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AUGER RECOMBINATION IN QUANTUM WELL LASER WITH PARTICIPATION OF ELECTRONS IN WAVEGUIDE REGION
A.A. Karpova1,2, D.M. Samosvat2, A.G. Zegrya2, G.G. Zegrya1,2 and V.E. Bugrov1
1Saint Petersburg National Research University of Information Technologies, Mechanics and Optics, Kronverksky Pr. 49, St. Petersburg, 197101 Russia 2Ioffe Institute, Politekhnicheskaya 26, St. Petersburg, 194021 Russia Received: May 07, 2018
Abstract. A new mechanism of nonradiative recombination of nonequilibrium carriers in semiconductor quantum wells is suggested and discussed. For a studied Auger recombination process the energy of localized electron-hole pair is transferred to barrier carriers due to Coulomb interaction. The analysis of the rate and the coefficient of this process is carried out. It is shown, that there exists two processes of thresholdless and quasithreshold types, and thresholdless one is dominant. The coefficient of studied process is a non-monotonous function of quantum well width having maximum in region of narrow quantum wells. Comparison of this process with CHCC process shows that these two processes of nonradiative recombination are competing in narrow quantum wells, but prevail at different quantum well widths.
1. INTRODUCTION nonequilibrium carriers is still located in the waveguide region. Nowadays an actual research field of semiconduc- In present work a new loss channel in InGaAsP/ tor optoelectronics is InGaAsP/InP multiple quan- InP MQW lasers is under consideration. It affects tum well (MQW) lasers, because their lasing wave- significantly the threshold characteristics of laser length is 1.3 – 1.55 micrometers and coincides with and leads to generation failure at high excitation transparency windows of optical fiber [1-5]. It is levels and temperatures. For such Auger recombi- known [2], that lasers based on InGaAsP/InP have nation the recombination energy of localized elec- some shortcomings: the threshold current strongly tron-hole pair is transferred to barrier carriers due to depends on temperature above the temperature of Coulomb interaction (Fig. 1). active region T > 60 °C. Also at high pump levels It is known, that the lifetime of electron localized generation fails. It is assumed, that this effect is in QW depends on three processes such as caused by carrier and lattice heating [2]. radiative recombination, nonradiative Auger recom- For laser structure with InGaAsP/InP quantum bination of confined electrons and holes [7,8] and wells considered in this paper the depth of QWs for nonradiative recombination of confined carriers in- electrons and holes are 0.08 eV and 0.464 eV, re- teracting with electrons in the waveguide region. spectively. At high excitation levels and high tem- Thus, the last process should be considered as well peratures electrons are partly ejected from the QW if the threshold characteristics of lasers are into the waveguide region [1]. Also as shown in [6], analyzed. We will use four-band Kane’s model [9], that even for structures with deeper QWs part of Corresponding author: A.G. Zegrya, e-mail: [email protected]
© 2018 Advanced Study Center Co. Ltd. 194 A.A. Karpova, D.M. Samosvat, A.G. Zegrya, G.G. Zegrya and V.E. Bugrov based on 8x8 Hamiltonian [10], which describes in the most accurate way the wave functions and en- ergy spectra of carriers in narrow-gap AIIIBV semi- conductors. The aim of the present work is to perform a de- tailed analysis of rate and coefficient for the studied process of Auger recombination, the dependence of its coefficient on QW width and a comparison of this process with CHCC for the model structure [2].
2. BASIC EQUATIONS Because of a small effective mass electron are just partly localized in a narrow QW, while holes are almost completely localized therein. As a result a carrier redistribution, which is equal to charge re- distribution, takes place over the active region con- Fig. 1. Two schemes of electron transitions for sisting of QWs and waveguide region. To approxi- CHCC-process and the studied AR in QW mate the electron density in the barrier region a self- consistent Poisson’s equation is used. An additional n = n/T, = -e/T, is electron potential, n and p equation that relates the quasi-Fermi levels for elec- are the quasi-Fermi levels for electrons and holes, trons and holes is: n and p are quantized energy levels of an electron and a hole in the conduction and valence bands p 2D n 2D n 2D , (1) QW QW b respectively, T is the temperature of electron–hole 2D 2D 2D plasma in terms of energy. The quasi-Fermi levels where nQW , pQW , and nb are 2D carrier concentra- tions of electrons and holes in QW and barrier re- for electrons and holes are measured from the band gion. Final expressions for 2D carrier concentrations edges as well as quantized energy levels of local- [2] are: ized carriers.
1/ 4 4 T 2. WAVE FUNCTIONS AND CARRIER n 2D N2D (T ) b c 3 E STATES IN QW AND BARRIER B REGION 3/2 d Let us consider a single QW with its width a, lo- 1 exp( u ) 0 c n (2) cated so that x-axis is its axis of symmetry (Fig. 2). 1/ 2 3/2 To find the rate and the coefficient of studied Auger d , process the wave functions of carriers are needed. 1 exp( u ) III V 0 c 0 n For most A B semiconductors the wave functions of the conduction band are described 6 represen- tation, and the wave functions of the valence band 2D 2D n n n N ln exp 1 , QW c (3) are described by and representations. The T 7 8 equations for the corresponding wave functions can be written in differential form. The following repre- 2D mhh 2D p p sentation of the basis functions is chosen, which p N ln exp 1 , QW c (4) seems to be more suitable for the present work [7]: mc T where mc, mc, and mhh, are effective electron masses s , s , x , x , y , y , in barrier region and in QW and effective heavy hole (5) 2D z , z , mass in QW respectively, N = mT/ 2 and c c N 2D = m T/ 2 are two-dimensional effective den- c c where s , x , y , z are s- and p-type Bloch sity of states for conduction band in barrier region functions with angular momentum 0 and 1 respec- and in QW, E = me4/2 2 is the Bohr energy in B c s tively. The arrows denote spin directions. barrier region, uc =Ec /T, Ec is conduction band The carrier wave function has following repre- discontinuity and the barrier height for electrons, sentation: Auger recombination in quantum well laser with participation of electrons in waveguide region 195
3. AUGER RECOMBINATION RATE AND COEFFICIENT With the account of the antisymmetrized form of the wave functions, the matrix element of the Auger process is the following:
Mfi MI MII , (8)
e2 MI 3 (r1,1)4 (r2,2 ) 0 r1 r2
(9) 1(r1,1)2 (r2,2 ) ,
e2 Fig. 2. A schematic representation of a single QW MII 3 (r1,1)4 (r2,2 ) of InGaAsP/InP heterostructure. 0 r1 r2
(10) (r , ) (r , ) , s s Ψ p , (6) 1 2 2 2 1 1 where s and are spinors. Near the point the where r and r are the carrier coordinates, and equations for the envelope functions s and in 1 2 1 2 the spherical approximation have a form [7]: are spin variables, e is a charge of electron, 0 is the static dielectric constant of the QW semiconduc- (E E) i Ψ 0, tor. c s It could be easily shown that the matrix element (E E)Ψ i s of direct Coulomb interaction MI is much larger than 2 the matrix element of exchange interaction ( 4 )(Ψ) 1 2 M :M>>M . 2m (7) II I II To estimate the matrix element of investigated 2 Auger process for a transition of an excited particle ( 1 2 2 )[[Ψ]] i[σΨ] 0. 2m into continuous spectrum the following approxima-
tion could be made: Ec, Ev<< Eg, what implies Here is the Kane matrix element, 1, 2 = 3 are k 2+q2>>k 2. It means that the total momentum of generalized Luttinger parameters, = /3, is the 4 1 so so an excited particle is much larger than that of a spin–orbit splitting constant of QW semiconductor, localized one. The matrix element could be calcu- E and E are the energies of the lower edge of the c v lated using the Fourier representation: conduction band and the upper edge of the valence band, m is a free electron mass, =( , , ) de- x y z 4e2 I (p)I (p) d p notes the Pauli matrices. M 23 14 , I 2 2 q1q 2q 3q 4 (11) For reasons of convenience, the energy of the 0 p q 2 upper edge of the valence band Ev is set equal to as it is shown in Fig. 2. * ipx I (p) (x) (x)e d x, (12) An explicit form of carrier wave functions and ij i j expressions determining their boundary conditions, energy spectra and dispersion equations could be 1, q 0, q (13) found in [7, 8]. The wave functions of electrons in 0, q 0. the barrier region have the same form with wave func- where q = |q - q | = |q - q | is the momentum tions of electrons localized in QWs. The only differ- 1 2 3 2 transferred in the plane of the quantum well during ence is replacement of quantity kc with quantity k =i for |x| > a/2. the Coulomb interaction, p is the transverse trans- c c ferred momentum. 196 A.A. Karpova, D.M. Samosvat, A.G. Zegrya, G.G. Zegrya and V.E. Bugrov
The matrix element could be divided into two parts The Auger recombination rate is given by: determined by integration areas. The first term con- tains the multiplier that decays exponentially away 2 2 G M f f k ,k ,k ,k 1 2 from an interface. The second part arises from inte- 1 2 3 4 (18) gration of the wave functions of carriers within the (1 f )(1 f )(E E E E ), quantum well. As a result, the matrix element could 3 4 3 4 1 2 be represented in a following form: where f1 and f2 are the Fermi distribution functions
of the carriers in the initial state, f3 and f4 are the 8e2 Fermi distribution functions of the carriers in the fi- M (M M ). 1 2 (14) nal state. 0 The Auger recombination coefficient C is related The mechanisms that lead to violation of the mo- to the rate G by the expression: mentum conservation law k1+k2k3+k4 are different G Cn n 2D p 2D , (19) for two terms M1 and M2. For the first term M1 the b QW QW momentum nonconservation is associated with the where G = G/S and S is a normalization area. scattering of carriers on the interfaces, and this To calculate the AR several additional approxi- matrix element corresponds to a true threshold-free mations can be made for simplicity: process. The reason why the conservation law breaks down for M is that the volume of integration 2 2mT 2m T with respect to x is restricted to the QW region, and c c q1 , q2 q3 ,q4 . carrier momentum has an uncertainty. Approximation q< 2 2 2 M 2 M M ( i ) M (i ) is a statistical sum, , fi i I II 3 4 2 (17) Re M ( i )*M (i ) 4 M , 2m T I II I hh qT . (i) (ii) where MI and MI are transition matrix elements with fixed spin states of carriers. Auger recombination in quantum well laser with participation of electrons in waveguide region 197 Integrating (20) over 2mcT q1 and considering the AR matrix element to be inde- pendent of this variable a following expression for coefficient might be obtained: 2 1 1 2 m 1 C M c . 3 2 (22) (2) L k (q ) 4 h Here B B B 1/ 4 Fig. 3. Dependences of C , C1 , and C2 on QW EB L a width a. B 4 T is a characteristic length associated with electron spreading in the subbarrier regions, a = 2/m e2 B s c is the Bohr radius in barrier region. Let us estimate the coefficient of the studied AR. The coefficient C2 for quasithreshold AR process looks like E 82 (q k q k )2 C B g h c c h 2 2 (q 2 k 2 )3 h h La a c (23) CHCC B sin2 (k k k k )a / 2 1 Fig. 4. Dependences of C and C on QW f 1 c h , 2 width a. (kf k1 kc kh ) k4 (qh ) where the peaks at the same QW width a45 Å. The de- B pendence for C2 on QW width maxima at QW widths a94 Å and a195 Å. One may notice, that g . 2mc Eg the relative contribution of threshold process corre- B sponding to C1 significantly exceeds the contribu- The coefficient C1 for threshold AR process could B tion of quasithreshold process associated with C2 . be represented in a form: Fig. 4 shows the dependences of coefficients CCHCC and CB on quantum well width a for CHCC 2 E 6 E E process of Auger recombination and the studied C 4 B g c 1 2 E process respectively. According to Fig. 4, two coef- g La a ficients reach their maximum values at different QW c widths. Thus, the coefficient CCHCC has a maximum (24) (k 2k q ) 1 at a40 Å and the dependence of coefficient CB on c h h . 2 2 2 2 QW width has a peak at a45 Å. (qh kh )(kc c ) k4 (qh ) 5. CONCLUSION 4. RESULTS AND DISCUSSION In this paper a new process of Auger recombination Fig. 3 shows the dependences of AR coefficient CB is suggested and discussed. For this process the B recombination energy of localized electron and hole for the studied process and its components C1 and B is transferred to the carrier in the waveguide region. C2 for threshold and quasithreshold processes on the quantum well width a at T = 20 °C. It can be It is shown, that in InGaAsP/InP MQW lasers the B B studied process competes with known CHCC proc- clearly seen, that dependences C and C1 reach 198 A.A. Karpova, D.M. Samosvat, A.G. Zegrya, G.G. Zegrya and V.E. Bugrov ess and might strongly affect the lifetime of [5] S.J. Sweeney // IEEE Photonics Technology nonequilibrium carriers in quantum wells and thresh- Letters 10 (1998) 1076. old current density. [6] Z.N. Sokolova, I.S. Tarasov and L.V. Asryan // Semiconductors 45 (2011) 1494. REFERENCES [7] G.G. Zegrya and A.S. Polkovnikov // JETP 86 (1998) 815. [1] G.G. Zegrya // Semiconductors 35 (2001) [8] A.S. Polkovnikov and G.G. Zegrya // Phys. 1001. Rev. B 58 (1998) 4039. [2] L.V. Asryan // Semicond. Sci. Technol. 15 [9] E.O. Kane // J. Phys. Chem. Solids 1 (1956) (2000) 1131. 82. [3] S. Seki // IEEE journal of quantum [10] P.C. Sercel and K.J. Vahala // Phys. Rev. B electronics 32 (1996) 1478. 42 (1990) 3690. [4] J.D. Evans // IEEE Journal of Selected Topics in Quantum Electronics 1 (1995) 275.