# Capacitance of gated GaAs/AlxGa1-xAs heterostructures subject to in-plane magnetic fields.

**ABSTRACT** A detailed analysis of the capacitance of gated GaAs/AlxGa1-xAs heterostructures is presented. The nonlinear dependence of the capacitance on the gate voltage and in-plane magnetic field is discussed together with the capacitance quantum steps connected with a population of higher two-dimensional gas subbands. The results of full self-consistent numerical calculations are compared to recent experimental data.

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**ABSTRACT:**We have studied the capacitance-voltage characteristics of an optically excited wide quantum well. Both self-consistent simulations and experimental results show the striking quantum contribution to the capacitance near zero bias which is ascribed to the swift decreasing of the overlap between the electron and hole wave functions in the well as the longitudinal field goes up. This quantum capacitance feature is regarded as an electrical manifestation of the quantum-confined Stark effect.Physical Review B 03/2001; · 3.66 Impact Factor - [Show abstract] [Hide abstract]

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Page 1

arXiv:cond-mat/9501041v1 11 Jan 1995

Capacitance of Gated GaAs/AlxGa1−xAs Heterostructures Subject to In-plane

Magnetic Fields

T. Jungwirth, L. Smrˇ cka

Institute of Physics, Acad. of Sci. of Czech Rep.,

Cukrovarnick´ a 10, 162 00 Praha 6, Czech Republic

(Received February 1, 2008)

A detailed analysis of the capacitance of gated GaAs/AlxGa1−xAs heterostructures is presented.

The nonlinear dependence of the capacitance on the gate voltage and in-plane magnetic field is

discussed together with the capacitance quantum steps connected with a population of higher 2D

gas subbands.The results of full self-consistent numerical calculations are compared to recent

experimental data.

PACS numbers: 73.40.LQ, 72.20.M

In modulation doped GaAs/AlxGa1−xAs heterostructures a charged channel is formed in GaAs near the interface

due to the electron transfer from donors in AlxGa1−xAs. If a metal Schottky gate is evaporated on the AlxGa1−xAs

surface the electron density N of the GaAs inversion layer can be controlled by the gate voltage Vg. In strong inversion

and at low temperatures the impurity depletion charges are essentially fixed and the charge induced by the variation

of the gate voltage goes entirely into the inversion layer. In this regime the heterostructure behaves like a capacitor

with one metal electrode and the second electrode represented by the channel in GaAs.

The differential capacitance (per unit area) of a gated structure, C = d|e|N/dVg, is traditionally [1], [4] written in

the form

C−1= C−1

b

+γz

ǫ

+

1

e2g.

(1)

Here Cb= ǫ/dbdenotes the barrier capacitance determined by the AlxGa1−xAs barrier width db and the dielectric

constant ǫ. The second term takes into account the distance z of the centroid of the inversion layer charge from the

interface. The numerical prefactor γ results from approximating (see e.g. [2] and [3]) the dependence of the subband

energy E0on N by dE0/dN ∝ z. In GaAs/AlxGa1−xAs, γ lies in the interval 0.5 - 0.7. The last term of equation (1)

expresses the dependence of the capacitance on the density of states g of the 2D electron gas.

Recently the capacitance of gated GaAs/AlxGa1−xAs structures has been measured with a high accuracy by Hamp-

ton et al [4], both as a function of the gate voltage and of the in-plane magnetic field. The main results of the

measurements are, with a kind permission of authors, reproduced in figure 1. The aim of this paper is to present

a detailed analysis of the data based on the full self-consistent numerical calculation, going beyond the validity of

equation (1).

A band diagram of the charged gated structure is sketched in figure 2. Boundary conditions for the electrostatic

potential Φ(La) = 0 and dΦ/dz(La) = 0 correspond to the assumption of fixed impurity depletion charges and to a

charge neutrality of the whole structure. The gate voltage Vg is a difference between the chemical potential Eg

of the gate and the chemical potential EF/|e| of the inversion layer. At the metallic gate the chemical potential

is determined essentially by the electrostatic potential Φg since the difference is a constant built-in voltage of the

Schottky barrier VB. We obtain from the Poisson equation

F/|e|

Φg=db+ z

ǫ

|e|N + K .(2)

The constant K is a contribution to the potential due to frozen out impurity charges and due to the discontinuity at

the GaAs/AlxGa1−xAs interface. Consequently, the gate voltage is given by

Vg=EF

|e|+db+ z

ǫ

|e|N + K + VB

(3)

and the reciprocal value of the capacitance reads

C−1= C−1

b

+ C−1

c

(4)

where

C−1

c

=

d

dN

?z

ǫN +EF

|e|2

?

.(5)

1

Page 2

Thus, the gated structure capacitance can be again considered as a barrier capacitance Cband a channel capacitance

Ccin series. The reciprocal value of the channel capacitance is a sum of two terms: the electrostatic one, related to

the concentration dependent position of the channel centroid in GaAs and the thermodynamical one originating from

the concentration dependent chemical potential in the semiconductor. Both these quantities can also depend on the

applied magnetic field.

Up to now a very general analysis of the gated structure capacitance has been carried out regardless the detail

properties of the inversion layer. In the next step we summarize basic formulae describing the layer electron structure.

In a zero magnetic field the standard envelope function approximation yields the Schr¨ odinger equation for a con-

duction electron

?p2

2m− |e|Vconf(z) − E

?

ψ (r) = 0(6)

where the confining potential Vconf is a sum of the electrostatic potential Φ, obtained within the Hartree approx-

imation, and the exchange-correlation term Vxc, describing the effects of electron-electron interaction beyond the

Hartree approximation. We use the simple analytic paramertization of Vxcwithin the local-density-functional formal-

ism, suggested by Ruden and D¨ ohler [5]. Thus, the self-consistent solution to coupled Schr¨ odinger equation (6) and

Poisson equation for the electrostatic potential Φ gives the electron structure of the conduction channel, including the

many-body corrections to E0and z [6].

Looking for the eigenfunction of the equation (6) in the form

ψ(r) =

1

√Sexp(ikxx + ikyy)φn(z) (7)

we obtain 2D subbands of allowed energies

E(kx,ky,n) =¯ h2

2m(k2

x+ k2

y) + En.(8)

Equations (7) and (8) are the mathematical expression for the separability of the free in-plane (x,y-direction) and

bound out-of-plane (z-direction) components of the electron motion.

Firstly, let us assume only the lowest subband to be occupied. Then, due to the parabolic dispersion of the in-plane

energy, the 2D density of states per unit area g is a universal constant m/π¯ h2and the 2D concentration of the electron

layer reads

N = g(EF− E0) (9)

where E0is the lowest energy of the bound out-of-plane motion.

The average position z0of the electron layer is determined by the z-dependent part of the wavefunction as

z0=

?

z|φ0(z)|2dz .(10)

Now, the attention can be turned back to the heterostructure capacitance. Due to (9) and (10) quasi-analytical

relations for both the thermodynamical and electrostatic parts of the reciprocal channel capacitance are available. The

electrostatic term is related exclusively to the out-of-plane component of the electron motion. In the thermodynamical

term both components of the motion play a role. However, according to (9) their influence is independent and we

obtain the channel capacitance in the form

C−1

c

=z0

ǫ

+N

ǫ

dz0

dN+

π¯ h2

m|e|2+

1

|e|2

dE0

dN

. (11)

Different 2D gas concentrations (or gate voltages) directly yield different shapes of the confining potential Vconf

and thus the bound state eigenenergy E0 is modified. For confining potentials which do not depart substantially

from an exactly triangular well one obtains the scaling laws z0∝ N−1/3and E0∝ N2/3[3], from them the relation

dE0/dN ∝ z0 and the equation (1) is regained. The quantitative deviations can be expected for more realistic

shape of the confining potential of standard gated structures and, in the case of gated symmetric wells or inverted

structures, the above mentioned scaling law is not valid at all. Therefore both dE0/dN and z0have to be determined

independently in the course of the self-consistency procedure.

2

Page 3

In magnetic fields parallel to the heterostructure interface the separability between the in-plane and out-of-plane

components of the channel electron motion is lost. Assuming the magnetic field B ? y the wavefunction φ(z) becomes

kx dependent. As discussed in more details in [7–9], this results in a charge redistribution of the 2D gas and in a

modification of 2D subbands. The centroid z0of the channel first shifts to the bulk GaAs and then, after the magnetic

field reaches a critical value, it returns back to the heterostructure interface. The deviation of the energy dispersion

E(kx) from the parabolic shape yields a density of states which form depends both on the magnetic field and the

electron concentration [10]. Finally, the subband edge is shifted to higher energies by the in-plane field. It follows

from the above mentioned effects on the electron structure that the electrostatic term and both contributions to the

thermodynamical term are modified by the in-plane magnetic field in a rather complicated way.

A sufficient amount of information about the heterostructure capacitance has been given to open the last problem

we want to deal with. Except the very general formula (5) previous conclusions are restricted to 2D electron systems

with one occupied subband. Now we extend the discussion to multi-subband systems since the filling or depleting of

the higher subband is reflected in the observed capacitance quantum steps. A heterostructure where the number of

occupied subbands is changed by the gate voltage in the absence of the magnetic field will be considered. A discussion

concerning the in-plane magnetic field induced depopulation of higher subbands would be analogous.

In the case of two occupied subbands we obtain the channel capacitance

C−1

2c=z

ǫ+N

ǫ

dz

dN+

π¯ h2

2m|e|2+

1

|e|2

dE

dN

(12)

in the form similar to (11) but with z0and E0replaced by

z =z0N0+ z1N1

N

, (13)

E =E0+ E1

2

.(14)

The quantities N0= (EF− E0) m/π¯ h2and N1= (EF− E1) m/π¯ h2are the concentrations of electrons in the first

and second subband, N0+ N1= N.

Two terms in equations (11), (12) do not contain the derivatives with respect to N. The first one, proportional to

the centroid coordinate, does not cause any step when (11) is replaced by (12) after reaching the critical concentration

Ncsince z(Nc+ 0) = z0(Nc− 0). The step in the second term is a universal constant and reflect the fact that the

density of states is doubled above N = Nc. The additional steps come from the terms with derivatives which can be

expressed for N > Ncsimilarly as for the case of single subband occupancy with a help of dzi/dN, dEi/dN, i = 0,1.

The previous analysis shows the principal reasons for the gate voltage or in-plane magnetic field dependency of the

one-subband channel capacitance as well as for steps in capacitance when a higher subband is populated. To obtain

results comparable to experimental data full self-consistent numerical calculations, treating the electron-electron

interaction beyond the Hartree approximation both in zero and non-zero magnetic fields, are unavoidable since the

simple scaling rule leading to (1) is quite unrealistic in the presence of the magnetic field and for the double subband

occupancy.

Figure 1 presents data of low temperature capacitance measurements on an AlxGa1−xAs/GaAs sample with x = 0.3

and channel concentrations from about 1×1011to 4×1011cm−2. Since the shape of the confining potential depends

also on depletion charges parameters which are not exactly known we performed numerical calculations for Na =

4 × 1014cm−3and several frozen-in depletions lengths La. Due to an unambiguous relation between the gate voltage

dependency of the capacitance, including the position of the quantum step, and the shape of the quantum well it is

possible to use the experimental curve measured at By= 0 for determining La. This way we obtained La= 550nm.

(We do not discuss the height of the quantum step since a current leakage occurred at the gate voltage Vg= +150mV,

which might quantitatively modify the measured voltage dependency of the capacitance.) As shown in figure 3, if

the theoretical and experimental gate voltage dependencies at the zero magnetic field are in a good agreement, the

theory predicts, with a very reasonable accuracy, the behaviour of the capacitance for non zero in-plane magnetic

fields. The impurity scattering of electrons together with the roughness of the interface are responsible for smearing

of the quantum step, we simulate their influence by introducing the Dingle temperature TD≈ 1K. The effect of the

magnetic field on both the electrostatic and thermodynamical terms in the reciprocal channel capacitance is shown

in figure 4 to illustrate the violation of the formula (1). Note that the term 1/e2dEF/dN cannot be simplified due to

the field and concentration dependence of the density of states.

In conclusion, we want to point out that the capacitance measurements provide a lot of information about 2D

electron gas properties but the data have to be interpreted very carefully. It is well known that in structures shown

3

Page 4

in figure 3 the 2D channel with higher concentration is confined closer to the GaAs/AlxGa1−xAs interface resulting

in a smaller distance between electrodes. However, this effect is not reflected directly in the capacitance due to

simultaneous shift of bound states energies. Similarly, the charge redistribution by the in-plane magnetic field is

accompanied with the shift of subband edges. Moreover, the modification in the density of states contributes to

the magnetic field induced changes in the capacitance. Finally, the discontinuities in the 2D density of states are

responsible only partially for the capacitance quantum steps. We have seen that also the ratio between the filling of

subbands and the relative positions of electron layers in each subband play an important role.

We are indebted to J. P. Eisenstein for stimulating discussions concerning the subject of this paper and for enabling

us to use their experimental data prior to publication. We thank also to A. H. MacDonald who turned our attention

to this problem. This work has been supported by the Academy of Science of the Czech Republic under Grant No.

110414, by the Grant Agency of the Czech Republic under Grant No. 202/94/1278, by the Ministry of Education,

Czech Republic under contract No. V091 and by NSF, U. S., through the grant NSF INT-9106888.

[1] F. Stern, unpublished internal IBM technical report, 1972

[2] F. Stern, Phys. Rev. B 5, 4891 (1972).

[3] G. Bastard, Wave Mechanics Applied to Semiconductor Heterostuctures, (´Edition de Physique, Paris, 1990), p. 189.

[4] J. Hampton, J. P. Eisenstein, L. N. Pfeiffer and K. W. West, Solid State Commun. (in press).

[5] P. Ruden and G. H. D¨ ohler, Phys. Rev. B 27, 3538 (1983).

[6] F. Stern and S. Das Sarma, Phys. Rev. B 30, 840 (1984).

[7] J. M. Heisz and E. Zaremba, Semicond. Sci. Technol. 8, 575 (1993)

[8] T. Jungwirth and L. Smrˇ cka, J. Phys. C.: Condens. Matter 5, L 217 (1993).

[9] T. Jungwirth and L. Smrˇ cka, Superlattices and Microstructures 13, 499 (1993)

[10] L. Smrˇ cka and T. Jungwirth, J. Phys. C.: Condens. Matter 6, 55 (1994).

FIG. 1.

determined by Hampton et al [4].

Experimental curves of the capacitance as a function of the gate voltage and of the in-plane magnetic field

FIG. 2. Schematic band diagram for a gated GaAs/AlxGa1−xAs heterostructure.

FIG. 3. Self-consistently calculated gate voltage and in-plane magnetic field dependencies of the relative capacitance. For

B = 0 the results for three depletion lengths of acceptors are presented, Na = 4 × 1014cm−3. Magnetic field dependencies are

plotted only for selected gate voltages and La = 550nm to make comparison of experimental and theoretical curves easier.

FIG. 4. An increment of the reciprocal channel capacitance δC−1

decomposed in a thermodynamical term 1/e2dδEF/dN and two contributions N/ǫdδz/dN, δz/ǫ to the electrostatic term.

c

= C−1

c (B) − C−1

c (0) (dotted line) in the magnetic field,

4

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