Quantum Wires and Quantum Dots in Heterojunction Devices with Field-E Ect Electrodes

110 Brazilian Journal of Physics, vol. 26, no. 1, March, 1996

Quantum Wires and Quantum Dots in Hetero junction

Devices with Field-E ect Electro des

W. Hansen, D. Schmerek and H. Drexler

Ludwig-Maximilians-Universitat, Muchen

Geschwister-Schol l Pl. 1, 80539 Munchen, Germany

Received July 21, 1995

We generate in esp ecially designed hetero junction crystals one-dimensional electron wires

and zero-dimensional electron dots with a eld-e ect controlled numb er of electrons o c-

cupying the lowest quantum states. Very strong and well controlled con nement of one-

dimensional quantum wires is achieved with patterned eld electro des, that allowusto

study the electronic prop erties of quantum wires as function of the sti ness of the con ne-

mentpotential. For exp eriments on smallest electron dots with very large energy spacing of

the quantum dot levels we study structurally con ned electrons in self-assembled InGaAs

islands that are grown in the Stranski-Krastanow mo de. The electron systems are char-

acterized with capacitance sp ectroscopy and with far-infrared absorption exp eriments that

prob e the electronic ground states and the fundamental excitations, resp ectively.

[3]

or quantum dots . Such systems are preferentially I. Intro duction

prepared from a high mobilitytwo-dimensional elec-

Manyphysical prop erties of an electron system are

tron system at the interface of a mo dulation-dop ed

critically dep endent on its dimensionality. This is

hetero junction crystal. Their geometry is usually de-

mainly due to mo di ed screening and di usion and the

ned either byetching patterns into the crystal surface

reduced density of states (DOS) in lower-dimensional

or by means of patterned eld-e ect electro des biased

electron systems. The latter arises from the fact that

to deplete the areas b eneath the electro des of mobile

[4]

con nement of the electrons reduces the degree of free-

electrons . Then electron strip es or discs are left in

dom for free motion and thus the dimensionality of the

the unetched areas or in the gaps b etween the elec-

quasi-continuous regions in phase space. Thus energy

tro des. Although these techniques have b ecome quite

levels of electron systems con ned in one or two di-

sophisticated and manyintriguing results are obtained,

mensions form two- or one-dimensional subbands. Of

imp ortant parameters like the energy quantization due

course, con nementinto all three spatial dimensions

to the lateral con nement and the numb er of electrons

leads to a fully quantized energy sp ectrum likeinan

o ccupying the one-dimensional subbands of the wires

atom. So-called two-dimensional electron systems are

or the energy levels of the dots is often not well known

usually generated in the plane of a semiconductor-

and determined quite indirectly if at all. Furthermore,

insulator or semiconductor hetero junction interface and

the in uence of the density of states on transp ort and

[1]

are well known for more than three decades . The

optical prop erties is quite involved, so that in general

tunability of the electron densityaswell as the advan-

it is imp ossible to determine this fundamental quan-

tages asso ciated with the reduced dimensionalityhave

tity from such exp eriments. For instance, in transp ort

given rise to a vast numberoftechnological applications

exp eriments electron lo calization and in optical tran-

among which the b est known are eld e ect transistors

sitions selection rules as well as collective e ects are

[2]

and heterostructure lasers .

imp ortant aside from pure density of states e ects.

Within the last decade growing interest has b een

The exp eriments on quantum wires and quantum devoted to the fundamental prop erties of one- and zero-

dots describ ed in the following are p erformed with dimensional electron systems, so-called quantum wires

present address: Center for Free Electron Laser Studies, University of California, Santa Barbara.

H. Hansen et al. 111

nm, an undop ed GaAs spacer layerandanundoped devices esp ecially designed to havevery well de ned,

barrier layer are grown. The highly dop ed GaAs con- strong con nement and a go o d control of the electron

tains an electron system of low mobility, which is con- density. We present capacitance sp ectra that clearly

ducting (sheet conductivitytypical 10mS) even at re ect the quantization of the electronic energies due

the low temp eratures and high magnetic elds of our to the lateral con nement. Indeed, capacitance sp ec-

exp eriments. It forms a back electro de buried in the troscopy has previously b een very successfully applied

heterostructure sample. The front barrier is made of an to two-dimensional electron systems in order to obtain

undop ed short p erio d AlAs/GaAs sup erlattice (SPS), information ab out the two-dimensional thermo dynamic

[58] [7]

eachlayer typically 10 epitaxial monolayers thick. Such DOS . There with a re ned metho d it was even

a barrier exhibits a lowleakage current and serves here p ossible to determine quantitatively many-particle cor-

as an undop ed insulator. The thickness of the SPS is rections to the single particle DOS, which are found to

typically 32 nm and covered with a 10 nm thickun- be very imp ortant at small electron densities or in high

dop ed GaAs layer to protect the AlAs against oxida- magnetic elds. For technical reasons, that will b ecome

tion. On this hetero junction crystal a metal gate (typ- obvious in the following, this re ned metho d cannot

ically 30 nm Ti) is thermally evap orated. b e applied to one- or zero-dimensional electron systems

and a sophisticated quantitative analysis is only p os-

sible with self-consistentnumelical calculations basing

on parameters that are less well controlled. Neverthe-

less, the substructures observed in the di erential ca-

pacitance can b e interpreted on basis of very simple

mo dels yielding a go o d notion ab out imp ortant param-

eters suchasquantization energies, electron o ccupa-

tion and typical widths of the structures. Furthermore,

the electron wires realized in our exp eriments are of

such good quality, that unlike in previous exp eriments

the substructures are clearly observed in the direct ca-

pacitance signal rather than in the derivativemaking

a comparison to self-consistentnumerical calculations

less ambiguous.

The parameters obtained from the capacitance sp ec-

tra are used to understand the high frequency conduc-

tivity of the quantum wires. These exhibit characteris-

tic resonances in the Far-Infrared (FIR) that are asso ci-

ated with transitions b etween the one-dimensional sub-

bands of the wire or the energy levels of the dot. Com-

parison of the subband spacing extracted from the ca-

pacitance data to the resonance energies tells us ab out

the collective contributions to the intersubband transi-

tions, and in the case of the quantum dots it substan-

tiates our interpretation of the capacitance data.

Figure 1. (a) Cross section through the epitaxially grown

layers of a MIS-dio de. The distances b etween back elec-

I I. Samples

tro de and hetero junction interface z as well as top gate z

b t

are indicated. The thickness of the GaAs spacer layer on

top of this electro de is typically z = 100 nm. (b) Sketchof

The samples of our exp eriments are basically metal-

the conduction band edge as function of the co ordinate in

[9]

insulator hetero junction (MIS) dio des . The hetero-

growth direction at gate voltage b elow and ab ove a thresh-

junction is grown by molecular b eam epitaxy.Atypical

old voltage Vtll as describ ed in the text.

epitaxial layer sequence and an idealized band structure

are sketched in Fig. 1a and 1b. On the [100] surface

Note that unlike in a conventional mo dulation-

of a GaAs substrate wafer an undop ed GaAs bu er, a

[10]

highly dop ed GaAs layer with typical thickness of 20 dop ed hetero junction our MIS- dio des generally do

112 Brazilian Journal of Physics, vol. 26, no. 1, March, 1996

not contain mobile electrons at the hetero junction in- strip es is 150 nm so that at V 6= 0 the p erio d of the

terface at zero gate voltage as indicated in Fig. 1b. surface p otential is a=500 nm.

Like in a Si metal-oxide-semiconductor dio de the car-

riers have to b e eld induced by a p ositive gate volt-

age applied b etween the back electro de and the front

gate. If the gate voltage V is larger than a threshold

voltage V the conduction band edge forms a mini-

mum at the interface b etween the GaAs spacer layer

and the front barrier into which electrons are injected

from the back electro de. In all exp eriments discussed in

the following charge injection through the shallowtun-

Figure 2. Sketch of the electro de con guration for the gen-

nel barlier formed by the undop ed spacer layer happ ens

eration of eld-induced quantum wires. The gate voltage V

is applied b etween the back electro de in the heterostructure

suciently fast that we can assume equilibrium with re-

crystal and one of the interdigital electro des. The voltage

sp ect to charge exchange b etween the electron system

V is applied b etween the twointerdigital electro des.

at the hetero junction interface and the back electro de.

The advantage of our MIS-dio des is that no intentional

dopants are contained in the layers b etween the eld

induced electron system and the front gate. This is im-

p ortantifwewant to generate very small electron sys-

tems with high mobility and precisely controlled, strong

lateral geometry at the hetero junction interface. In

a conventional mo dulation-dop ed heterostructure large

areas would have to b e depleted from mobile electrons

in order to de ne small electron wires or dots. The re-

maining dopants in the front barrier are left unscreened

and thus seriously deteriorate the p otential in the wires

[11]

or dots .

Beneath a homogenous large gate electro de a two-

dimensional electron system is formed at the hetero-

junction interface at V >V . If the gate is patterned

g th

Figure 3. (a) Cross section through a MIS-dio de with an

in the plane of the crystal surface the geometry of the

interdigital gate. (b) Potential sup erlattice induced bythe

electron system will represent an according replica of

interdigital gate in the plane of the hetero junction interface.

Electrons are injected from the back electro de into the p o-

this pattern. In Fig. 2 the gate pattem, that we use for

tential minima if the gate voltage V is suciently high. The

the generation of eld-induced quantum wires, is de-

corresp onding lo cation of the electron wires is indicated in

(a).

picted together with the back electro de. The so-called

interdigital gate consists of two electro des that can b e

biased at di erentvoltages with resp ect to the back The interdigital gate o ers a go o d control of the

elect o de. As indicated in Fig. 2 they form a grating of p erio dically mo dulated surface p otential of our hetero-

parallel metal strip es with every second strip e biased at junction devices. At the op en crystal surface in the

avoltage V with resp ect to the back electro de and with gaps b etween the metal strip es the surface p otential

avoltage di erence V to the adjacent strip es. Wede- is not well known. The unknown b oundary condi-

ne the geometry of our interdigital gates by electron tions for the op en surface regions cause considerable

b eam lithography with a bilayer PMMA resist. The arbitrariness in mo del calculations of the p otential b e-

[12;13]

metal strip es are prepared by thermal evap oration and neath split gate devices However, with split gate

a lift-o pro cess. As indicated in Fig. 3 the width of devices the mo dulation amplitude of the surface p o-

[12]

the metal strip es is 100 nm, the gap b etween adjacent tential is xed, if stress e ects caused by di erent

H. Hansen et al. 113

thermal contraction of the electro de and semiconduc- growth of a strained InGaAs layer on a GaAs sub-

tor material can b e neglected. In Fig. 3 a cross section strate leads to sp ontaneous formation of semiconductor

through a MIS-dio de with a interdigital gate and the islands with very small sizes. This so-called Stranski-

sup erlattice p otential in the plane of the hetero junc- Krastanowgrowth mo de emerges after growthofafew

tion interface are indicated. The Fourier comp onents monolayers as is judged by the re ection high-energy

[15]

of the surface p otential decay rapidly in the z -direction electron di raction (RHEED) pattern .At this p oint

according to sinh[q (z z )]=sinh[qz ]. Here q = n2=a; the InGaAs layer sp ontaneously relaxes into dislo cation

t t

(n =1; 2; :::); is the wavevector of the corresp onding free, lens-shap ed InGaAs islands randomly distributed

Fourier comp onentandz is the distance of the plane and emb edded in a very thin InGaAs wetting layer.

in which the sup erlattice p otential is calculated from Insp ection with scanning an atomic force microscop e

the plane of the gate. However, since in our structures (AFM) and a transmission electron microscop e (TEM)

the distance d = z z between the hetero junction reveal that at optimized growth conditions the islands

t b

interface and the surface is small (typically d =40 have a diameter of typically 20 nm in the plane of the

nm) a considerable fraction of the mo dulation ampli- wetting layer and are ab out 6 nm high in the direc-

tude is still present in the plane of the electron system. tion p erp endicular. Furthermore, it is found that the

We estimate that the rst Fourier comp onentofthe size distribution is remarkably narrow with the diam-

unscreened sup erlattice p otential is reduced byabout eter uctuating only by ab out 10% and the heightby

[15;17]

50% at the hetero junction interface. At this p ointit only one monolayer . The dot densities are ad-

9 2 10

b ecomes apparent that mo dulation dop ed hetero junc- justable b etween typically 10 cm and several 10

tions with very large spacer layers in the front barrier cm .

to preserve high mobilityeven at low densities are in-

adequate for our purp oses. The separation b etween the

planes of the electron system and the front gate would

b e to o large to ensure strong p otential mo dulation.

Thus the amplitude of the sup erlattice p otential is

controlled bythevoltage V as long as the p otential

landscap e at the hetero junction interface is not lled

with electrons. The gate voltage V controls the align-

ment of the p otential landscap e with resp ect to the

Fermi energy in the back electro de. Once the p otential

minima are lled with electrons as indicated in Fig. 3

the external p otential induced by the gate electro des

is partly screened. The distinction b etween the so-

called bare p otential induced by the external electro des

and the e ective potential, that includes the electron-

electron interaction and thus determines the subband

spacing, will b ecome imp ortant for the discussion of the

high frequency resp onse of the quantum wires.

Figure 4. (a) Cross section through a MIS-dio de contain-

Although zero-dimensional electron discs can b e

ing a thin InGaAs quantum well with lens shap ed Stranski-

[14]

also de ned bypure elde ect we shall discuss in the

Krastanow islands at a distance z from the back electro de.

[15]

(b) Conduction band edge along a line in the growth direc-

following a recently employed metho d to generate

tion that intersects the InGaAs-island of the structure in

extremely small quantum dots that takes advantage of

(a). Two energy levels are indicated schematically in the

quantum well of the InGaAs-island.

the so-called Stranski-Krastanow growth mo dus. Here

a coherently strained InGaAs layer is grown on top of

After formation of the InGaAs islands a 5 nm thick the undop ed GaAs spacer layer of our MIS-dio des, as

[16]

GaAs spacer layer is dep osited b efore the growth is n- is depicted in Fig. 4 . It is well known that the

114 Brazilian Journal of Physics, vol. 26, no. 1, March, 1996

ished with the SPS barrier and the cap layer of the

MIS-dio de. Here the GaAs acts like a barrier in which

almost ideal typ e I InGaAs quantum b oxes are embed-

ded. As will b e demonstrated in the following the o c-

cupation of the electronic energy levels in the InGaAs

dots can b e nicely controlled by means of the gate bias

applied b etween the homogeneous front gate and the

back electro de of our MIS-dio des.

I I I. Exp eriments

Wewould like to rst discuss capacitance measure-

ments on quantum wires de ned with interdigital gates.

[18]

In Fig. 5wedepicttypical results obtained on a

MIS-dio de with d=42 nm and z =142 nm. Each elec-

tro de of the interdigital gate consists of 300 strip es.

Each strip e is 180 m long. Strip e width W and p erio d

a are as indicated in Fig. 3. The di erential capac-

Figure 5. Capacitance of a wire array as function of the gate

voltage V . The exp erimental arrangement is indicated in itance is measured by slightly mo dulating the voltage

the inset. Di erent traces have b een recorded at di erent

Vg at a frequency f=lOkHz and with an amplitude of

magnetic elds applied p elp endicu la r to the sample surface.

The capacitance scale refers to the top trace that has b een

dV = 5 mV. The di erential capacitance is calculated

recorded at zero magnelic eld. The eld has b een increased

from the excited ac current through the sample.

in steps of 1 T in the lower traces, which are o set for clarity.

The voltage bias b etween the electro des of the interdigital

If all the electro des involved would b e p erfectly

gate is V =1.0 V. As describ ed in the text the indices at

the top trace denote the one-dimensional sub-bands that metallic the capacitance would b e entirely determined

are lled in the corresp onding gate voltage ranges. A sub-

by the geometric arrangement of the electro des. In our

structure in the sp ectra, that is discussed in the text, is

highlighted by an arrow. (From Ref.[18]). case, however, the center electro de formed by the one-

dimensional wires is not p erfectly screening the elec-

Wethus exp ect that the capacitance signal in Fig. tric eld due to the nite DOS in the electron sys-

5 re ects the density of states of the one- dimensional tem. Tocharge the wires with electrons the conduc-

subbands in the wire. At the threshold voltage V = tion band edge has to b e lowered so that the electro-

V 1:13 V the capacitance signal increases rapidly, chemical p otential of the electron wires is kept aligned

re ecting the generation of the quantum wires which with the chemical p otential in the back electro de. Con-

represent a third electro de b etween the front gate and sequently,netcharge has to b e simultaneously induced

the back electro de. At larger gate voltages pronounced in the back electro de and an electric eld is maintained

structures are apparenteven if no magnetic eld is ap- between the back electro de and the wires. The de-

plied. The strong steps numb ered with indices in Fig. vice capacitance thus deviates from the value that pure

5 re ect the one-dimensional subband onsets as can b e metallic electro des would have. This fact is well known

veri ed by their dep endence on the voltage di erence from similar measurements on two-dimensional electron

[58]

V and on a magnetic eld applied p erp endicular to the systems, where e g. in a magnetic eld the forma-

wires. A straight forward quantitativeanalysisofthe tion of Landau levels in the DOS is clearly re ected. In

capacitance data at zero magnetic eld is hamp ered the reduced DOS b etween the Landau levels results in

in our case by the fact, that a parasitic capacitance pronounced minima of the capacitance signal at integer

between the front electro de and the backcontact con- lling factors.

H. Hansen et al. 115

tributes to the signal even when the electron wires are

4=3

4=3 2 2

formed. We exp ect the wires to widen with increas- U = ( + ! ) ;

ing electron o ccupation, so that the parasitic capaci-

2m = ~=c . In Fig. 6 the exp eri- where = eg

w s

tance as well as the geometric contribution of the wire

mental values obtained on a sample with di erentvolt-

capacitance are not constant when the gate voltage is

ages V applied b etween the electro des of the interdig-

changed. Indeed, the overall increase of the capacitance

ital gate are presented together with b est ts to the

signal in Fig. 5 re ects this b ehavior. Hence, the ca-

data according the ab ove equation. As exp ected at zero

pacitance can only b e mo delled with a self-consistent

magnetic eld the voltages U increase with increasing

numerical calculation that still needs to b e p erformed.

V , i. e. with increasing sti ness of the con nement

However, a simple mo del can b e applied to approach

potential. The data are surprisingly well describ ed by

a quantitative analysis from the magnetic eld dep en-

the ab ovemodeleven at high magnetic elds. From the

dence of the capacitance data. We assume that the ef-

ts to the data we derive subband spacings, whichare

fective con nementpotential V (x) in the lateral direc-

depicted in the inset of Fig. 6, and a wire capacitance

tion can b e describ ed by a harmonic oscillator p otential

of c = 150pF =m; whichdoesnotchange signi cantly

2 2

with characteristic frequency : V (x)= m x =2.

with V :

Here m =0:07m is the e ective conduction hand

mass in GaAs. Furthermore, we assume that this ef-

fective con nementpotential do es not change consid-

erably in the density range where only the lowest one-

dimensional sub-band is o ccupied. Accordingly,we de-

rive the number n of carriers that has to b e lled into

the lowest one-dimensional subband when the chemical

p otential crosses the onset of the second subband from

the ideal single particle DOS

3=2

g 2m !

n = ;

where g = 2 is the spin degeneracy of the GaAs

conduction band. The separation b etween the rst

and the second one-dimensional subband increases with

2 2 1=2

magnetic eld according to ~! = ~( + ! ) ; and

! = eB =m is the cyclotron frequency. The steps in

the capacitance signal at least partly arise from the

DOS maxima at the subband edges. Since b oth the sub-

Figure 6. Exp erimental data of the voltage di erences

U between the onsets of the rst and the second one-

band separation and the DOS within the subbands rise

dimensional subband in a quantum wire array measured

with the magnetic eld it is plausible that the voltage

with di erentvoltages V applied b etween the electro des of

the interdigital gate. The exp onents of U and B are chosen

separation b etween the subband onsets also increases

such that from a simple mo del describ ed in the text a lin-

with the magnetic eld.

ear relation is exp ected. Best ts to the mo del are depicted

by straight lines. The inset shows the subband spacings

Assuming that the wire capacitance CW p er unit

obtained from the ts to the mo del.(From Ref.[19]).

length do es not change in the densily range considered

As apparent from the inset of Fig. 6thesub- we also have n = c U , where U is the separation

ql w ql ql

band spacings increase linearly with V with a slop e between the onset voltages of the rst and the second

of 1.4 meV/V. Surprisingly, the extrap olation of the one-dimensional subbands. The magnetic eld dep en-

data to zero gate voltage gives a nite energy of 3.2 dence of the voltage di erence b etween the subband

meV rather than zero. Atpresent this result is not onsets thus can b e describ ed by

116 Brazilian Journal of Physics, vol. 26, no. 1, March, 1996

understo o d. It may indicate non-parab olicityoftheef-

fective con nement p otential or a nite p otential mo d-

ulation at V =0: Such a p otential may arise from

charges trapp ed in the gaps of the interdigital gate or

[12]

from stress e ects .

Reduction of the temp erature at which the capac-

itance is recorded to 100mK do es not alter the sp ec-

tra signi cantly except that the spin p olalization of

the subbands b ecomes noticeable at smaller magnetic

[19]

elds . Clearly the exp erimental capacitance sp ec-

tra are broadened and do not re ect the singularities

of the ideal one-dimensional single particle DOS. Im-

purities as well as inhomogeneities of the con nement

p otential may considerably broaden the DOS. Indeed,

the overall length of the wire prob ed in Fig. 5 is ab out

5.4 cm and it is interesting to investigate how the sp ec-

tra might alter, if the length of the wire is consider-

ably reduced. A 5.4 cm long wire has b een used for

the exp eriment in Fig. 5 to guarantee a large signal

Figure 7. Capacitance sp ectrum recorded on a wire of

150,um length. Here the signal was recorded in a capaci-

to noise ratio and to reduce the in uence of parasitic

tance bridge arrangement with an imp edance transformer

contributions to the signal from the measurement set-

attached close to the sample. The steps at the threshold

voltage V 0:95 V corresp ond to an increase of the ca-

up. Fig. 7 demonstrates, however, that it is p ossible

pacitance of ab out (8 2)fF . The top trace is recorded at

to record with a re ned measurementtechnique capac-

B=0, the magnetic eld is increased in the lower traces in

steps of 1 T. The traces are o set for clarity.

itance sp ectra on wires with length that is reduced by

more than a factor of 300. Here the capacitance signal

Wewould like to note, that in Fig. 5aswell as

is detected with a bridge arrangement using an unbi-

in Fig. 7 a substructure is clearly visible in the ca-

ased MIS-dio de as reference. In order to minimize the

pacitance sp ectra which has not b een discussed so far.

in uence of parasitic capacitances the signal is picked

It o ccurs close to the threshold voltage in the one-

up with an imp edance transformer mounted very close

dimensional quantum limit, i. e. at gate voltages,

to the sample. A similar technique has b een used by

where only the lowest one-dimensional subband is o c-

Asho ori and co-workers to measure capacitance sp ectra

cupied. Such a substructure is not observed on two-

[20]

of a single quantum dot .

dimensional electron systems generated, e. g., in MIS-

Metal strip es enclosed in tuning-fork shap ed elec- dio des with a homogeneous gate. It b ecomes more

tro des form the gate of the sample in Fig. 7. The pronounced at higher magnetic elds as is obvious in

tuning fork is biased bythevoltage V with resp ect b oth gures. Spin splitting can b e excluded as origin

to the strip es to control the con nement p otential. In of the structure. At elds b eyond 7T an additional

spite of the considerable reduction of the wire length structure arises at higher gate voltages, that can con-

the sp ectra of Fig. 7 are qualitatively very similar to sistently b e asso ciated to the spin p olarization of the

[18]

those in Fig. 5. Obviously, the mechanisms that lead to one-dimensional subbands . Reduction of the tem-

[19]

a broadening of the capacitance traces act on a length p erature do es not signi cantly alter the substructure

scale b elow100m, whereas uctuations on a macro- and at higher temp eratures it remains even visible when

scopic length scale do not play a signi cant role. Aside the steps arising from the size quantization are consid-

[21]

from the very pronounced signatures of the size quan- erably temp erature broadened . The structure has

tization this proves the high quality and homogeneity b een tentatively asso ciated to a rapid renormalization

[18]

of our MIS-dio des with interdigital gates. of the wire p otential once it b ecomes o ccupied, but

H. Hansen et al. 117

an elementary excitation in the electron system. If the due to lacking self-consistent calculations this interpre-

[22]

relaxation time for heat conduction from the electron tation remains sp eculativeuptonow .

system into thermal baths like the sample substrate or From the strong size quantization veli ed with

contacts is suciently long, a signi cant mo dulation of ab ove capacitance measurements we exp ect a drastic

the temp erature in the electron system will o ccur. This mo di cation of the high frequency conductivityofour

results in charge ow, if the thermo dynamic densityof wires as compared to a two-dimensional electron sys-

states is not constant. tem. In particular, if the electric eld is p olarized

p erp endicular to the wires the conductivityisassoci-

ated with transitions b etween the one-dimensional sub-

bands, so that the zero frequency conductivityvan-

ishes and resonances, so-called intersubband plasmons,

are exp ected at nite frequencies in the FIR regime.

One-dimensional intersubband plasmons have b een in-

tensively studied with conventional Fourier-transform

sp ectroscopy on large wire arrays with an overall wire

[4;23]

length of several dozens of meters Hence in those

exp eriments the wire arrays had to b e fablicated with

metho ds in which the con nement p otential and the

subband o ccupancy are less well controlled. In Fig. 8 a

measurement technique is describ ed with whichwe are

able to p erform FIR absorption sp ectroscopyeven on

[24]

our relatively small wires in interdigital MIS-dio des .

Figure 9. (a) Far-infrared absorption as function of a mag-

netic eld detected by the photo-current of a MIS-dio de with

interdigital gate. Di erent traces are recorded with di erent

voltages V between the electro des of the interdigital gate.

The gate voltage V is adjusted close to the threshold voltage

Figure 8. Measurement set-up for photo-current measure-

so that the quantum wires are in the one-dimensional quan-

ments. Charge ow from the wire system is detected with

tum limit. The sample is illuminated with FIR radiation of

a current-voltage converter (I/U) and an ampli er (lo ck-in)

constantwavelength =118 m and the magnetic eld is

phase lo cked to the mo dulated laser radiation.

used to tune the resonance condition to the laser frequency

as indicated in (b). (b) Resonance energies calculated in a

harmonic oscillator mo del assuming characteristic energies

The exp erimental set-up sketched in Fig. 8 resem-

of ~ =7:8meV, ~ =9.8 meV and ~ =0 for the full,

0 0 0

bles the set-up for measurements of di erential capac-

the dashed and the dotted line, resp ectively. The dashed

dotted line indicates the energy of the laser radiation. The

itance sp ectra like those in Fig. 5. However, rather

inset presents the zero eld resonance energies ~ derived

than mo dulating the gate voltage here the sample is

from the exp erimentversus V .

irradiated from a FIR-laser with intensity-mo dulated

radiation. The sample is e ectively heated by the ab- The photo-current on the MIS-dio de of Fig. 5 is de-

sorb ed FIR if the laser wavelength is in resonance with picted in Fig. 9a as function of a magnetic eld applied

118 Brazilian Journal of Physics, vol. 26, no. 1, March, 1996

[24]

p erp endicular to the wires . The data nicely demon- late the resonance energies from the p olarizability in-

[27]

strate that the photo-current is resonantly dep endent cluding many-particle e ects .Itiswell known from

on the magnetic eld and the resonance condition can calculations in random phase approximation, that col-

b e tuned by the gate voltage V . The b ehavior can lective contributions result in signi cantly larger reso-

b e most easily understo o d with the generalized Kohn nance energies than the single particle level spacings.

[25;26]

theorem . This theorem bases on the assumption This, again, is qualitatively consistent with our exp er-

that the bare p otential, i. e. the p otential induced by imental ndings. Wewould liketonotethatrecent

the extemal electro des, is of parab olic shap e. For the conventional FIR transmission exp eriments on wire ar-

sake of simplicitywe pursue this assumption although rays with non-parab olic con nementpotentials have re-

it is not p erfectly consistent with the assumption of a vealed that in such wires the overall magnetic eld dis-

parab olic elfective p otential that wehave used in ab ove p ersion of the intersubband-plasmons still b ehaves sim-

[28;29]

analysis of the magneto-capacitance data. In an elec- ilar to the disp ersion in Fig 9.b However, there are

tron wire con ned by a parab olic bare p otential the distinct magnetic eld regimes, where pronounced split-

radiation eld couples only to the center of mass mo- tings of the resonances are observed and exp ected from

[2830]

tion whereas the intemal electron distribution of the theory Corresp onding exp eriments on quantum

wire remains unchanged. Consequently, the resonances wire arrays generated byinterdigital gates remain to

are exp ected to b ehavelike single particle transitions b e p erformed in future.

in a harmonic oscillator p otential. If the bare p oten-

tial is characterized by the frequency , the magnetic

eld dep endence of the transition energies is given by

Measurements of the di erential capacitance have

+ ! as depicted in Fig. 9b. In the inset of Fig. 9

also b een very successfully applied to investigate very

we depict the energies ~ as determined from the com-

0 small zero-dimensional electron systems. As describ ed

parison of the exp erimental resonance p ositions with

in the previous section we generate such electron dots

the intersection.s of the calculated resonance disp er-

in InGaAs islands that are randomly distributed on a

[16]

sions with the laser energy.

plane emb edded in the GaAs spacer . The di erence

We note, that like in Fig.6 the energies almost lin-

between capacitance sp ectra obtained on samples with

early rise with the voltage V .However, b oth the zero

and without InGaAs islands is demonstrated in Fig. 10.

voltage intercept of 6.2 meV and the slop e of 1.7 meV/V

The broken line represents the di erential capacitance

are considerably larger than the values in Fig. 9. This

of a sample with an InGaAs layer that is to o thin for the

is exp ected, since according to our mo dels the data in

relaxation in three dimensional growth, so that no In-

Fig. 9 re ect the unscreened bare p otential, whereas

GaAs islands are formed. The capacitance signal b elow

the data in Fig. 6 re ect the e ective electron p o-

gate voltage V =0.6 V re ects the capacitance b etween

tential at a medium value of the electron densityin

the homogeneous top gate and the back electro de. The

the one-dimensional quantum limit. The bare p otential

steep rise of the capacitance at V =0.65 V re ects the

and the e ective p otential are equal only at very small

generation of a two-dimensional electron system at the

electron densities close to the threshold voltage where

hetero junction interface. A very similar b ehavior is ob-

the Coulomb repulsion b etween the electrons canbe ne-

served in a sample containing self-assembled InGaAs

glected. From the quantization energy ~ =7:8meV

dots as demonstrated by the full line in Fig. 10. In this

at V =1:0V we determine a typical wire width of 24

sample an In Ga As well was grown until relaxation

0:5 0:5

nm in this case. As so on as more electrons are lled into

into islands of ab out 20 nm diameter with a densityof

10 2

the wire the Coulomb repulsion results into an increased

ab out 10 dots/cm o ccured. Again a rapidly increas-

wire width as determined by the e ective con nement

ing di erential capacitance at gate voltage V=0.65 V

p otential. From the quantization energy ~ =4:4 meV

indicates the threshold of a two-dimensional electron

we determine an e ective wire width of 31 nm.

system. This is further veri ed bytheShubnikov-de

A more sophisticated mo del would havetostart Haas oscillations in the magneto-capacitance depicted

with a self-consistent electron p otential and then calcu- in the inset of Fig. 10.

H. Hansen et al. 119

collective contributions to the resonance energy is neg-

ligible in dots o ccupied with very few electrons weob-

tain a level spacing of 41 meV. The Coulombenergy

of the electrons in the dot is estimated from the self-

capacitance C =4 d of a disc-shap ed electro de with

diameter d=20 nm as determined from the AFM or

TEM insp ections: e =C = 18 meV. With these energies

we calculate the voltages, at whichcharge exchange b e-

tween the dots and the back electro des takes place and

thus an increased capacitance is exp ected. In Fig. 11a

and 11b wepresent exp erimental sp ectra and results

of such a mo del calculation, resp ectively.For compar-

ison with the exp erimental sp ectra in Fig. 11b a phe-

Figure 10. Capacitance sp ectra of a reference sample with

nomenological broadening is intro duced. Broadening

avery thin strained In Ga As layer (dashed line) and

0:5 0:5

is exp ected to mainly result from uctuations of the

a sample containing InGaAs islands (full line). The anow

denotes 20% of the capacitance b etween gate electro de and

[16]

threshold voltages of di erent dots in the sample .

back electro de. The distance b etween these electro des is

z 95 nm and the InGaAs well is separated from the back

The only parameters adjusted for comparison b e-

electro de by z =50 nm. The inset depicts the magneto-

tween the exp erimental sp ectra and the mo delled ca-

capacitance of the dot-sample measured at a gate voltage

beyond the threshold of a two-dimensional electron system.

pacitance is the center value and the width of a Gaus-

sian distribution accounting for the distribution of the

threshold voltages of the prob ed dots. Whereas a small However, in the dot-sample pronounced capacitance

broadening of 5 mV width would makethecharging maxima are observed in the sub-threshold regime. As

with every single electron discernible the exp erimen-

will b e further veri ed in the following we asso ciate the

tal sp ectra are b etter describ ed with a broadening of

two p eaks observed in this regime with the charging of

[16

25 mV, where the Coulomb energy is not resolved any the twolowest quantum levelsinthequantum dots .

more. The resolved maxima re ect the size quantiza- We exp ect the lowest s-like energy state of the quantum

tion into the twofold and fourfold degenerate dot lev-

dots to b e twofold spin degenerate and the next level to

els and their separation describ es surprisingly well the be p-like and fourfold degenerate due to the lens-shap e

separation of the capacitance maxima in the exp eri- of the InGaAs islands. Maxima arise in the capacitance

mental sp ectra. Fulthermore, the magnetic eld dep en- signal if charge exchange b etween the dots and the back

dence calculated in our simple mo del accounts well for

electro de is energetically favourable, i. e. if the total

the b ehavior of the exp erimental data. As apparentin energies of the system with N and N + 1 electrons in

Fig.11 the p-level splits into two branches while the s- quantum dots are degenerate. In a simple but widely

[31]

level is hardly a ected by the magnetic eld. Again,

pursued mo del the total energy of electron systems

for the sake of simplicitywe estimate the magnetic

in the quantum dots is divided into a sum of twocontri-

eld disp ersion of the energy levels in the dot from butions one regarding the single-particle level spacing

a parab olic con nement mo del. In Fig. 11b the sep- due to the size quantization of non-interacting electrons

aration of the branches from the s-level is calculated

and a second term considering the Coulombinteraction

according to the harmonic oscillator p otential mo del:

of the electrons. Weestimatevalues for these two con-

0:5~( + ! ! )with~ = 41 meV and no fur- 4 tributions from indep endent measurements and deter-

c 0

ther adjustable parameter. mine from these the exp ected voltage spacing of the

capacitance p eaks.

The exp eriments demonstrate that we can eld-

The single-particle level spacing b etween the two e ect control the numb er of electrons that o ccupythe

lowest quantum states is determined from FIR reso- six lowest quantum states of the InGaAs dots. Capaci-

nances of the quantum dots measured with conventional tance and FIR sp ectra provide a rather comprehensive

[16]

Fourier-transform sp ectroscopy . Assuming that the picture ab out the level spacing b etween the

120 Brazilian Journal of Physics, vol. 26, no. 1, March, 1996

The voltage separation of the p eaks is consitent with

a Coulombcharging energy of 24 meV. Measurements

on single self-assembled dots as well as on dot systems

with controlled spatial arrangementremaintobeper-

formed in the future. Although prepared in situ dur-

ing epitaxial growth without any lithography the self-

assembled InGaAs quantum dots so far representthe

smallest man-made zero-dimensional electron systems

with a tunable numb er of eleclrons. If there is found a

way to arti cially control the lo cation of the dots, e.g.

by creation of nucleation centres, they representvery

[34]

interesting systems also for p ossible applications .

IV. Summary

Very well de ned electron wires and dots are eld-

e ect induced in esp ecially designed MIS-dio des as

demonstrated by the capacitance and FIR sp ectra dis-

cussed in this article. With interdigital gates electron

wires are investigated in the quantum limit, i.e. at elec-

tron densities where only the lowest one-dimensional

subband is o ccupied. Parameters like the subband spac-

ing and the one-dimensional electron density are deter-

mined in simple analytical mo dels for the con nement

Figure 11. Exp erimental capacitance sp ectra of a MIS-dio de

with InGaAs quantum dots at di erent magnetic elds ap-

p otential. For the subband spacing we ndtypical val-

plied p erp endicula r to the crystal surface. The sample pa-

ues b etween 4 meV and 6 meV dep ending on a voltage

rameters are z =50 nm and z 95 nm as in Fig. 10. The

b t

vertical scale refers to the trace recorded at B=0, the other

with whichwe control the con nementpotential.

traces are o set for clarity (b) Capacitance traces calcu-

lated in a simple mo del for parameters E =41 meV and

Well resolved onset voltages for the rst and the sec-

e =C = 18 meV. As describ ed in the text, Gaussian broad-

ond one-dimensional subband in capacitance measure-

ening with standart deviations of 5 mV and 25 mV are used

for the dashed and the full lines, resp ectively.(From Ref.

ments and a narrow FIR resonance even at low elec-

[16]).

tron densities in the quantum limit demonstrate the

high quality of the electron systems Unlikeinprevi-

lowest energy states. Avariety of exp eriments are ous works on electron wires and dots where the struc-

[32]

presently under way that further prob e the optical tures of interest have b een only resolved with a deriva-

[35;36]

and the electronic prop erties of these interesting quan- tivetechnique, here the exp erimental data can b e

[33]

tum systems .For example in Fig. 12 we present more directly compared to results of more involved self-

more recent data demonstrating that with improved consistent mo del calculations. Such calculations, how-

growth conditions it is p ossible to resolve the spin de- ever, remain to b e done for our devices. The elementary

generacy of the s-state in the charging sp ectra. The excitations in the wires are prob ed with FIR absorption

gate voltage regime depicted in Fig.12 retlects the sub- exp eriments with a new measurementtechnique. This

threshold range with two capacitance maximare ect- technique is very sensitiveeven if the wire is in the

ing the charging of the s-andp-states. However, unlike one-dimensional quantum limit. E orts are under way

in Fig.10 and Fig.11 the rst maximum at V =0.1V to considerably reduce the length of the wire system

exhibits a substructure indicating that it actually con- prob ed in capacitance and FIR absorption exp eriments

sists of two p eaks. These we asso ciate with the charg- as is demonstrated by capacitance data obtained on a

ing of the rst and the second electron into the s-state. single 150 m long wire.

H. Hansen et al. 121

fruitful collab oration with M. Holland, D. Leonard, S.

Manus and A. Schmeller. M. H. and D. L. prepared

the very go o d epitaxial heterostructure material and

A. S.contributed very much to the developmentofthe

interdigital gates. S. M. help ed us improving the mea-

surement set-up and gave imp ortanthints to the in-

terpretation of the capacitance data. Furthermore, we

would like to particularly thank J. P. Kotthaus, P.M.

Petro , A. O. Govorov and V. Dolgop olovformany

very valuable advises and discussions. Financial SUp-

pOlt of the Deutsche Forschungsgemeinschaft and Es-

prit Basic Research is gratefully acknowledged.

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conductor Structures, Fundamentals and Applica-

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