly the energy of the band gap between E and E , which is: hv = E E . We have too the last characteristic 2 1 21 2 − 1 of the laser: the coherence, because all the photons emitted will be in phase and reinforcing.
But only this mechanism of stimulated emission is not enough to have a laser.
Under condition of thermal equilibrium, light emission is balanced by the light absorption.
To overcome this loss of photons, some additional concentration of excited electron that exceeds the equilibrium that has to be established in the system [10]. In other terms, the population of the upper levels of energy must be greater than the lower ones. Usually, under thermal equilibrium, all the electrons has fallen down the lower levels by spontaneous emission, and so the population at the lower levels are bigger. This is only the natural explication of the mathematical formulation of Fermi levels. So what we want is the non- natural case, called population inversion because of the previous explained reasons.
1.1.1 Population inversion
We have already talk about the term population. Some equations to explain it.
At the equilibrium, the following equation relates electrons, n, and holes, p, concen- trations:
2 np = (ni) (1.1)
32 Where ni is the intrinsic concentration that is temperature dependent. If some charge carri- ers are injected into the semiconductor, the equilibrium is disturbed because this excessive concentration cannot be maintained by thermal excitation. Because this non-equilibrium situation we have new Fermi levels, FC and FV , to describe the new concentrations in the conduction and valence band respectively. Translating what we have explain about population inversion, is obvious, that to have this effect, we need:
f (E ) f (E ) (1.2) Ce C ≥ V e V
This means, that the electron population probability in the conduction band state fCe having an energy EC is higher than that of the valence band with energy EV . Now, taking into account that holes and electrons complement each other, that is, empty electron state is occupied by holes and vice versa, the electron fV e(EV ), and the hole, fV h(EV ), occupation probabilities in the valence (equivalent for the conduction) band are connected in the following way:
f (E ) = 1 f (E ) (1.3) V e V − V h V
Using the latter expression with the well-known Fermi equations, it can be have finally the following equation which sets the condition of inverse population in semiconductor:
F F E E (1.4) C − V ≥ C − V
The interpretation of eq.1.4 says that the energy separation of electron and hole Fermi levels has, at least, to be larger than the band gap of a semiconductor. Another frequently used term appears here, transparency current. This one is the injected current on the laser that provides the minimum level of pumping (provides sufficient carriers to have inverse population), which must be exceeded to begin the device lasing.
33 1.1.2 Threshold Conditions & Optical Gain
The laser structure basically consists on the region where inversion population takes place, which is the light-amplifying region, and a positive optical feedback made with a suitable structure, a mirror at each side of the amplifying medium in general. With this structure, the light (photons) is confined within an optical cavity and reflected back-to-back before having enough energy to leave the cavity.
To explain the mechanism of lasing, lets have a look at figure 1.1. The steady-state light intensity inside the cavity is reached when the optical gain balances all the optical loss in the cavity. Therefore, the laser threshold condition in terms of gain and loss is:
”optical gain at threshold caused by light amplification in the active region of the lasers is equal to the sum of output loss and internal loss”. Focusing on fig.1.1, lets assume light propagation from the left to the right facet. As initial light (φ0) passes through the amplifying medium, it will be amplified and decayed by the exponential coefficient of Gα
(gain-losses), over the length L of the cavity. On each rebound at each facet, this quantity is multiplied by the reflectivity of the corresponding facet.
So finally, at the steady-state condition at threshold, the round trip yields in no change of the light intensity. Thus:
φ = R R φ exp[2(G α )L] (1.5) 0 1 2 0 th − i or equivalently, 1 1 G = α + ln (1.6) th i 2L R R 1 2 A lot more can be said about the functioning of the laser, but it is not the aim of this thesis, and in this chapter just some basic concepts are summarized.
34 Figure 1.1: Light propagation and its intensity in different phases of round-trip cycle inside the Fabry-
Perot cavity [9].
1.2 Quantum wells
We have talked about discrete energy levels in the band gap arising from doping, and a continuum of allowed states in the valence and conduction bands. A third possibility is the formation of discrete levels from electrons and holes as a result of quantum-mechanical confinement.
In order to achieve quantum confinement, we can think of a quantum well imple- mentation as a thin layer of semiconductor with a given energy gap, sandwiched between two slabs of another semiconductors with higher energy gap [15]. This is the simplest way, as it is shown in the figure 1.2. The thickness of the narrow gap semiconductor must be smaller or at least on order of the carrier mean free path between collisions with impu- rities of phonons. Hence, the quantum well thickness usually ranges from a fraction of a nanometer to a 10 or 20 nm. This thickness should be also smaller than the DeBroglie
35 Figure 1.2: Quantum Well layers scheme wavelength: h λ = (1.7) p where p is the electron momentum. For example, for GaAs at room temperature, we have
λ = 24nm. In relation with the mean free path for electrons, restrictions are less heavy, as a typical range for λmean for the same example is λmean 146nm [15]. ∼ This structure has the property of confining electrons and holes in a very thin layer, so they behave according to the well-known problem of quantum mechanics for a particle in a potential well. Therefore, instead of having a continuous available states in the con- duction band, the conduction band electrons in the narrow-gap material are confined to discrete quantum states. The same happens in the valence band.
Approaching to energy levels in quantum wells, it must be taken into account the motion of carriers in the direction perpendicular to the heterointerfaces, because this mo- tion is quantized, involving descrete quantum energy levels. In the framework of quantum mechanics, it can be demonstrated that the lowest energy levels for a square potential well
36 can be estimated as: 2 2 π ¯h 2 Ej Ec = 2 j (1.8) − 2mnd
Here j is the quantum number labeling the levels, and d is the thickness of the quantum well. In order to have a real quantum well, that is, to have quantization in the structure, the following relation must be satisfied:
π2¯h2 2 kBT (1.9) 2mnd
This expression means that the difference between the levels should be much larger than the termal energy kBT .
37 1.3 Quantum dot
The concept of Quantum Dot (QD) has been previously introduced. Let’s define it now more in a proper way. As previously said, QD is the ultimate example of size quantization in solid, an it represents a semiconductor crystal with a size of only several nanometers, coherently inserted in a larger badgap semiconductor matrix. A QD mimics the basic properties of an atom providing a geometrical size allowing the practical application of atomic physics to the field of semiconductor devices.
When the motion of a charge carrier in a crystal is limited to a very small volume (it is imporant that the matrix material provides a larger bandgap than the QD material and that the potential wells are attractive both for electrons and hole) the energy spectrum of the charge carrier is quantized. This case is similar to the case of electron quantization in the attractive Coulomb potential of an atomic nucleus. In the simplest case, when the particle is ”locked” in a rectangular potential box, and the electron potential is equal to zero at the ground level, and tends to infinite elsewhere, that is, the ideal three-dimensional confinement, the energy quantized levels are given by:
2 2 2 2 2 π ¯h nx ny nz Enx,ny,nz = 2 + 2 + 2 (1.10) 2me Lx Ly Lz where me is the electron effective mass, Lx,y,z are the box lengths, and nx,y,z = 1, 2, 3, ... are the quantum numbers.
38 1.4 Density of States: DOS
The number of allowed states (sublevels) within a given band is roughly equal to the number of atoms inside the crystal. However, these states are not distributed uniformly within the energy band. To characterize this distribution, a function called Density of
States or DOS is used. DOS is a function of energy and it is a measure of allowed electronic states within unit energy interval per unit volume of crystal. This is:
1 dN ρ(E) = ( ) (1.11) V dE
DOS has dimensionality of energy−1 volume−1 and measures how many states are · available within unit energy interval around some energy E in a crystal of unit volume.
The shape of the DOS depends significantly on the chemical nature of the crystal host and its structure, and its calculation is not easy. A comparative of the influence of the structure is given in fig.1.3, and so the size quantization strongs determine the DOS shape.
In the following chapters DOS of QDs will be described in detail.
39 Figure 1.3: Active region of a diode lasers representing layer of (a) bulk semiconductor, (b) several QW,
(c) array of quantum wires (d) QDs. Each one with its corresponding DOS [10].
1.5 A New Device: Photonic Crystal
New, as are in recent years when its predicted properties are becoming a reality, and exciting devices are being developed and studied nowadays: photonic crystals. Photonic
Crystal (PC or PhC) are dielectric and periodic structures characterized by a photonic band gap. This means that are structures that allows the propagation of light inside them only for a selected frequency range. That is, preventing light from propagating in certain directions with specified energies [3].
Photonic crystals can be considered as a generalization of dielectric Bragg mirrors to two or three directions of space. They presents strong analogies with crystals in the fact that light propagation in any direction is allowed within energy bands and forbidden
40 in others.
2D photonic crystals in a waveguide geometry are a recent concept for light propagation control which has seen its main promises verified in the optical range. While
3D photonic crystals are, from the basic physics point of view, the ideal material for light control, their fabrication is still a challenge, at opposed to their two-dimensional counterparts, as witnessed by the successful realization of controlled 2D photonic crystals by standard e-beam and dry etching lithography techniques. This means in simple terms that the degree of light control appears sufficient for integrated optics application, and this includes QDs embedded into 2D-PC structures as it will be shown in chapter 4.
Three good reasons for PC use in the field of photonic integrated circuits are:
photonic crystals allow us to have high performance integrated optics structures • with very small sizes (a possibility opened by photonic crystals due to their lossless
guiding properties even in the presence of sharp bends or imperfections).
photonic crystals can be made through highly-simplified and low-cost fabrication • routes (based on the capacity of photonic crystals to achieve many optical func-
tions by a single epitaxy, lithography-and-etch sequence, yielding at once mirrors,
waveguides, resonators, and more complex functions).
photonic crystals allow novel functions due to their special physical properties or • diffraction effects. They also allow to revisit some older concepts which have been
dropped because of their impracticality using the usual optoelectronic integrated
circuits routes, due to the ease of cascading photonic crystal-based elements such as
41 for multi-segment coupled-cavity laser.
What is more, thinking on the useful relation between PC and QDs, now we now that, in photonic crystals, emission of light is modified because the refractive index varies spatially on length scales of the order of the wavelength. Hence, light interference by Bragg diffraction causes characteristic stopbands leading to direction-dependent spectra, that on the end modifies the LDOS (Local Density of States) relative to free space and hence the spontaneous emission rate of embedded QDs emitters. So, tuning the parameters of a PC, it might be possible to tune the LDOS of a QD emitter. The real posibilities of this idea will be studied on chapter 4.
42 Chapter 2
Quantum Dot Lasers
Originally the idea to ”exploit quantum effects in heterostructure semiconductor lasers to produce wavelength tunability” and achieve ”lower lasing thresholds” via ”the change in the density of states which results from reducing number of translational degrees of freedom of the carriers” was introduced in 1976 by Dingle and Henry.
The history of quantum dot lasers is strongly correlated with the history of the general development of semiconductor lasers, beginning from bulk lasers, continuing with heterostructure lasers and ending with quantum based lasers. Lasers structures have been permanently improved from the first bulk lasers to the quantum dot separate confinement heterostructure lasers. So we can have finally not only macroscopic design rules of carrier transport but also the possibility of density of states engineering in a semiconductor material by realizing nanometer structures in the active region of a laser.
QD lasers represent an ultimate case of the application of the size quantization concept to semiconductor heterostructure lasers, an approach which previously resulted
43 in the successful development of QW devices [6]. Of particular interest are the following expected main advantages over the conventional quantum well lasers: the narrower gain spectra, significantly lower threshold currents and the weaker temperature dependence of the latter.
As a consequence of the already said quantum confinement in all the three dimen- sions, the energy spectra of carriers in QDs are discrete. For this reason, structures with
QDs have generated much interest as a new class of artificially structured materials with tunable (through varying the composition and size) energies of discrete atomic-like states that are ideal for use in laser structures [13].
Amongst the anticipated advantages of QD lasers were [22]:
decreased transparency current, •
increased material gain, •
large characteristic temperature T (which characterizes the stability of the threshold • 0 current with temperature increase),
increased differential gain, •
decreased chirp (shift of the lasing wavelength with current). •
The original predictions were generally based on simplified assumptions of:
infinite barriers, •
ideal QDs of identical shape, •
temperature-insensitive homogeneous broadening, •
44 one confined electron and hole level, •
bimolecular electron hole recombination, •
ultrafast energy relaxation of injected carriers, •
equilibrium carrier distribution, •
lattice matched heterostructures and similar confinement volume for electrons and • holes.
In realistic devices these assumptions had to be replaced in recent years by different ones:
finite barriers, •
size and shape dispersion of QDs, •
many electron and hole levels and the impact of the continuum states, •
monomolecular (excitonic) recombination, •
non-equilibrium carrier distribution, •
strained heterostructures with completely different potential wells for electrons and • holes.
The motive for these previous assumptions not to came true are related to additional effects such as carrier transport and relaxation, the dots size-fluctuation, and other non- linear effects. All this effects will be discussed in this and the next chapter.
Key parameters in laser applications are the laser threshold Ith and its characteris- tic temperature T0, the quantum efficiency η and its characteristic temperature T1. These
45 parameters determine the temperature range in which a laser device can be used. Further- more, modulation speed and maximum output power and last but not least beam quality are important parameters for communication applications [16]. All these parameters will be discussed too in this chapter.
46 2.1 Quantum Dots fabrication
A lot of efforts were undertaken to develop QD fabrication technology suitable for device applications. The challenge is to prepare the semiconductor structures, typically 10 nm in size, in an efficient and reproducible way. Currently, there are a number of in situ and ex situ QD fabrication methods. In situ methods are actually better, and they are based on making quantum dots during a growth process. After that, they may be covered epitaxially by host material without any crystal or interface defects [9].
For the fabrication of low dimensional lasers a certain number of techniques have been developed. A list of fabrication techniques to realize dot structures is [16]:
1. Lithographical techniques:
Overgrowth of patterned strained layers (buried stresors) •
Overgrowth of etched V-grooves (an extension of the buried crescent laser tech- • nique to nanometer sizes). V-groove and migration growth (AlGaAs/GaAs ma-
terial system) along (111)A-planes for V-grooves in the [011]-direction.
Lithography based patterning (dry etching and overgrowth) techniques. Dry • etching and control of surface recombination and dead layer effects.
Local implantation enhanced intermixing (IEI) techniques. •
Focused ion beam (FIB) technique. •
2. Growth techniques:
Growth on vicinal substrates (FLS fractional layer super-lattices) by MOVPE. •
Growth on misoriented substrates (tilted super-lattices TSL) by MBE. •
47 SSL (serpentine super lattices) technique. •
3. Self assembled growth techniques (self-assembled quantum dots= SAQD or SAD):
Growth of dots •
It turns out that in general, artifically patterned nano-structures based on implan- tation or etching techniques show a non-vanishing contribution of defect effects. These defects seriously mask quantization effects and therefore clear single dot effects could hardly be observed. Epitaxially grown dots by self-organization principles, on the other hand, already showed nice low dimensional effects. Despite this success, there is an intrinsic disadvantage of SAD structures as dots are made statistically distributed in size and area, and at this point, non-desired dots size fluctuation is a critical aspect to take into account into the growth and postgrowth processes of the quantum dots. Size and average distance between dots can be controlled only to a certain degree inside the Gaussian distribution due to the epitaxial growth conditions.
The Gaussian distribution width shows a two-fold effect [16]:
Size non-uniformity reduces the available gain in a certain energy range and can • even result in an inferior device performance of a dot laser compared to a QW laser.
Distribution of dot-to-dot distances doesn’t allow the realization of an optical filter • like a DFB or DBR by SAD growth solely. A post epitaxial realization of a DFB
grating is of course always possible, but would not meet the very attractive possibility
to realize complex coupled DFB lasers with dot arrays directly.
.
48 Some examples of QD laser structures are shown in fig.2.1
Figure 2.1: Schematic diagrams of QD Structures for laser applications fabricated by (a) etching and (b)
Stranski-Krastanow self-organized growth. [7]
2.1.1 Lithography Based QD Laser Fabrication
Mainly two techniques can be distinguished as mentioned in the previous section: direct epitaxial growth with reduced flexibility and lithography based as implantation and dry etching with enhanced flexibility, but for some cases poorer material quality.
Electron Beam Lithography
The realization of small, arbitrarily shaped crystal structures to obtain quantum size effects can be realized by patterning techniques based on electron beam resist masks in the nanometer range. A quite flexible technique to realize small masks is based on a local variation of the chemistry of the mask material by a well defined local energy supply. The different possibilities to supply energy defines the kind of lithography (photons, electron beams, ion beams).
49 Thanks to its flexibility, commonly used are electron beam lithography (e-beam) to define artificially shaped nano-structures and also to define mask for the above mentioned lithography techniques. The resolution of the e-beam process not only depends on the minimum electron beam width, but also on the detailed parameters of the development process as the resist characteristics. The resist thickness should be large to ensure digital patterns of wires or dots and its sensitivity is also an important parameter. Here sensitivity means the number of electrons needed to define a pixel of 10nm2, which is more or less the desired dot size [16].
Dry Etching
After lithography, different technologies can be applied to transfer the lithography pattern to the semiconductor substrate. Most common are dry etching techniques or implantation techniques, but with the latter, no real dot features at room temperature could be achieved.
After the definition of masks with nanometer sizes, the structures can be transfered by dry etching in conjunction with epitaxial overgrowth as we can see in the figure 2.2.
Many results are reported based on this fabrication method. For example, elec- troluminescence from QD-like structures prepared by holographic photolithography, wet chemical etching, and liquid-phase epitaxy (LPE) regrowth was first reported in 1984.
But the electroluminescence from the dots at 1.4µm was observed at 77 K under pulsed ∼ conditions. However, lasing threshold could not be attained, probably due to excessive non radiative recombination at the regrown interfaces [7]. Later, in 1994, Hirayama [8] propose a tensile-strained, buried-heterostructures InGaAsP QD lasers fabricated using electron-beam lithography, dry etching, and OMCVD growth and regrowth. This struc-
50 Figure 2.2: Principal process steps of lithography based dot laser production using dry etching methods for dot array patterning [16] tures allow the achievement of 1.5 µm wavelength emission with a QW layer that is thicker than for corresponding compressively strained structures, thus increasing the optical con-
finement factor. The dots were 30 nm in diameter, with a 70 nm center-to-center spacing on a square lattice; the optical confinement factor was estimated to be approximately 0.3% for a single dot layer. Lasing from the QD region was observed under pulsed conditions at
77 K at 1.26 µmwavelength. The relatively large blue shift (shift to shorter wavelength) of the lasing wavelength was attributed to large band filling in the dots due to the small optical confinement factor and dot size fluctuations.
Finally, it can be said that this technique allows a most flexible dot definition with the additional advantage to use the dot array directly as an optical filter. However, this
51 fabrication method is determined strongly by the damage level induced by etching, spe- cially during dry etching. Nevertheless, if low damage etching techniques in combination with epitaxial overgrowth are applied, structures down to 50nm with optical functionality can be realized.
2.1.2 Stranski-Krastanow method
Since the mid-1990s, there has been considerable work on the direct synthesis of semicon- ductor nanostructures by applying the phenomenon of island formation during strained- layer heteroepitaxy, a process called the ”Stranski-Krastanow growth mode”. In the orig- inal paper by Stranski and Krastanow (1938) island formation in heteroepitaxial ionic crystals was investigated. Nowadays this growth mode describes island formation on a wetting layer in heteroepitaxial systems with different lattice constants. In InAs/GaAs superlattices, Goldstein et al. (1985), observed for the first time island formation in a semiconductor system. Originally, this instability leading to island growth was considered detrimental for the growth of quantum well systems in strained heteroepitaxy, and efforts were undertaken to avoid it.
However, from 1990 onwards this topic has gained considerable attention, as it was realized that such islands may represent zero-dimensional semiconductor nanostructures.
It was found that, for suitable growth conditions, the size distribution (height and lateral size) could be substantially narrowed, which reduced inhomogeneous line broadening and thus made the determination of optical properties of quantum dots much more reliable.
It was soon established that the quasi-zero-dimensional density of states in InAs islands
52 embedded in GaAs could be used to realize devices such as injection lasers, which exhibit properties in some respects superior to strained III-V quantum well systems [21].
The self-organized growth of uniform nanometer-scale islands it is based on the knowledge on that a layer of a semiconductor material having a lattice constant different from that of the substrate, after some critical thickness is deposited, may spontaneously transform to an array of three-dimensional islands (clusters). For the first atomic layers, the atoms arrange themselves in a planar layer called the ”wetting layer”. The wetting layer acts like a charge reservoir for the quantum dots. Since the QDs appear spontaneously during growth, they are said to be ”self-assembled”. An schematic view, comparative of different semiconductor growth modes is shown on Fig.2.3, just to realized the difference between SK process and other similar techniques.
The island size can be tuned by the epitaxial growth conditions from which the most important are the substrate temperature and the composition of (Si)Ge or In(Ga)As material deposited that defines the mismatch strain. The change in size corresponds to an energy shift of the emitted light, which is due to the altered carrier confinement and strain in the dots. Smaller dots emit photons of shorter wavelengths, corresponding to higher energies. From typical atomic force microscopy (AFM) images, the size and surface density of Ge islands on Si surface can be estimated. The islands are about 20 nm in lateral size an 2 nm in height [9]. These are measures given to take into account about the extremely small size of this devices.
The most important benefits of such an epitaxial self-assembly are:
A giant number of nanostructures is formed in one simple deposition step. •
53 The synthesized nanostructures can reveal a high uniformity in size and composition. •
They may be covered epitaxially by host material without any crystal or interface • defects.
Other benefits are: the cluster (dots) formation is energetically favorable and there is no need of using masks or lithography and etching in the whole process.
Figure 2.3: Schematic diagrams of the growth modes for semiconductor systems: (a) Frank-van der Merwe
(FM, layer by layer); (b) Stranski-Krastanow (SK, islands with wetting layer); (c) Volmer-Weber (VW, islands). [9]
It was demonstrated for the example of InGaAs QDs on a GaAs surface that there exists a range of deposition parameters, where the islands are small ( 10 nm), have a ∼ similar size and shape and form dense arrays. Due to the strain-induced renormalization
54 of the surface energy of the facets, an array of equisized and equishaped 3D islands may represent a stable state of the system (so called Size-Limited Island Growth, SLIG), as manifested by the process reversibility studies upon temperature ramping experiments.
Interaction of the islands via the substrate makes also their lateral ordening favorable.
But this process must be very accurate because a deviation from the optimal range of growth conditions results in ultrafast ripening of the islands within only a few seconds.
Interaction of the equilibrium islands via the substrate also allows their lateral ordering.
At the same time, the existence of the equilibrium size is extremely important in practice, as it allows slow ripening of QDs, giving more room for manipulations with them. If strained InGaAs islands are covered with a thin GaAs layer, islands in a second sheet are formed over the dots in the first sheet, resulting in a three-dimensional ordered array of QDs, either isolated or strongly vertically coupled. The size and shape of the InGaAs islands can be changed by replacing InAs by InGaAs or InGaAlAs and by changing the deposition mode [6].
Controlling the strain fields and nominal growth parameters, enable various types of ordered array structures, achieve long-wavelength emission is possible. Lets see different parameters and its effects on the fabrication process and therefore, on the QDs resulted from the latter.
Temperature
Increase in the substrate temperature during the InAs deposition results in a decrease in the volume. The SLIG island’s density and the lateral size as well as PL (photolumi-
55 nescence) spectra do not depend on the growth interruption time in the range of 10-40 s before the islands are capped with GaAs. If the substrate temperature is reduced after the deposition, the density of islands increases, the average volume of the single island increases, while their lateral sizes undergo shrinkage [23]. This effect is illustrated in figure
2.4 where the dots ramped from 500 to 450◦C within 90-120 s and covered at the final temperature demonstrate bright PL emission at room temperature near 1.3 µm.
Figure 2.4: Plan view and cross-section. TEM images of InAs QDs
Vertically stacking
Nowadays, there is another direction of work in order to achieve better uniformity, and it is called ”the modified Stranski-Krastanow”. This growth mode takes place during deposition of multiple layers of such islands. For island layer separations comparable
56 to the lateral island size, the elastic local strain fields surrounding the islands extend to the surface during multilayer growth and the influence the formation of subsequent island layers. At the end, we have the critical thickness for island nucleation reduced, and islands are self-ordered in vertical stack. So we have dots coupled and significant Coulomb interaction of discrete localized electron levels within the QDs. Electronic coupling causes a significant long wavelength shift of the PL emission with increase in the number of stacks, even in the case where very small islands are used as stacking objects. Using this approach it is possible to fabricate QD structures emitting in the range 1.3-1.44µm. This technique will be explained further in the section dedicated to VCSELs.
Activated Alloy Phase Separation
A promising way of achieving long-wavelength emission from GaAs-based QDs is to use activated alloy phase separation. It is based on the controlled increase of the volume of the initially small strained island by alloy overgrowth and was demonstrated for InAs QDs on
GaAs substrates covered by a (In,Ga)As layer. Initially, an array of small coherent InAs stressors is formed [24]. If these InAs islands formed by InAs deposition are covered by
GaAs, the resulting QDs have a high density, but a rather small lateral size and height and may be revealed only in a strong beam TEM conditions, figure 2.5. Once the dots are covered by InGaAs alloy, it is energetically favorable for InAs molecules to nucleate at the elastically relaxed islands, where the lattice parameter is close to that in the unstrained
InAs[6].
This techniques allows 1.32 µm at room temperature.
57 Figure 2.5: Large InAs QDs formed by activated alloy phase separation
Defect reduction technique
The formation of coherent InAs islands is accompanied in many cases by formation of dislocated clusters. As these clusters attract Indium atoms more effectively as compared with desired clusters, in consequence, dislocated objects size is larger. So, it is possible to select the undesired islands by height. This technique cover the InGaAs islands by a thin GaAs layer with a thickness sufficient to overgrow smaller coherent islands, but leaving the large (dislocated) islands uncovered. With this, the dislocated islands have the strongest lattice mismatch with GaAs and cause the strongest repulsion of the latter upon
58 overgrowth. After the covering, the substrate temperature is increased. The whole process is detailed on the figure 2.6. During this procedure the InGaAs acumulated in dislocated clusters evaporates and redistributes through the GaAs surface forming a second wetting layer. This approach led to the formation of defect-free long-wavelength GaAs-based QDs
Figure 2.6: Defect reduction technique
using MOCVD, their successful stacking and realization of room-temperature gain and stimulated emission at wavelengths up to 1.38µm.
59 2.2 Density of States: DOS
Calculating energy states in semiconductor quantum dots is made particularly challenging by a number of factors [18]:
1. The QDs contain a large number of atoms. A typical self-assembled quantum dot has
a base of 300A˚ and a height of 50A˚. Therefore the dot itself may contain 105 ∼ ∼ ∼ atoms. This dot then needs to be surrounded by a barrier material to isolate it from
other dots. A representative system containing both the dot and barrier therefore
typically contains 106 atoms. ∼
2. The valence band maximum in III-V semiconductor materials is three-fold degener-
ate, therefore an realistic approach must describe at least the band mixing between
the three valence band edge states.
3. In a zero-dimensional InAs/GaAs quantum dot system, the charge carriers are ar-
tificially confined inside the dot, which is typically smaller than the bulk excitonic
radius (this is called the ”Strong Confinement” regime). This dramatically enhances
the Coulomb interaction between charges in the dot and strongly modifies the di-
electric screening.
There are different ways to approach to this problem, and the most used is to adapt effective mass based techniques and their more sophisticated extension, by the technique called the k p method. This method has been extremely successful in explaining a range · of properties of bulk semiconductors. In essence, this method expand the single particle wave functions of the system in a basis of bulk Bloch orbitals derived from the Brillouin
60 zone center (τ point). If a sufficient number of basis states are included, this expansion provides an excellent description of the band structure of the bulk material close to the
τ point. It has been proven a successful model in describing spectroscopic and transport properties in both three-dimensional bulk systems and two dimensional quantum well structures. Also has demonstrated to provide at least a qualitative picture of the energy states in zero dimensional systems [11].
However, considering self assembled dots with ”lens” shaped structure (what it is a normal shape for self-assembled QDs), DOS function can be calculated easily using an adiabatic approximation [41], where the Hamiltonian for the electrons can be separated into a direction perpendicular to the growth plane and a radial direction.
Let’s proceed with the calculus using the Schr¨odinger equation, written in cylindrical coordinates, for describing the motion of a single electron:
¯h2 1 ∂ ∂ ∂2 ∂2 − − r r + + V (r, z) ψ(r, θ, z) = E ψ(r, θ, z) (2.1) 2m r2 ∂r ∂r ∂θ2 − ∂z2 · where the assumption of symmetry of the lens shaped dot about the z-axis means that it can be used a wavefunction ψ(r, θ, z) as:
1 ψ(r, θ, z) = ejmθφ(r, z) (2.2) √2π
This factorization of the 3D wavefunction into an exponential with an angle dependence
(θ), and the 2D new wavefunction φ(r, z) is very useful, as now it can be applied the adiabatic approximation, and finally have another factorization into two one-dimensional wavefunctions. So, finally:
1 jmθ ψ(r, θ, z) = e gr(z)fm(r) (2.3) √2π
61 It must be taken into account that this approximation is valid in the case of lens shaped
QDs, as them allows a carrier wavefunction strongly confined to the lowest energy level in the z direction and the potential changes relavitely slowly in the radial axis r.
Finally, the new one-dimensional wavefunctions satisfy the following system of equa- tions: 1 ∂ ∂ − r r m2 + E (r) f (r) = Ef (r) (2.4) r2 ∂r ∂r 0 m m − ∂2 2 + Ve(r, z) gr(z) = E0(r)gr(z) (2.5) ∂z where E0 is the effective lateral confining potential. A more deep inspection of equation
2.4 reveals that it represents a one-dimensional finite quantum box in the z direction, which is a classic quantum mechanics problem, solved by matching the wavefunction and its derivative at the two boundaries. Equation is solved with a transfer matrix approach.
Finally, applying boundary conditions and values for each case, equation 2.3 can be solved and the DOS obtained. To obtain the DOS for an ensemble of QDs, it must be taken into account that, owing to the nature of the self-assembly process, each dot shape is unique and the differences in dot shape mean each dot has slighly different quantum confined energy levels and therefore the ensemble of QD forms a broad spectrum of allowed energy states.
This latter argument can be easily thinked and corroborated by a simple inspection of eqs. 2.4-2.2, where the dependence on radius is clear. This effect of DOS broadening of
QDs is clearly observed on fig.2.7, where an ensemble of 5000 different dot sizes (enough for having a higher number a negligible effect in DOS function) and assuming a Gaussian distribution of dot radii, with a standard deviation of 1 nm, 2 nm and 3 nm. In this
62 17 3 Figure 2.7: Electron DOS function for ensemble with typical quantum dot density 1x10 cm− with 5000 random dot sizes for three dot size distributions [41] picture, the lowest energy level at 0.21 eV (relative to the bottom of the quantum dot potential) is clearly observed for all three structures despite the broadening due to the different dot shapes. However, the higher energy levels have merged to form a broad peak for the structure with 1 nm deviation in the radius, this peak becomes a broad shoulder for the structures with large deviations in the radius. As expected, if the variation in dot sizes is increased, the DOS function is changed leading to a less favorable distribution, which is broadened and reduced in its peak magnitude [41].
63 2.3 Rate Equations of QD Laser
The rate equations should be the main tool for describing the QD Laser, its performance, behavior and characteristics. A theoretical analytical model must be capable of describing both non equilibrium carrier distributions in the QD zero-dimensional levels at low temperature, as well as a quasi-equilibrium distribution for higher temperature operation.
Also, in the analysis, the wetting layer must be specifically included in the rate equation to globally couple the QDs. Quantum mechanics plays a critical role in determining the correct form of the rate equations, since zero-point fluctuations in the reservoir modes must be included to correctly relate the spontaneous emission, stimulated emission, and absorption rates [26].
A typical stripe laser geometry schematic illustration is shown in Fig.2.8. The elec- tron and hole levels of the wetting layer (WL) are assumed to be driven by a current source. Several assumptions are made in order to simplify the analysis and clarify the relevant physics. First, it is assumed that the WL states contains only low concentrations of electrons and holes. This assumption can be done because it is a typical situation on operating conditions of QD lasers, specially for steady-state operation. The WL carriers are then treated using the Boltzmann approximation, with the continuum of WL states replaced by an effective DOS. The last assumption is that each QD contains only single electron and hole levels (ground state) with a degeneracy of two due to spin. Equal electron and hole relaxation rates is assumed too. The effect of carriers escapes and capture from and to the excited state and wetting layer are considered too. In the structure, another
64 assumption for simplicity is made: ideal facet reflectivity
Figure 2.8: (a) Schematic illustration of the broad-area stripe geometry QD laser diode. (b) Schematic showing the Al, Ga, and In content in the laser waveguide region [20].
Let’s begin with the energy band diagram for the SOA QD in the active region, shown in Fig.2.9 [27]. As previously said, the two dimensional WL is taken into account.
The carriers injected to the separate confinement layer (SCL) relax into the wetting layer and then into the excited and ground states (GS). The recombination of these carriers can be in a radiative or non-radiative way (as in every recombination process). All the different processes, as recombination, capture, escape, etc, are considered on Fig.2.9. The different temporal variables associated to recombination or scaping processes collected on
Fig.2.9 are:
τ , recombination lifetime in the SCL. • sR
65 τ , difussion lifetime in the SCL. • d
τw2, electron relaxation time from the Wl to the excited state. •
τ , electron escape time from the excited state to the wetting layer. • 2w
τ , spontaneous radiative lifetime in wetting layer. • wR
τ , electron relaxation time form the excited state to the ground state. • 21
τ , electron escape time from the ground state (GS) to the excited state. • 12
τ , spontaneous radiative lifetime in the QDs. • QR
τ , electron escape time form the wetting layer. • WE
Figure 2.9: Energy band diagram of the QD SOA and the main processes inside the active region
The photon rate equation is given by
1 ∂S = g0(fn + fp 1)S αg0S (2.6) vg ∂z − −
66 where vg is the group velocity, g0 is the maximum modal gain, S is the photon density,
α the material loss, and z is the longitudinal direction of the device, its direction of light propagation. fn and fp are the electron and hole occupation probabilities of the GS state, respectively, and expressions for them can be found in bibliography [28].
The rate equation for electrons in the SCL can be written, just taking into account the processes that takes part in this layer, if they implicates gain or loss on electron concentration, and the involved relaxation constants. We have,
∂N J N N s = s s (2.7) ∂t eLs − τd − τsR where Ns is the electron concentration in the SCL, and as usual, J is the current density, e is the electron charge, t is time and Ls is the thickness of the SCL.
The rate equations for electrons in the wetting layer, excited state and ground state can be written, respectively [28],
∂N N L N (1 h ) N h N w = s s w − n + w n W (2.8) ∂t τdLw − τw2 τ2w − τwR
∂N hn Nw(1 hn) Nwhn N (1 fn)hn N (1 hn)fn Q = − Q − + Q − (2.9) ∂t τw2 − τ2w − τ21 τ12
∂NQfn NQ(1 fn)hn NQfn(1 hn) NQ = − − fnfp vgg0(fn + fp 1)S (2.10) ∂t τ21 − τ12 − τQR − − where the new constants are: NQ is the surface density of QDs where its typical value
10 −2 is 5 10 cm , N is the electron density in the wetting layer, Lw is the effective ∼ · W thickness of the wetting layer of the QD stacked layers, and hn is the electron occupation probability of the excited state.
67 Due to better confinement in the dots, another assumption is made: the radiative recombination rate in the dots is assumed to be proportional to the joint probability f f , which reflects the discrete nature of the energy states of the dots. It is well-known, n × p that the electron and hole occupation probabilities is not equal and so are not in QD active layers, so a relation can be established between fn and fp in the way fp = γfn + η.
So it is a linear relation where γ and η are constants parameters. Using this relation and operating [27] with eqs.2.7-2.10, it can be obtained the following equation, aiming to find an expression for the photon density:
S S(1 η) γf 2 + f η + = i + − (2.11) n n S S (1 + γ) sat0 sat0
Ssat0 is an intrinsic parameter related to the output saturation intensity of the amplifier
(and will be usefull later for making comparisons). With this latter equation and the photon rate equation 2.6, it can be found an analytic solution by eliminating S and solving the differential equation in term of fn. So finally, the photon density is:
i ηf γf 2 S = S (1 + γ) − n − n (2.12) sat0 (1 + γ)f 1 + η n − i is the normalized current of the amplifier. It must be realized that S and fp are position dependent (i.e. S(z)) and the rest of parameters are constants.
The optical gain of the amplifier will be ratio between the photon density at the input, and the output one: S(z = L) G = (2.13) S(z = 0)
The output saturation intensity can be calculated from the point of photon density at which the optical gain, G, is reduced by 3 dB (but not only this). It can be obtained
68 that the output saturation intensity increases with increasing applied current [27], unlike happens on bulk SOAs.
A couple of figures is given to illustrate the behaviour of the optical gain in relation with the current on the device fig.2.10(a) and in relation with the saturation photon rate in fig.2.10(b).
Figure 2.10: (a)Optical gain as a function of normalized output photon density, (b) Optical gain as a function of normalized current for different values of input photon density [27].
69 2.4 Threshold current
Ideally, threshold current of a QD laser should remain unchanged with the temperature and the characteristic temperature should be infinitely high. This would be so indeed if the overall injection current went entirely into the radiative recombination in QDs and the charge neutrality in QDs were the case. In fact, because of the presence of free carriers in the OCL, a fraction of the injection current is wasted therein. This fraction goes into the recombination processes in the OCL (in the barrier regions). Hence, recombination in the
OCL gives rise to one more component of the threshold current. The latter component, associated with the thermal excitation (leakage) of carriers from QDs to continuous spectrum states in the OCL, depends exponentially on the temperature. Besides, violation of the charge neutrality in QDs, causes the threshold current component, associated with the radiative recombination in QDs, to be temperature dependent. As a result, the total threshold current should become temperature-dependent, especially at high temperatures.
Hence the characteristic temperature T0 is not infinite as predicted but finite [37].
The threshold current of a QD laser is temperature dependent, but this effect comes from the dependence on the radiative recombination rate inside the dots and in the separate confinement layer. From the aforesaid, it is evident that the threshold current density is the sum of the current densities associated with the radiative recombination in QDs and in the OCL.
To be able to obtain a equation that describe the behaviour of the threshold current,
first must be distinguished between the case of equilibrium filling of carrier levels in QDs
70 or non-equilibrium case. For the distinguishment a temperature Tg must be taken into account [37].
At the non-equilibrium case (T < Tg), the characteristic times of thermally ex- cited electrons and holes that escape from a QD, are small compared with the radiative lifetime in QDs, τQD. Redistribution of carriers from one QD to another occurs, and quasi- equilibrium distributions are established with the corresponding quasi-Fermi levels. As a consequence of this redistribution, the level occupancies (and numbers of carriers) in vari- ous QDs will differ. Having no time to leave a QD, the carriers recombine there. Evidently, this case of operation is not desired.
At the equilibrium case (T > Tg), complete state filling is achieved, and Fermi levels well established. Under the conditions of the thermal equilibrium between the electrons
(holes) confined in QDs and free electrons (holes), and going a little into the equations, we have the following expression for the threshold current:
eN f f j = s f f + ebBn p p n (2.14) τ n p 1 1 (1 f )(1 f ) QD − n − p where all the variables have been presented in previous sections but b, which is the OCL thickness, and B, the radiative constant for the OCL. As previously said, the threshold current is the sum of the current densities of QDs and OCL, and this result is shown on the equation 2.14, as the terms of the sums corresponds to the contribution of the spontaneous radiative recombination in QDs and in the OCL, respectively.
The population inversion in QDs required for lasing can be written as:
min NS fn + fp 1 = (2.15) − NS
min where NS is the minimum tolerable surface density of QDs required to attain lasing
71 at given total loss β. This equation too relates the mean level occupancies in QDs. The dependence of the threshold current on NS is nonmonotonic. In the case of equilibrium
filling of QDs (in which we are working), the minimum threshold current density has been shown to be [37]: 2 min 1/2 min eNS 1/2 jth = + (ebBn1p1) (2.16) τQD The temperature dependence is given in the variables n1 and p1 as they are related in the way e−1/T . In the special case of symmetric structure (f = f ), ∼ n p min 1 NS fn,p = 1 + (2.17) 2 NS in which the dependence on NS is clearly reflected. Equation 2.17 can be easily rewritten
min min just changing L and L instead of NS and NS, respectively, to express the dependence on the cavity length. This dependences are shown in fig. 2.11. As it can be seen in that
figure, NS tends to its minimum value, which is the same case of the cavity length tending to Lmin, the mean electron and hole level occupancies in QDs tend to unity (occupation probabilities closed to 1), which demands infinitely high free-carrier densities in the OCL.
As a result, the threshold current increases infinitely. As previously said, the jth of a
QD laser must be temperature-independent and the characteristic temperature must be infinitely high [1]. The expression for T0 is:
∂ln(j ) −1 T = th (2.18) 0 ∂T This independence would be so indeed if the overall injection current went entirely into the radiative recombination in QDs. In fact, because of the presence of free carriers in the OCL, a fraction of the injection current is wasted therein. This fraction goes into the recombination processes in the OCL (the second term in 2.14).
72 Figure 2.11: Threshold current density versus (a) the normalized surface density of QDs, (b) cavity length [13].
Always in the case of equilibrium filling of QDs, and giving the notation of jOCL to the current component associated with the recombination in the OCL and the second term (of the right hand-side) in 2.14, it depends on T exponentially. As a result, jth must become temperature dependent, especially at high T . Hence TO must become finite.
On the other hand, and naming the first term in 2.14 jQD, let’s consider the charge neutrality in QDs were the case (fn = fp). In this case, jQD, would be temperature- independent. Examination of the problem shows (dependences on NS and n, p1) that the electron and hole level occupancies in QDs at the lasing threshold become temperature- dependent if the violation of the charge neutrality in QDs is taken into account properly.
Thus, correct consideration of the QD charge reveals the T-dependence of jQD.
But the T-dependences of fn,p are much weaker compared to that of the exponential dependence of jOCL. Consequently, jQD increases with T much more slowly than jOCL does and this is collected on fig. 2.12.
73 Figure 2.12: Threshold current density and its components versus the temperature. The inset shows jQD
and jOCL on an enlarged (along the vertical axis) scale. The broken line depicts jQD calculated assuming the charge neutrality in QDs [13].
Giving more examples, on Fig. 2.13 shows the threshold temperature dependence from 79 K to 324 K of and experiment made by Deppe [20]. The maximum temperature is set by the upper limit of the probe station. Spectral data at three different temperatures are shown as insets. Let’s comment the figure.
At 79 K, the threshold current density is 20 A/cm and is nearly independent of temperature from 79 K (20 A/cm ) to 220 K (22 A/cm). It then increases sharply with increasing temperature from 240 K to 324 K. The threshold decrease at 180 K (20 A/cm) is due to thermalization to the wetting layer, which increases the gain as compared to the inhomogeneously broadened gain region. So we can conclude that coupling with the wetting layer results in slow pulsation frequencies ( 1GHz). The spectral insets show that ∼ lasing occurs on the ground state from 79 K to 280 K and on the first excited state at
74 higher temperatures. At 79 K, carrier capture in individual QDs is independent with a low probability of escape (relative to the spontaneous recombination rate), and the broad lasing spectrum must be due to spectral hole burning [28]. At 240 K, the number of lasing
Figure 2.13: Threshold current density versus temperature, along with spectral insets for temperatures of 79 K, 240 K, and 280 K. An abrupt decrease in the multimode spectral linewidth is observed above 230
K and indicates coupling of the QDs with the wetting layer [20].
modes decreases due to thermal coupling of the QD ensemble through the wetting layer, as it can be confirmed too in [30]. At 280 K, the spectrum measured just above threshold shows both states lasing at once. Above 280 K, the first excited state is the first to come to threshold. This is because of carriers in the higher energy levels have a higher nonradiative recombination rate due to an increased number of recombination paths. The solid curve in Fig.2.13 is the calculated threshold current using the analysis presented above in this chapter. The dashed curve shows the calculated threshold current density assuming that
75 the nonradiative recombination rate equals zero.
These results predict quite low threshold current density at room temperature if nonradiative recombination is eliminated from the QDs (what would be one of the ideal predictions made for QD lasers).
76 2.5 Differential Gain
In classical bulk semiconductor laser, the differential gain relates the gain to the carrier density N by the simple equation g = g (N N ) [40], where N is the transparency 0 − tr tr carrier density. g0 describes the rate of change in gain at the lasing wavelength to changes in the carrier density occupying states at the corresponding energy, this is mathematically expressed g0 = dg/dn. But of course, the situation is more complicated on lasers based on nanostructures as SAQDs.
One of the main problems is a basis difference on the possible lasing states. This is, in SAQD based lasers, there is a finite steady-state carrier population of reservoir states so that induced carrier perturbations are distributed over a range of energies and contribute only partially to the gain at the lasing energy. And the energy separation between these reservoir and lasing states is often large, complicating the coupling process between the two regions. On the other hand, these laser structures, may also have a measurable carrier population of excited states. Any carrier density perturbation will be fed to several states other than the lasing state (usually the ground state, as it have be seen), resulting effectively in a reduced differential gain.
It is obvious that the differential gain degrades when the ground-state population increases and for small energetic spacing between the various transition.
Fig.2.14 shows the state filling reduction of the conduction term at room tempera- tures. In this figure, the solid lines are used to represent the ground state. The separation between the wetting layer and the second excited state was chosen to be the same which
77 fits reasonably well to luminescence measurements in cases where the wetting layer emis- sion is clearly detected. It can be seen that for twice the room temperature thermal energy
(which is the choosen unit for measure state spacing) and ground state occupation of 0.85, the differential gain drops by one order of magnitude.
Large differential gain in room temperature conventional SAQD lasers may be pre- dicted based on the conduction band filling reduction whose peak values may exceed considerably the valence band peaks as seen in the large spacing cases os fig.2.14
Figure 2.14: Differential gain reduction in SAQD due to state filling versus the electronic ground-state occupation probability. Each of solid line denote a different energy spacing between the available states, characterized in units of room temperature [40].
However, the most easier and useful way to calculate the differential gain (gdiff ) is:
∂ln(G a) g = L−1 o (2.19) diff ∂i
a where L is the length of the amplifier and Go is the unsaturated optical gain. The depen-
78 dence of gdiff on the normalized current i can be seen in the equation 2.19 and reflected on the fig.2.15. It is clear that the differential gain is inversely proportional to the applied current. Other results obtained by Qasaimeh [27], concludes that for a current slightly above the transparency one, the differential gain is approximately 1.5 times larger than the maximum modal gain.
Figure 2.15: Differential gain as a function of applied current for different values of input photon density
(excitation intensity) [27].
More about the differential gain will be explained in the next chapter.
79 2.6 Q-Switching
Pulsed solid-state lasers are among the most important and widely deployed commercial laser systems. Its applications cover a wide range, from light source emitters in commu- nication systems, to useful devices capable of marking, cutting, drilling, and as helpful tools in range finders, retinal surgery, etc. For this reason, a technique called Q-switching is used to produce very short pulses for these types of applications.
The most simple method that can be envisioned for producing pulsed laser is to switch the gain of the medium on and off by switching the pump energy to the lasing medium on and off. When pump energy is sufficient to allow laser gain to exceed the threshold, an output beam appears. But this scheme has a problem: output pulses made will be quite rounded, since there is a delay as population inversion and hence gain builds in the laser. This also set limits on the pulse length and repetition rate for the laser. So this technique must be improved.
In a Q-switching technique, the laser output is switched by controlling loss within the laser cavity. More precisely, Q-switching is ”loss switching” in which a loss is inserted into the cavity, thus spoiling it for laser action [11]. To understand the physical process involved in passive Q-Switching and the improvement achieve in this technique as the laser is a QD based, it can be observed the magnitude of the gain and absorption in the laser cavity during turn on and subsequent pulse emission on fig. 2.16.
On this figure 2.16, it is shown the carrier and photon density time evolution in the laser cavity during a Q-switched pulse; also shown is the peak gain in the structure and
80 Figure 2.16: Carrier and photon density time evolution in the lasing cavity during a Q-switched pulse.
Also the peak gain in the structure and the absorption in the cavity [41] also the absorption in the cavity at that wavelength. With the support of the figure, let’s explain the process for passive Q-switching.
The applied current causes carriers to build up in the gain section increasing the gain in the cavity. Initially there are no carriers in the absorber section so the absorption in the cavity is at a maximum. The number of carriers continues to increase until there is sufficient gain to overcome the intrinsic cavity losses plus the absorption in the cavity due to the absorber section. When the losses are overcome, the photon density in the cavity dramatically increases. Some of these photons are absorbed causing the carrier density in the absorption to increase until the absorption is bleached.
Eventually the increasing photon density in the cavity causes the carrier density to decrease in the gain section until the gain becomes insufficient to overcome the cavity losses and the photon density reduces as the laser switches off. The presence of absorption in the
81 cavity means that the number of carriers at turn-on is much higher than under normal operating conditions, therefore the number of photons increases due to the increased gain.
The best operation conditions appears when the photon density during the output pulse are as bigger as possible. To be maximized, it must be remembered equation 2.6, where it is shown that the carrier density in the gain section decreases and the carrier density in the absorber section increases. This is achieved buy making as large as possible the following ratio [41]: dα dna γ = dg (2.20) dng where na and ng are the photon density throughout the laser cavity and the density carrier at the absorber section. Therefore, the change of both absorption and gain with carrier density is vital for understanding Q-switching.
The saturation of gain is more prominent in quantum-dot structures than in other quantum-confined (wire or well) and of course, than bulk structures. This is due to the discrete nature of the states associated with individual dots: once a dot state is filled by two opposite spin carriers no further transitions associated with that energy state can occur.
And once all the dots states are occupied with carriers no further increase in gain can be achieved; any further injected carriers occupy states outside of the dots (for example at the wetting layer). Of course, this is undesired. So finally, we have that saturation should occur at as high gain as possible to ensure that the cavity losses in the laser structure can be overcome at a point where the differential gain is still appreciable.
In his report of some experiments, Owen [41] concludes that the greater the pro- portion of absorber length in the cavity the larger the current required to overcome the
82 cavity losses and absorption. Increasing the absorber length also increases the peak power current gradient due to the fact that at turn on of the pulse there is a larger number of carriers in the cavity available to produce photons as the absorption bleaches. Thus the high threshold cavities with the greatest absorption produce the highest power pulses.
It is seen that the considerable gain saturation associated with QD lasers results in excellent Q-switching characteristics.
83 2.7 VCSELs
Vertical Cavity Surface-Emitting Lasers or VCSELs are a nowadays necessary, especially emitting at 1.3µm. The currently existing FP (Fabry-Perot) and distributed feedback
(DFB) devices grown on InP substrates are rather expensive. In contrast, GaAs-based
VCSELs are proven to have high reliability, very low cost and superb parameters, such as low beam divergence, high wall-plug efficiency and high temperature stability of the emission wavelength and threshold current. Another advantages are the possibility of on- chip testing and integration, a significant reduction of lateral size, lateral arrays and beam steering are advantageous. Thus is clear the actual huge interest on VCSELs emitting at wavelengths around 1.3 or 1.55 µm, because these devices can clearly substitute the very expensive and sophisticated edge emitting DFB lasers in glass fibre optics, as VCSELs should have similar monomode behavior at much lower price. However, there are basic material problems making the development of such lasers difficult. Structures based on
InP wafers can emit the required wavelength fairly well. However, VCSELs made of these materials lattice matched to InP would need a huge number of Bragg mirrors, because the potential mirror material InP and GaInAsP posses a very small difference of their refractive index. Therefore, many alternative routes are currently being investigated.
At the same time GaAs-based VCSELs emitting in the first window of optical com- munications (850 940nm), are extremely cost-efficient and have reached a high state of − development, but operate only at distances below 300 m. In these lasers, the optical feed- back is provided by highly reflective epitaxially grown multilayers acting as top and bottom
Bragg mirrors around the active zone of a laser.
84 An additional advantage of VCSELs is the possibility of vertical integration of a device with wavelength modulators (e.g. for chirp compensation), intensity modulators and photodetectors, which is very important for advanced applications in wavelength division multiplexing (WDM) and dense WDM (DWDM), data transmission links and ultrafast data traffic using electronically wavelength tunable VCSELs. Both electronic
(electrooptic) and membrane tuning of the wavelength can be used for DWDM and, of course, WDM (CWDM). The development of optical fibres, which do not contain the water absorption hump between 1.3 and 1.55 µm range, further extends the DWDM range and potentially merges DWDM and CWDM applications [6].
In one of his articles [25], Ledentsov gives some examples of the state of the art of
VCSELs emitting at 1.3 µm. Arrays of VCSELs allow a possibility to reach up to 1 W in continuous wave (CW) laser operation, while having a very low beam divergence. On the other hand, VCSELs using the previous technique of activated alloy phase separation to fabricate the QDs. The process consists on growing directly on GaAs substrates and when fabricated include selectively oxidized Al(Ga)O current apertures, intracavity metal contacts, and distributed Bragg reflectors (DBRs), see figure 2.17. Devices operate at room temperature and above with threshold currents below 2 mA and differential slope efficiencies of about 40 percent in the CW mode and shows an output power of 0.65 mW at room temperature. Degradation tests demonstrate sufficiently long operation lifetime of this device. From direct observation, for lasers with 10 fold stacked QDs emitting at
1.27 µm, cut-off frequencies larger than 10 GHz have been observed.
85 Figure 2.17: VCSEL structure with self-assembled InAs dots and GaAs/AlO Bragg mirrors (left). Light output and efficiency versus driving current for this device (right). The laser spectrum is depicted in the inset of the right figure. [42]
It is clear from all these results that QDs lasers (VCSELs) are practically useful for applications in optical transmitters with direct modulation. At the end of this chapter some results achieved nowadays on the construction of VCSELs are shown in the figure
2.18.
86 Figure 2.18: State of the art of VCSELs fabrication by different researcher teams all over the world (year
2000)
87 Chapter 3
Non-linear Effects
The performance of SOAs (Semiconductor Optical Amplifiers) can be improved by incorporating arrays of quantum dots in the active layer of the device as it has been shown in the previous chapter. Using QDs, the saturation intensity of the amplifier is increased and the optical gain of the system is improved. Also, it has been shown that QDSOAs exhibit reduced heating effects and lower relative intensity noise, which improve the efficiency of the amplifier[31]. But non-linear effects related mainly with the nonuniformity of the dot size must be taken into account. These nonuniform dots degrade the performance of QD devices, and hence its ideal predicted properties like: low threshold current, high gain, etc. Some of these effects have been already mentioned in the previous chapter, as DOS function broadening, worse Q-switching performance, etc.
From the point of view of practice, low-power high-gain SOAs are useful in today’s optoelectronic systems, because they can perform many functions ranging from linear am- plification to ultra-fast signal processing (where large nonlinearity is necessary to provide
88 switching and wavelength conversion).
89 3.1 Homogeneous broadening
Homogeneous broadening means simply that all the energy-decay and dephasing mecha- nisms we have discussed act on all atom or oscillator in a collection in the same fashion, so that the response of each individual oscillator is broadened equally [29]. In a classical review, the homogeneous Lorentzian linewidth (FWHM) is:
2 ∆ωa = γ + (3.1) T2 where γ is the total energy decay rate, also called in this case lifetime broadening and the inversion of T2 is the rate at which ”dephasing events” occur, whatever may be the origin of these dephasing events. If dephasing effects are absent, only the lifetime remains. If in addition all non radiative mechanisms are turned off, only radiative decay will be left, and the linewidth will take on its minimum possible value, called purely radiative lifetime broadening, which is impossible to remove. In a collection of real atoms, the transition at frequency ωij between two energy levels Ej and Ei with total decay rates γj and γi, respectively, will generally have a lifetime-broadening contribution that is given in a more exact analysis by 2 ∆ωa = γi + γj + , (3.2) T2,ij where 2 is the dephasing rate appropriate to that particular transition. The main T2,ij point here is that in most cases the γ term in the classical oscillator linewidth is replaced by the sum of the upper-state and lower-state energy decay rates γi + γj. Some values for strong visible-wavelength atomic transitions will be ranging from a few MHz to a few tens of MHz.
90 In a Quantum Dot Laser, the problem is that at low temperatures, dots with different energies start lasing independently due to their spatial location as well as a delta function- like single-dot optical gain, while at room temperature, the dots contribute to a narrow-line lasing collectively via homogeneous broadening of the optical gain of a single dot.
To study this effect, a new model for the linear optical gain of the quantum-dot active region with a volume density of dots ND is given [34]:
σ 2 ∞ Pcv g(E) = K | | [fc(E ) fv(E )] G(E Ecv)Bcv(E E )dE (3.3) c,v Ecv −∞ − · − − where each variable
subscript c, v: discrete state in conduction and valence band, respectively; •
E : energy of interband transition; • cv
fc,v(E ): occupation function of the conduction (valence) band discrete states, with • interband transition energy of E;
P σ: transition matrix element; • cv
K: a constant that groups many other ones not interesting in our case. •
This equation 3.3 takes into account both inhomogeneous broadening due to the dot size fluctuation and homogeneous broadening due to polarization dephasing in terms of a convolution integral. For the inhomogeneous broadening of G(E), is taken a Gaussian distribution function (this will be studied in depth in the next section). The homogeneous broadening has a Lorentz shape as
¯hτcv π Bcv(E E ) = (3.4) − (E E)2 + (¯hτ )2 − cv
91 whose full-width at half-maximum (FWHM) is given as 2¯hτcv, where τcv is the polarization dephasing or scattering rate.
Once known all variables and terms of eq.3.3, it can be realized that the optical gain at a given energy, E, is the result of the individual contribution of all dots within the homogeneous broadening around that energy. In the case of the homogeneous broadening function tending to a delta function shape, dots with different energies have no correlation to each other and so contribute to the optical gain independently since they are spatially isolated from each other.
Figure 3.1: Model for the lasing spectra, showing the relationship between lasing spectra and homoge- neous broadening of single-dot optical gain. (a) Homogeneous broadening tending to a delta function. (b)
Homogeneous broadening comparable to inhomogeneous broadening [34]
As it can be seen in figure 3.1, when homogeneous broadening is comparable to inhomogeneous broadening, lasing-mode photons receive gain not only from energetically resonant dots but also from other non-resonant dots that lie within the amount of homogeneous broadening.
92 By giving appropriate values to the parameters at rate equations 2.8 - 2.10, Sakamoto
[34] obtains different light emission spectra for a fixed inhomogeneous broadening of
20meV , and shows its results on figure 3.2.
For the case (a), which is for ¯hτcv = 1meV (temperature of 80K), the lasing emission rises from the top of spontaneous emission spectrum as the current increases, leading to a broad lasing emission over the range of 25meV at 10 mA. As the measured (but out of range in the figure) spontaneous emission at 0.5mA is a Gaussian function with
FWHM = 20meV (due to dot size fluctuation), it can be quickly realized that the emission band is wider than this FWHM. This effect can be explained by the fact of all the groups of dots having an optical gain larger than the threshold gain start lasing independently.
Figure 3.2: Calculated emission spectra up to well above the lasing thresholds for (a)¯hτcv = 1meV ,and
(b)¯hτcv = 10meV [34]
On the other hand, in fig.3.2(b), for ¯hτcv = 10meV (room temperature, 300k), lasing
93 occurs from the top of the spontaneous emission, leading to a very narrow spectra due to an almost single-mode lasing, which is maintained also at higher currents. This is clearly the desired spectrum for a Quantum Dot Laser.
The explanation for these spectra is found in the delta-function like or at least negligibly small homogeneous broadening of optical gain in case Fig.3.2(a), and in the case of room temperature Fig.3.2(b), with the homogeneous broadening comparable to the inhomogeneous one.
On conclusion, the polarization dephasing rate increases as temperature increases and the homogeneous broadening of optical gain connects spatially isolated and energeti- cally different quantum dots, leading to collective lasing of the dot ensemble.
94 3.2 Inhomogeneous broadening
Instead of what occurs with homogeneous linewidth broadening, inhomogeneous broadening, in a collection of nominally identical atoms may, for various reasons, have slightly different resonant frequencies ωa, such that the ωa values for different atoms are randomly distributed about some central value ωa0. So we have small but different shifting amounts for each atom in the collection. An applied signal passing through such structure will then see a total response due to all the atoms. If the random shifting of the individual center frequencies is sizable compared to the linewidth ∆ωa of each individual response, any measurement of the overall response from all the atoms in the collection will then give a broadened summation of the randomly shifted responses of all the individual atoms.
Finally, the overall response of the collection of atoms will be substantially broadened, and the response at line center will be substantially reduced in amplitude.
In a laser, atoms at different sites in a crystal may see slightly different local surroundings, or different local crystal structures, because of defect, dislocations, or lattice impurities. This produces slightly different values for the exact energy levels of the atoms, and thus slight shifts in transition frequencies. To the extent that the local lattice surroundings are similar for every atom but vibrate rapidly and randomly in time, they produce a dynamic homogeneous photon broadening. To the extent that the surroundings are different from site to site but static in time, they produce a static inhomogeneous lattice broadening or strain broadening.
Highly uniform dot arrays are essential for low-chirp high-speed optical devices since
95 the gain spectrum of such active layers exhibits near-singular DOS, which provide symmet- ric and sharp gain peaks. Unfortunately, as previously commented, present technologies do not provides the desired accuracy on dots shape. For example, the size of λ = 1 1.3µm − InGaAs/GaAs QD fluctuates by 14%, which broadens the emission spectrum and de- ∼ creases the modal gain of the active layer. The value of inhomogeneous line broadening for the ground state, due to size fluctuation, is 30meV (to a maximum of 70meV for other ∼ sizes) [31]. Due to a 15% size fluctuation (and some fluctuation in the dot shape), the ∼ photoluminescence (PL) from self-organized QD active layer is inhomogeneously broad- ened. Some works shows that PL linewidths can be reduced by postgrowth annealing and by overgrowth of a two-dimensional layer [32],[33]. And despite the disadvantages of size
fluctuation, some researches have taken advantage of the inhomogeneous broadening in the gain spectrum of the laser. For example, a wide range tunable laser with a tuning range exceeding 200 nm [35].
Giving an analytical analysis of inhomogeneous line broadening, we begin with the fact that due to the small size of the quantum dot, the energy of electrons and holes of the QD is limited to discrete states. The energy levels of pyramidal dots can be obtained by solving the Schr¨odinger wave equation [36]:
¯h2 1 Ψ(x, y, z) + V (x, y, z)Ψ(x, y, z) = E Ψ(x, y, z) (3.5) 2 m∗(x, y, z) i − ∇ ∇ where Ψ(x, y, z) is the wave function, Ei is the energy, V (x, y, z) is the confining potential, and m∗(x, y, z) is the carrier effective mass. We are going to use the solutions proposed by O. Qasaimeh, in his paper [39], who at the same time uses the method proposed by
Califano on his article [36]. In this solutions, Ψ(x, y, z) is developed as a set of cubical dots
96 with infinite barrier height, and good agreement between numerical data from this model and the peak positions of experimental PL spectra has been obtained. The energy levels of the electron, Ee and hole ground state Eh, are calculated using this method as a function of dot size. It can be obtained the following formulas that shows clear understanding of the effect of size nonuniformity on the optical characteristics of a QD laser:
L Ee = Ae + Be exp − (3.6) Le
L Eh = Ah + Bh exp − (3.7) Lh where Ai, Bi and Li are constants, and L is the pyramidal base length of the QD. The subindex i stands for either electron or holes. Ee and Eh are evaluated with respect to the GaAs conduction band and valence band, respectively. The validity of this model for electron and hole states is shown in the figure 3.3, where the calculated and fitted data are presented as a function of L.
Normally inhomogeneous line broadening is caused by size fluctuation, which is inherited from inaccuracy in controlling the growth and fabrication steps as we have pre- viously said [37], [38]. Since this process is random, we assume that a Gaussian distribution function of probability can describe the fluctuation in the dot size,
1 L L 2 f = exp − a (3.8) L 2 2 √2πσL − 2σL where La is the average value of L (statiscally La = L ) and σ is the standard deviation L (statiscally σ 2 = (L L )2 . Working with eq. (3.8) and statisticals mathematics we can L − a obtain many other statistical variables that can make understoodable the effects of dot
97 Figure 3.3: Energy levels of the electron and hole ground states as a function of dot size (GaAs-InAs)
[36] size fluctuation. And so, as L, the base length of a pyramidal quantum dot, is a random variable, we can express the probability density function of Ee caused by size fluctuation as ∂L fEe = fL (3.9) ∂Ee
A similar expression to (3.9) can be obtained for the hole energy level. Applying funda- mental definitions of statistical mathematics, we can have the average and the standard deviation of Ee and Eh, using previous equation (3.6) we have
∞ 2 La σL Ei = EifEi dEi = Ai + Bi exp − + 2 (3.10) −∞ L 2L i i ∞ 2L σ 2 σ 2 σ 2 = (E E )2f dE = B 2 exp − a + L exp L 1 (3.11) Ei i − i Ei i i L 2 2 − ∞ i Li Li
Where in these equations, Bi is a constant dependent on the average value La. It is clear from (3.10) and (3.11) that both E and σ increase as σ increases. For small size i Ei L
98 fluctuation, that means i.e, σ Li, the average electron/hole energy level is approximately L equal to E E (L = L ). i ≈ i a If we assume that the laser is lasing from the ground state, the optical transition energy is, therefore, equal to E = Eg + Ee + Eh, where Eg is the bandgap of the GaAs material. Thus, to know the average optical transition energy we must take the average of this equation, taking into account that Eg is a fixed well-defined energy, and so, the average is equal to
E = E + E + E (3.12) g e h where E and E are obtained from (3.10), and finally, the standard deviation of E, e h using the equation (3.11) to calculate respectively, the standard deviation of electron and holes, is given by
2 2 2 σE = σEc + σEh + σEc σEh (3.13)
It is known from statistical mathematics that the calculated deviation in eq. (3.13) is the parameter desired. It represents the inhomogeneous line broadening caused by size
fluctuation. It is clear from these equations that both E and σ are exponential functions E of σL.
Starting from this point, it is possible to make some deductions [39]. Suppose that the gain peak of inhomogeneously broadened QD laser is located at a point called E = Emax, and this point has the highest transition probability. If it is represented the probability density function of the optical transition energy fE. Although E = Emax has the maximum probability, it is clear that depending on the shape of the function f , E will be more E e or less equal to E = Emax. Since both Ee and Eh, as we can see in equations (3.6) and
99 (3.7), are monotonically decreasing functions of the random variable L, one can express
the probability density function fE as it has been done before with fEe on equation 3.9, and so: ∂L fEe = fL (3.14) ∂E
The calculated probability distribution is shown in figure 3.4 for a typical value of La =
130A˚ and different values of σL. As depicted from the figure, the shape of fE strongly depends on σL, as the latter increase, the probability function exhibits wider distribution and long tail at high energy. In addition, it should be noted that the peak of fE, which corresponds to the highest probability transition energy, previously defined as Emax, shifts to lower energy when increases. The shift is mainly associated with the group of the dots that have higher transition density and are less sensitive to size fluctuation. Since large dots are less sensitive to size fluctuation compared with small dots (since fE is proportional to a negative exponential in relation with L, as it has been shown on previous equations), they are responsible for the red shift in E . For very small size fluctuation (σ L ), the max L i shape of fE is reduced to a Gaussian distribution with the highest probability transition occurs at E (L = La).
In the inset of figure 3.4 it is shown the dependence of inhomogeneous line broadening on σL. As the deviation increase, the inhomogeneous line broadening increase too, and this occurs in a quasi-linear way, with more non-linearity as σL became larger
3.2.1 Gain and Differential Gain
Unfortunately, size fluctuation in QD arrays, with the current technologies, prevents QD devices from achieving a true zero-dimensional quantum effect and so obtaining its so de-
100 Figure 3.4: Probability distribution of the optical transition energy for different values of QD size fluc-
tuation. The inset shows the inhomogeneous line broadening as a function of σL [39] sired characteristics as no-chirping effects and high-speed modulation. The inhomogeneous broadening of typical QD laser due to size fluctuation is about 30 to 70 meV. In order to get true zero-dimensional quantum effects it is essential to reduce size fluctuation of
80%. To study the effect of σe on the gain and the differential gain, we need first to derive an expression for the QD electron and the hole densities. The Fermi-Dirac function for electrons and holes stands from semiconductor basis and is given by,
E µ −1 P = 1 + exp i − i (3.15) i KT where K is the Boltzmann constant, T is the absolute temperature and µi is the elec- tron/hole quasi-Fermi level. Given this equation, to calculate the concentration of electron and holes in QDs we just multiply the dot density to its respectively Fermi function, so
101 we have
n = N P (3.16) QD · e
p = N P (3.17) QD · h for the QD electron and hole densities. As it depends on Ei, Pi (i = e, h) is not a de- terministic variable, although a random one, so it has its probability density function fP , that can be obtained in the same way as done for equatio (3.9) by applying statistical mathematics. So we have f = f ∂E , and going on with the same procedure done Pi E ∂Pi
Figure 3.5: Probability distribution of the electron and hole Fermi function due to size fluctuation. The
inset shows the average value of the optical transition energy as a function of σL.
above to calculate E and all its derivative parameters, it can be calculated after P i i in the same way as done in Eq. (3.10). After that, and using a simple algebra, it can be
102 demonstrated [39] that Pi is
1 σ 2 P P [1 ( E ) (1 P )(2P 1)] (3.18) i ≈ i0 − 2 KT − i0 i0 − where P is the value of P when σ = 0 and σ is given by eq. (3.13). i0 i L E
Going back to the probability distribution fPi of the electron and the hole, results are shown in Fig. 3.5. It can be clearly observed how as holes have heavier effective mass than electrons, their distribution is found to be narrower than electrons one. And also, the width of both functions is observed to decrease as the applied bias increases where they can be considered as a delta functions, which occurs when Pi 1. →
Focusing finally on modal gain, we can use the expression obtained by Jiang and
Singh for better understanding of the effect of size fluctuation,
∞ Γ (E) = Γ H(E E)f (E)[P (E) + P (E) 1]dE (3.19) g g0 − E e h − −∞
where g0 is a constant, Γ is the optical confinement factor, and H(E E ) is the ho- − mogeneous broadening function. For small homogeneous broadening linewidth, the latter term can be replaced by a delta function, and the optical gain can be approximated by
Γ (E) Γ [P + P 1]f . Apparently, taking in mind the expressions for P and P g ≈ g0 e h − E e h previously calculated , the modal gain is inversely proportional to σ ( ∂Ee + ∂Eh ). In L × ∂L ∂L Fig.3.6 for different applied bias, the gain peak shifts to higher energy as the applied bias increases indicating filling higher energy states. The shift here depends on size fluctua- tion. In the inset of this figure, is the corresponding shift in the transition energy (Emax)
103 Figure 3.6: Modal gain as a function of energy for different applied bias. The inset shows Emax as a
function of average electron density for different value of σL.
for different values of σL. It is shown that for small size fluctuation, the value of (Emax) remains almost constant.
On the other hand, the maximum gain, calculated as a function of n , is on the
figure 3.7. It can be seen that the maximum gain saturates for large size fluctuation, but for low applied bias ( n < 2 1010cm−2), the maximum gain for larger size fluctuation · is higher than the one smaller size fluctuation. A reason for this occur can be found in the fact that dots with large size have lower transition energy and higher transition probabilities while dots that have small size have higher transition energy and lower transition probabilities. On resume, large dots reach transparency sooner than small dots and yield net gain at lower bias.
104 The differential gain is evaluated as a function of energy and for different values of injection density. Looking at the inset of Fig.3.7, we found that the peak of the differential gain occurs at 12meV above E for σ = 7A. This means that the optimum ∼ max L detuning of the laser should be 12 meV to achieve zero linewidth enhancement factor and large differential gain. As σL increase too the detuning to 28 meV as shown in the inset.
Figure 3.7: Maximum modal gain as a function of dot density for different values of σL. The inset shows the detuning between differential gain peak and modal gain peak as a function of dot density.
Other results are collected from the experiment of Dery [40] in the figure 3.8. In it, differential gain versus inhomogeneous broadening, in SAQDs with conduction band en- ergy spacing of twice the room temperature thermal energy is shown. The upper, middle,
105 and lower sets denote different populations for the central transition dots (E0) is consid- ered here the central transition frequency for the ground state level. Dashed and solid lines denote, respectively, results taking into account a constant value for the state filling variable or not, and as it can be appreciate, without sensitive effect on results.
The figure include also an axis for FWHM, to obtain the results, it has been changed from 5 to 100 meV, leading to a reduction of a factor of 4 in both the gain and the differential gain. This specific reduction stems from the fact that for a wide inhomogeneous broadening, the homogeneous width can be viewed as a delta function (as we already has seen in the previous section) and hence, the differential gain scales as σ−1. This behavior dominates the right side of the figure 3.8. For the opposite extreme case, where hypothetical identical dots are considered, the gain inhomogeneity is basically a delta function, and dominates the left-hand side of the figure. This two explanations give sense to the fact that the two edges differ by a factor of four. Same results were obtained for other energy spacing values.
It is clear that inhomogeneous line broadening causes early gain saturation and has strong effect on the differential gain and the optimum operating point of the laser.
3.2.2 Threshold Current
Going back to the expression of Eq.3.18 fairly matches experimental behavior. It is clear from it that large size fluctuation reduces Pi and consequently increases the threshold current of the laser. The threshold current of a QD laser depends on the radiative re- combination rate inside the dots and in the separate confinement layer. And the radiative
106 Figure 3.8: Differential gain versus inhomogeneous broadening in SAQDs with conduction band energy spacing of twice the room temperature thermal energy. Upper, middle, and lower sets denote different populations for the central transition dots [40] recombination is proportional to the product of the two concentration densities, this is
P P . Another effect is that the functions becomes wider and more dependence on σ e × h L as the point of work in which the applied current is close to the transparent one, leading to strong influence on the threshold conditions of the laser. This latter result was obtained by Qasaimeh on the calculation data for the Fig. 3.5.
The results of experiments done in [43] are also included in fig.3.9, where values taken into account for the elaboration of the figure are depicted (the calculation process is not explained because it does not take sense after the analytical study done in previous paragraphs). The figure is just enclosed to confirm the dependence of the threshold current on size fluctuation.
107 Figure 3.9: Threshold current density versus QD size fluctuations [43]
3.3 Cross-Gain Modulation
Cross-gain (XG) wavelength converters, which are an essential component in wavelength- division multiplexed systems and photonic networks, are very simple to realize and have shown impressive performance in previous techniques as QW an bulk SOA. With the benefits inherited to the incorporation of QD in the active layer moreover the less heating effects due to lower operating current, improvement in the amplifier wavelength conversion efficiency at low input power is achieved. So we have a non-linear effect on QD based amplification that is desired, and its basis of operation are explained.
To study this effect [44] it can be used a pump, named S0, and probe signals S1 injected into the QD SOA. Both consists of a continuous wave (CW) with small signal s0 and s1 respectively. The interaction between the probe and pump signal via cross-gain modulation causes modulation of the probe signal, being the total intensity the sum of signals (S = S0 +S1 +∆s, where ∆s = s0 +s1). Due to this modulation, it can be assumed
108 that, defining f as the electron occupancy probability of the Ground State (GS) and h the same for holes,
f = fo + ∆f (3.20)
h = ho + ∆h (3.21) where ∆f and ∆h are the perturbing complex signals. The XG conversion efficiency is defined as, s (L) S (L) η = 1 0 (3.22) s0(0) · S1(L)
In his calculations, O. Qsaimeh [44] obtains, under quasi-Fermi equilibrium condition, some interesting results. Part of this calculations are collected on Fig.3.10. On it, XG conversion efficiency η as function of different relaxation times can be seen. This relaxation times are:
τ21, which is the electron relaxation time from the excited estate to the GS, and in the inset, τeff , which is the effective lifetime of carrier depletion and depends inversely on
τ21 and directly on f0 and h0. In the figure, fast relaxation time is accompanied with fast escape time which reduces stimulated emission and consequently decreases XG modulation via decreasing τeff . In the inset, the effect of τ21 over τeff is taken into account. Being Ω0 the normalized frequency of operation, some conclusions can be extracted:
for Ω < Ω , τ has no effect on τ . This is because under this range of operation, • 0 21 eff the effective lifetime is equal to the spontaneous lifetime.
for Ω > Ω∞, τ depends strongly on τ21 (and so on τ12) • eff
Finally, it can be said that the XG efficiency shows strong dependence on the es- cape/relaxation lifetime.
109 Figure 3.10: Cross-gain conversion efficiency as a function of detuning for different relaxation lifetimes
(τ21 and τeff )
110 Chapter 4
Applications to Photonic Crystal
Structures
Over the past decade, photonic resonators with increased LDOS (Local Density Of States) have been exploited to enhance emission rate and to improve numerous quantum optical devices. Whereas more attention has been given to increase the emission rate, the reverse is also possible in an environment with a decreased LDOS. Taking this into account, and the fact that the emission rate is directly proportional to the LDOS, if we use a modified photonic crystal structure in which the QDs are embedded, we can demonstrate that it is possible to control the generation or the enhancement of the Spontaneous Emission (SE).
To achive this, the kind of crystal that we have to use is an structure with a modified single-defect cavity.
111 The characteristics of spontaneously emitted light depend strongly on the envi- ronment of the emitter; both the total rate of emission and the direction-dependent spectra are affected. The LDOS counts the available number of electromagnetic modes in which photons can be emitted at the specific location of the emitter, and can be interpreted as the density of vacuum fluctuations. Radiative decay rate is described by the Fermi’s golden rule (see Appendix), and leads the way to the LDOS calculation. The latter, and hence the emission rate largely depend on the emission frequency ω and on the position of the emitter, but not on the emission direction. On the other hand, an introduction to photonic crystals has already been done in chapter 1. In photonic crystals, emission of light is modified because the refractive index varies spatially in regions of length of the order of the wavelength. Hence, light interference by Bragg diffraction causes characteristic stopbands that do lead to direction-dependent spectra; these, at end modifie the LDOS relative to free space and hence the spontaneous emission rate of embedded QDs emitters. The frequency of emission rate is modified if Bragg stop bands exclude a substantial solid angle of the crystal [47].
In order to confirm the already mentioned influence of Bragg diffraction, angle- resolved measurements [47] of QD emitters embedded in a titania inverse opal PC (which is a face-centred cubic structure of air spheres in a titania backbone material) have been performed. Results are collected in Fig.4.1. In this figure, the angle axis is limited to
α = 60◦ as no stopbands appear there, and the angular dependence of the relative intensity for three different lattice parameters are shown. From an inspection of the figure, it is evident that for a sample with a = 370nm (which is a small lattice parameter), the intensity
112 decays monotonically with the increasing angle. On the other hand, for a = 420nm and a = 500nm broad stopbands appear, centred at α = 0◦ and α = 25◦, respectively, as result of the photonic effects with a strongly reduced emission. The continuous color curves are the predictions with no adjustable parameters, of the theory that describes the propagation of light that is diffused by ubiquitous disorder present in real PC. Excellent agreement between this theory and the experiment is found, confirming that the emission of the embedded quantum dots is strongly affected by the photonic crystal. Moreover, it can be said that for small lattice parameter, no photonic effects are present. Theoretically,
Figure 4.1: Measured (points) and calculated (curves) angle-dependent intensities at fixed frequency for three samples with different lattice parameters. The intensities are divided by the intensity measured at
α = 60◦, where no stopbands appear.[47]α is the angle relative to the (111) lattice plane.
in photonic crystals, the LDOS and (as a result) the decay rate ideally vanish over a frequency range known as the photonic bandgap. But in real PC (finite dimensions ones), the LDOS is never zero, but strongly modulated pseudogaps are anticipated. In that gaps, vacuum fluctuations are expelled from the crystal, and the spontaneous decay of an
113 excited emitter is inhibited. Away from a gap, the LDOS is increased and spontaneous emission is enhanced. This theoretical discussion relies on the results shown in Fig. 4.2.
In (a), luminescence decay curves of QD embedded in the same three different PC used in Fig. 4.1 are shown, and in (b), decay rates of the excited QDs for different sizes and its consequent different emission frequencies.
In (a) two different decay rates are remarked: fast ( 1ns) and slow (> 10ns), due • ∼ to titania and QDs respectively. The latter are not simple exponentials shapes but
to realize this additional measurement over longer decay time is needed (and these
results are not enclosed in this picture). This is explained by fluctuations in the en-
vironment surrounding the QDs, which induce a broad distribution of non-radiative
decay channels, and effectively reducing the quantum efficiency. But here the de-
cay curve is investigated just up to 20ns, which is least sensitive to background ∼ counts and so it can be said that it shows a single exponential behavior. The ob-
served lifetimes in the inverse opals are all systematically shorter than the lifetime of
QDs inside of them, which is attributed to the reduced quantum efficiency, and this
causes limitation in the amount of enhancement and inhibition observed. Coming
back to the figure, it shows a strong variation in the lifetime by a factor of two for
photonic crystal with different lattice parameters.
In (b) Here, the frequency dependence of the total decay rate has been carried out. • It is well known that the total decay rate is the sum of both radiative and non-
radiative ones. For each sample, a strong variation of the total decay rate with the
emission frequency of more than 50% is observed. This indicates that the radiative
114 decay strongly depends on the emission frequency (and so on the size of the dot).
Focusing on each sample, for the photonic crystal with a = 500nm, a pronounced
inhibition of emission of 30% is observed in a wide bandwidth exceeding 10%. On
the sample with a = 420nm, a strongly enhanced emission is measured, extending
the decay rate by more than 40% relative to the reference value provided by the
sample with non photonic effect (a = 370nm). The decay rates for the a = 580nm)
sample are all enhanced over a broad frequency range.
All these measurements demonstrate that photonic crystals offer an effective way of controlling the radiative lifetime through the lattice parameter a.
A strong Purcell effect (see previous chapter section) is important in a single-photon source for improving the photon outcoupling efficiency (ηext) and the single-photon gener- ation rate. In the end, we want the deterministic generation of single photons from isolated quantum emitters, and to have a good external efficiency, we have to locate QDs within optical cavities and use the Purcell effect to funnel single photons into a single optical mode for collection.
Cavities realized using photonic crystals provide maximum flexibility to tune the
LDOS over a much wider bandwidth and achieve full control the spontaneous emission via the strength of the local vacuum field fluctuations. Furthermore, strongly localized modes in photonic crystals combine a planar geometry with high quality factors (Q = ωτphoton) and small effective mode volume, a significant advantage for achieving strong Purcell enhancement and realizing efficient QD based single photon emitters [46].
115 Figure 4.2: (a) Luminescence decay curves of QDs inside three different PC (variable parameter a).
Curves are overlapped after 5ns. Contribution of Titania is negligible from then on. (b) Decay rates of the excited states of QDs in the same different PC in (a) and for different frequencies due to QDs diameter size (Large, Medium and Small). Curves enclosed are just to guide the eye. Dashed line: decay rate for a homogeneous medium. [47]
To give a scientific basis to these properties of QDs embedded in PC described above, the experiment realized by Englund, Arakawa, et al. [45] is taken as main reference as they achieved the first demonstration of single photon emission from photonic crystal cavity-coupled QDs.
The total spontaneous emission (SE) of a QD situated on a point described by the position vector CrA, and spectrally detuned from the cavity wavelength of resonance by
λ λ , can be expressed as the sum of rates into the cavity as, − cav
τ = τcav + τPC (4.1)
116 When the strong-coupling regime is still not achieved, that is, in the so called ”weak- coupling” regime, the cavity decay rate exceeds the QD-cavity coupling strength, and so, the SE can be calculated by the simple application of Fermi’s Golden Rule (see appendix
1). Using expressions obtained for the elements found in the equation 4.1, as τcav follows a
Lorentzian and τPC is related to the LDOS, which in the PC band gap is reduced relative to bulk semiconductor, we can have the following expression, (no longer valid as the system approaches the strong coupling regime)
2 τ EC (Cr ) Cµ 1 = F A · + F (4.2) cav C 2 PC τ0 Emax Cµ 1 + 4Q2 λ 1 | || | λcav − Where τ0 is the bulk semiconductor emission rate, Cµ is the vector position of the emitter dipole, EC is the field in the cavity and Fcav is the cavity Purcell factor (see previous 2 E-(-rA)·-µ 1 chapter). The factors and 2 describe the spatial and spectral |E-max||-µ| 1+4Q2 λ −1 λcav mismatch of the emitter dipole Cµ to EC . Due to the not perfect matching of the spatial and angular position between Cµ and EC , the enhancement generally does not reach the maximum value F + F . F τ is the base SE rate in the photonic crystal without cav PC PC ≡ τ0 cavity. And Fcav is given by 3 λ3 Q Fcav 2 3 (4.3) ≡ 4π n Vmode where Vmode is the cavity mode volume.
Using the previous structure obtained by Vuckovic [48], and making some modifi- cations, the single-defect cavity in the 2D PC designed for SE enhancement is shown in
figure 4.3. The structure was patterned on a 160 nm-thick GaAs membrane by a combina- tion of electron beam lithography and etching. For the design of the high Q, a simulation program (as RSoft) working in FDTD (Finite Difference Time Domain) mode is used.
117 The scope is to maximize the first term of the first matrix of the equation 4.2, that is,
EC (Cr ) Cµ, to have maximum QD to field interaction. The cavity resonance frequency was A · chosen to fall near the middle of the QD distribution. Using an appropriate measurement
Figure 4.3: FDTD-assisted design of the photonic crystal cavity. The periodicity is a = 0.27λcav, hole radius r = 0.3a, and thickness d = 0.65a. (a) Electric field intensity of X-dipole resonance. (b) SEM image of fabricated structure. [45]
system based on photoluminesce (PL) spectroscopy, detailed on figure 4.4. The fraction of photoluminescence that originates from the cavity is enhanced over the background in two ways: through enhanced emission rate, and an increased coupling efficiency between cavity mode and collection optics. The latter efficiency was estimated by FDTD analysis of the cavity mode radiation pattern and equals 0.07 for an objective lens with NA = 0.6. A ∼ QD spectrum shows discrete lines at low pump power. This enhancement allows us to map out the cavity by pumping the QDs at high intensity, resulting in a broad inhomogeneous emission spectrum that mimics a white light source. The cavity Q then follows directly from a fit to the Lorentzian in equation 4.2. PL measurements made with the detailed system of fig. 4.4 shows the disadvantage that spectral coupling is unlikely with higher
118 Figure 4.4: Apparatus for photoluminescence and autocorrelation measurements. The incident Ti:saph laser beam (continuous or 160 fs-pulsed) pumps QDs at 5 K [45]
Q (about 104), so focusing on easier coupling to low-Q cavities (in the range 102 103) − for the same cavity design of Fig.4.3, better results are obtained. PL measurements are made for different Q, to structures numbered 2, 3 and 4. The results obtained are shown on Fig.4.5 and commented here,
Fig.4.5(a).At high pump intensity, the polarized dipole cavity with Q=5000. As • previously commented, coupling is difficult and this is observed on no single exciton
lines matching the cavity frequency. In the inset the Bulk QD spectrum is shown,
and different peaks of interest marked. Peak 1: GaAs bandgap transitions; 2: GaAs
free and impurity-bound exciton emission; 3: Broadband QD emission
Fig.4.5(b,c). As the Q factor is diminished to 250 and 200 respectively for structures • 2 and 3, the insets (at high pump-power spectra) reveal these resonances. At low
power spectra, coupled single exciton lines match the cavity polarization, validating
previous arguments.
119 Fig.4.5(d). For this structure, a single exciton line coupled to x-dipole cavity mode • with Q = 1600 is obtained. To increase spectral coupling, a temperature-tuning is
made ranging from 6 to 40K, as result it allows improved spectral alignment.
Figure 4.5: PL measurements. (a) Structure 1 at high pump intensity (1.5 kW/cm2), polarized x-dipole cavity, Q=5000. No coupling. Inset: Bulk QD spectrum. (b,c): The high pump-power spectra (insets) reveal resonances with Q=250 and Q=200, respectively. The low-power spectra show coupled single exciton matching the cavity polarization. (d) Structure 4, single exciton line A coupled to dipole cavity mode with very low Q (inset). Temperature-tuning from 6 to 40K allows improved spectral alignment (shaded line).
In all measurements, the Ti-Saph laser was pulsed at 160 fs with λ = 750nm [45]
Off-resonant dots see up to five-fold lifetime enhancement. To explain these increased lifetimes, we again turn to Eq. 4.2. The first term for emission into the cavity mode vanishes as the QD exciton line is far detuned from λcav. The second term FPC is reduced below unity due to the diminished LDOS in the PC for emission inside the PC bandgap, leading
120 to longer lifetime. This SE rate modification is illustrated in Fig. 4.6.
Due to spatial misalignment between the QD and cavity, most of the resonant dots presented here have moderate emission rate enhancement. This misalignment is estimated at about one lattice spacing based on the cavity mode pattern Fig.4.3. Considering the low probability of QD-cavity coupling, these modest rate enhancements are not surprising.
With better spatial and spectral matching, Eq.4.2 predicts that a rate enhancement of up to 400 is possible. But it must be taken into account that this equation 4.2 is no longer ∼ valid as the system approaches the strong coupling regime, this is, the cavity decay rate is more or less equal to the QD-cavity coupling strength.
Figure 4.6: Illustration of the predicted SE rate modification in PC as function of normalized spatial and spectral misalignment from the cavity (a is the lattice periodicity). This plot assumes Q = 1000 and polarization matching between the emitter dipole and cavity field. The actual SE rate modification varies significantly with exact QD location.[45]
Now, we are going to explain briefly the results obtained by another research group, this of Kress [46], where a new method is proposed in order to tune the local density of
121 optical modes of QDs and provide the control of the spontaneous emission dynamics of
SAQDs. In this experiment, a single layer of nominally In0.5Ga0.5As QDs were incorpo- rated into the center of a GaAs waveguide core as an internal light source. A 2D photonic crystal is defined by fabricating a triangular lattice of air holes in the GaAs waveguide.
The thickness of the Air-GaAs-Air slab waveguide consists is d = 400nm. Furthermore, ultra low mode volume microcavities were formed by introducing single missing hole point defects to realize H1 resonators (resonance at the fundamental frequency). The periodicity of the PhC was of a = 300nm and the air hole radius (r) was varied to tune the cavity mode energies through the inhomogeneously broadened spectrum of QD ground states and control the width of the photonic bandgap. Only a few dots are taking into account in realistic approximation of the number of dots both spectrally and spatially coupled to the cavity modes.
With an appropriate system for optical measurement, some interesting results were obtained. For the dots totally coupled to high Q, it is observe a very large Purcell enhancement of the emission rate ( 6x). An analysis of the spectral dependence of the ≥ decay rate as a function of emitter-cavity detuning shows that the maximum enhancement may be as large as 20, close to the limit required for the light-matter interaction to say ∼ to be in the strong coupling regime. What is more, for the dots out of resonance with the cavity mode, a strong suppression ( 2x) of emission decay time over a very wide ≥ bandwidth (∆λ 50nm). The reason for this effect to appear arise from the presence ∼ of a TE-polarized 2D PH bandgap that efficiently suppresses spontaneous emission into leaky modes of the system. Another result obtained is that as the ratio r/a is reduced, the cavity modes couple more weakly to the dielectric band continuum resulting in the
122 observed enhancement of the cavity mode Q-factor. In addition, surface roughness effects become less significant for smaller r/a, contributing further to the observed enhancement.
Another result are enclosed in this thesis just to have a clear idea of the good results obtained with QDs embedded on PC, in this case with a square lattice microcavity. The results are on the figure 4.7.
Figure 4.7: PL from a typical single-hole-defect, square-lattice microcavity of lattice constant a=300 nm and r/a = 0.37, showing a high Q mode at 926 nm. The InAs QD emission is between 870 and 970 nm.
High-resolution PL is displayed in the insets for devices with a=300 nm and r/a=0.38, fit to Lorentzians of Q=4000 (a) and 3600 (b).[49]
In conclusion, it has been shown that by designing a suitable photonic crystal en- vironment, it can be significantly modified SE rate of embedded quantum dots. For QDs
123 coupled to a PC defect cavity, it is observed up to 8 times faster SE rate and further- more, demonstrated antibunching. This coupled system promises to increase out-coupling efficiency and photon-indistinguishability of single photon sources. On the other hand, off-resonant QDs show up to five-fold decrease of emission rate compared to QDs in bulk semiconductor as their LDOS is diminished in the photonic crystal, and this through the lattice parameter a. This extended lifetime may find applications in QD-based photonic devices (e.g., switching), including quantum information processing (e.g.quantum mem- ory). It also shows that reported QD lifetime reduction due to surface proximity effects should not limit the performance of PC-QD single photon sources. Good agreement be- tween these results and FDTD simulations of SE in the photonic crystal are found, so
FDTD simulations are an ideal tool to study these structures behavior.
124 Conclusions
In many cases, the advantages of QDs are still not even well understood. There are many papers and works on the field of QD laser, but a whole complete theory which give response to all the questions set up by this new technology, has not to be reached yet. In consequence, in this thesis it has been taken into account different studies and points of view for the same features, aiming to give a better understanding of the theory and its difficulties.
Considerable progress has been achieved in recent years in the development of con- cepts and technologies relevant to QD lasers. On the field of fabrication, the new class of growth methods utilizing self-ordering permits the investigation of lasers incorporating dots whose sizes and densities are beyond the reach of current lithography techniques, even though the lithography-based techniques have much improved, and it is still a field to develop.
However, the performance of QD diode lasers has not reached yet what has been predicted by early theoretical models of these devices, that emphasized the role of the narrower DOS distribution in improving the static and dynamic laser properties.
125 Affected by the lower dimensionality, there are many other effects non-predicted by the early models of QD lasers, and so, nowadays, new and more realistic models are continuously been established. But what is clear, is that the realization of QD lasers with features approaching those of an ideal system requires significantimprovement in the uni- formity of quantum dots, to avoid mainly the undesired strong inhomogeneous broadening.
Considering the temperature problem, many considerations has been done in this thesis, but what is clear is that the predicted quite low threshold current density at room temperature, is reachable if nonradiative recombination is eliminated from the QDs.
Another thing to take into account is the necessity of a reduction in the optical cavity losses due to the extremely small effective volumes of the active region, that provides the ultimate control of the carrier and photon features via quantum carrier and photon confinement.
But not everything are problems. Lasing characteristics already improved (although theoretical predictions are far to reach in many cases) are: optical gain larger almost one order of magnitude, large optical saturation gain, less dependence of threshold current at low temperature and reduced chirp. The development of VCSELs with QDs layers inside, is a quite mature technique. And first devices lasing at wavelengths of telecommunications interest have been developed, with high reliability and low cost.
The idea of embed single QD light emitters in a three dimensional photonic bandgap structure offer the possibility of independently engineering the carrier and photon states
126 via the design of the semiconductor bandgap and refractive index distributions. What is more, it has been shown that by designing a suitable photonic crystal environment, it can be significantly modified SE rate of embedded quantum dots. For QDs coupled to a PC defect cavity, it is observed up to 8 times faster SE rate and furthermore, demon- strated antibunching. This coupled system promises to increase out-coupling efficiency and photon-indistinguishability of single photon sources. On the other hand, off-resonant QDs show up to five-fold decrease of emission rate compared to QDs in bulk semiconductor as their LDOS is diminished in the photonic crystal. This extended lifetime may find ap- plications in QD-based photonic devices (e.g., switching), including quantum information processing (e.g.quantum memory). It can also be concluded that reported QD lifetime reduction due to surface proximity effects should not limit the performance of PC-QD single photon sources. Good agreement between these results and FDTD simulations of
SE in the photonic crystal are found, so FDTD simulations are an ideal tool to study these structures behavior.
127 Appendix
Some abreviatures used on this thesis:
AFM= Atomic Force Microscopy •
TEM= Transmission Electron Microscopy •
FDTD= Finite Difference Time Domain •
4.1 Appendix 1: Fermi’s Golden Rule
One of the prominent failures of the Bohr model for atomic spectra was that it couldn’t predict that one spectral line would be brighter than another. From the quantum theory came an explanation in terms of wavefunctions, and for situations where the transition probability is constant in time, it is usually expressed in a relationship called Fermi’s golden rule.
In general conceptual terms, a transition rate depends upon the strength of the coupling between the initial and final state of a system and upon the number of ways the transition can happen (i.e., the density of the final states). In many physical situations the transition probability is of the form showm below, although there are many ways of
128 presenting this equation, 2π λ = M ρ , (4.4) if ¯h | if | f
The transition probability λ is also called the decay probability and is related to
1 the mean lifetime τ of the state by λ = τ . The other terms are: ρf as the density of final states, and M is the Matrix element for the interaction. The general form of Fermi’s | if | golden rule can apply to atomic transitions, nuclear decay, scattering ... a large variety of physical transitions.
A transition will proceed more rapidly if the coupling between the initial and final states is stronger. This coupling term is traditionally called the ”matrix element” for the transition: this term comes from an alternative formulation of quantum mechanics in terms of matrices rather than the differential equations of the Schrodinger approach. The matrix element can be placed in the form of an integral where the interaction which causes the transition is expressed as a potential V which operates on the initial state wavefunctionψi.
The transition probability is proportional to the square of the integral of this interaction over all of the space appropriate to the problem.
∗ Mif = ψf V ψidv (4.5)
This kind of integral approach using the wavefunctions is of the same general form as that used to find the ”expectation value” or expected average value of any physical variable in quantum mechanics. But in the case of an expectation value for a property like the system energy, the integral has the wavefunction representing the eigenstate of the system in both places in the integral.
The transition probability is also proportional to the density of final states ρf . It
129 is reasonably common for the final state to be composed of several states with the same energy - such states are said to be ”degenerate” states. This degeneracy is sometimes ex- pressed as a ”statistical weight” which will appear as a factor in the transition probability.
In many cases there will be a continuum of final states, so that this density of final states is expressed as a function of energy [50].
130 4.2 Appendix 2: Smith-Purcell Effect
Recently it have been found new ways to manipulate the light-matter interaction by modi- fying the photonic components, for instance by enhancing the optical field with feedback in cavities. This second ‘knob’on the light-matter interaction manipulates the optical density of states. Besides simply enhancing the interaction by locally amplifying the electromag- netic fields, the modified optical density of states produced by photonic structuring allows emission and absorption rates to be enhanced or suppressed, now known as the Purcell effect.
Many more information can be found in [51]
131 4.3 Appendix 3: Measurements Tools
4.3.1 AFM
Atomic Force Microscope measures the force between the sample surface and a fine tip with a typical radius of less than 10 nm. The force is measured either by the bending of a cantilever on which the tip is mounted scontact moded, or by measuring the change in resonance frequency due to the force stapping moded. With a typical resolution of several nm laterally and several vertically, AFM is ideally suited to characterize the shape of self-assembled islands. For large scan sizes up to 100x100 µm2, the lateral arrangement and correlation of island positions can also be obtained. With AFM any surface can be investigated.
4.3.2 TEM
Although the preparation of thin specimens is more elaborate than for other techniques, which makes TEM essentially an ex situ characterization technique, it is widely used due to its very high spatial resolution and sensitivity composition. Transmission electron microscopy can be performed either on thin slices parallel to the sample surface splan-view
TEM or on crosssectional slices. Hence buried islands can be well examined by TEM, with some restrictions due to specimen preparation: in many cases, the lateral island diameter is comparable to the slice thickness. The image contrast depends on different quantities, material scompositiond but also strain, as TEM images are obtained from diffraction patterns of high-energy electrons. Therefore image analysis is often not straightforward but requires elaborate image analysis techniques and/or model calculations. Compared to
132 other techniques, usually very small areas are investigated, so that no statistically averaged values can be obtained [52]
133 Acknowledgments
This thesis have been possible thanks to the Erasmus scholarship, but without the support and help of many people, the scholarship would had not been enough. At Rome, these people I want to say thanks to, are: prof. Gabriella Cincotti, for believing in me for this thesis, for her constant help and the flexibility she gave me, and Michela, for her partnership, advices and conversation. I do not forget prof. Schettini, because with his
first support and facilities I could did my exams at Rome.
I can’t forget my parents, without their help nothing would have go. And I want to end remembering all the friends I made during the career, they now who they are, and without their friendship my career would not have been the same.
134 List of Figures
1 Evolution of quantum confinement applied to semiconductor devices [4]...... 2
1.1 Light propagation and its intensity in different phases of round-trip cycle inside the Fabry-
Perot cavity [9]...... 35
1.2 Quantum Well layers scheme ...... 36
1.3 Active region of a diode lasers representing layer of (a) bulk semiconductor, (b) several
QW, (c) array of quantum wires (d) QDs. Each one with its corresponding DOS [10]. . . 40
2.1 Schematic diagrams of QD Structures for laser applications fabricated by (a) etching and
(b) Stranski-Krastanow self-organized growth. [7] ...... 49
2.2 Principal process steps of lithography based dot laser production using dry etching meth-
ods for dot array patterning [16] ...... 51
2.3 Schematic diagrams of the growth modes for semiconductor systems: (a) Frank-van der
Merwe (FM, layer by layer); (b) Stranski-Krastanow (SK, islands with wetting layer); (c)
Volmer-Weber (VW, islands). [9] ...... 54
2.4 Plan view and cross-section. TEM images of InAs QDs ...... 56
2.5 Large InAs QDs formed by activated alloy phase separation ...... 58
2.6 Defect reduction technique ...... 59
135 17 3 2.7 Electron DOS function for ensemble with typical quantum dot density 1x10 cm− with
5000 random dot sizes for three dot size distributions [41] ...... 63
2.8 (a) Schematic illustration of the broad-area stripe geometry QD laser diode. (b) Schematic
showing the Al, Ga, and In content in the laser waveguide region [20]...... 65
2.9 Energy band diagram of the QD SOA and the main processes inside the active region . . 66
2.10 (a)Optical gain as a function of normalized output photon density, (b) Optical gain as a
function of normalized current for different values of input photon density [27]...... 69
2.11 Threshold current density versus (a) the normalized surface density of QDs, (b) cavity
length [13]...... 73
2.12 Threshold current density and its components versus the temperature. The inset shows
jQD and jOCL on an enlarged (along the vertical axis) scale. The broken line depicts jQD
calculated assuming the charge neutrality in QDs [13]...... 74
2.13 Threshold current density versus temperature, along with spectral insets for temperatures
of 79 K, 240 K, and 280 K. An abrupt decrease in the multimode spectral linewidth is
observed above 230 K and indicates coupling of the QDs with the wetting layer [20]. . . 75
2.14 Differential gain reduction in SAQD due to state filling versus the electronic ground-state
occupation probability. Each of solid line denote a different energy spacing between the
available states, characterized in units of room temperature [40]...... 78
2.15 Differential gain as a function of applied current for different values of input photon
density (excitation intensity) [27]...... 79
2.16 Carrier and photon density time evolution in the lasing cavity during a Q-switched pulse.
Also the peak gain in the structure and the absorption in the cavity [41] ...... 81
136 2.17 VCSEL structure with self-assembled InAs dots and GaAs/AlO Bragg mirrors (left). Light
output and efficiency versus driving current for this device (right). The laser spectrum is
depicted in the inset of the right figure. [42] ...... 86
2.18 State of the art of VCSELs fabrication by different researcher teams all over the world
(year 2000) ...... 87
3.1 Model for the lasing spectra, showing the relationship between lasing spectra and homo-
geneous broadening of single-dot optical gain. (a) Homogeneous broadening tending to a
delta function. (b) Homogeneous broadening comparable to inhomogeneous broadening [34] 92
3.2 Calculated emission spectra up to well above the lasing thresholds for (a)¯hτcv =
1meV ,and (b)¯hτcv = 10meV [34] ...... 93
3.3 Energy levels of the electron and hole ground states as a function of dot size (GaAs-InAs)
[36] ...... 98
3.4 Probability distribution of the optical transition energy for different values of QD size
fluctuation. The inset shows the inhomogeneous line broadening as a function of σL [39] . 101
3.5 Probability distribution of the electron and hole Fermi function due to size fluctuation.
The inset shows the average value of the optical transition energy as a function of σL. . 102
3.6 Modal gain as a function of energy for different applied bias. The inset shows Emax as a
function of average electron density for different value of σL...... 104
3.7 Maximum modal gain as a function of dot density for different values of σL. The inset
shows the detuning between differential gain peak and modal gain peak as a function of
dot density...... 105
137 3.8 Differential gain versus inhomogeneous broadening in SAQDs with conduction band en-
ergy spacing of twice the room temperature thermal energy. Upper, middle, and lower
sets denote different populations for the central transition dots [40] ...... 107
3.9 Threshold current density versus QD size fluctuations [43] ...... 108
3.10 Cross-gain conversion efficiency as a function of detuning for different relaxation lifetimes
(τ21 and τeff ) ...... 110
4.1 Measured (points) and calculated (curves) angle-dependent intensities at fixed frequency
for three samples with different lattice parameters. The intensities are divided by the
intensity measured at α = 60◦, where no stopbands appear.[47]α is the angle relative to
the (111) lattice plane...... 113
4.2 (a) Luminescence decay curves of QDs inside three different PC (variable parameter a).
Curves are overlapped after 5ns. Contribution of Titania is negligible from then on. (b)
Decay rates of the excited states of QDs in the same different PC in (a) and for different
frequencies due to QDs diameter size (Large, Medium and Small). Curves enclosed are
just to guide the eye. Dashed line: decay rate for a homogeneous medium. [47] . . . . . 116
4.3 FDTD-assisted design of the photonic crystal cavity. The periodicity is a = 0.27λcav,
hole radius r = 0.3a, and thickness d = 0.65a. (a) Electric field intensity of X-dipole
resonance. (b) SEM image of fabricated structure. [45] ...... 118
4.4 Apparatus for photoluminescence and autocorrelation measurements. The incident
Ti:saph laser beam (continuous or 160 fs-pulsed) pumps QDs at 5 K [45] ...... 119
138 4.5 PL measurements. (a) Structure 1 at high pump intensity (1.5 kW/cm2), polarized x-
dipole cavity, Q=5000. No coupling. Inset: Bulk QD spectrum. (b,c): The high pump-
power spectra (insets) reveal resonances with Q=250 and Q=200, respectively. The low-
power spectra show coupled single exciton matching the cavity polarization. (d) Struc-
ture 4, single exciton line A coupled to dipole cavity mode with very low Q (inset).
Temperature-tuning from 6 to 40K allows improved spectral alignment (shaded line). In
all measurements, the Ti-Saph laser was pulsed at 160 fs with λ = 750nm [45] . . . . . 120
4.6 Illustration of the predicted SE rate modification in PC as function of normalized spatial
and spectral misalignment from the cavity (a is the lattice periodicity). This plot assumes
Q = 1000 and polarization matching between the emitter dipole and cavity field. The
actual SE rate modification varies significantly with exact QD location.[45] ...... 121
4.7 PL from a typical single-hole-defect, square-lattice microcavity of lattice constant a=300
nm and r/a = 0.37, showing a high Q mode at 926 nm. The InAs QD emission is between
870 and 970 nm. High-resolution PL is displayed in the insets for devices with a=300 nm
and r/a=0.38, fit to Lorentzians of Q=4000 (a) and 3600 (b).[49] ...... 123
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