Superlattices and Microstructures, Vol. 9, No. 1, 1991 23

MAGNETOQUANTUM EFFECTS IN TWO-DIMENSIONAL ACCUMULATION LAYERS OF SINGLE-BARRIER TUNNEL STRUCTURES

K.S. Chan*, F. W. Sheard, G.A. Toombs and L. Eaves

Department of Physics, University of Nottingham, Nottingham NG7 2RD, England

(received 13 August 1990)

A two-parameter variational wave function is used to calculate the properties of a two-dimenslonal accumulation layer in an (InGa)As/InP single-barrier tunnelling heterostructure under an applied bias. This model is used to describe the effect of a quantizing magnetic field applied perpendicular to the semiconductor layers. Using a Gaussian- broadened density of states to describe the Landau-level structure, the magnetoquantum oscillations in the sheet density, Fermi energy and tunnel current are calculated. The model accounts for the general features of the observed oscillations. The contribution of the density of states to the magnetocapacitance oscillations in tunnel structures is also discussed.

In Fig. i we show the conduction-band edge of a n-(InGa)As/InP/n+(InGa)As structure under bias in which, at liquid-hellum temperatures a degenerate two-dimensional electron gas (2DEG) is formed in the accumulation layer adjacent to the InP barrier. The 2DEG can be studied by using the small tunnel current flowing through the barrier as a probe [1-3].

When a high magnetic field B is applied perpendicular to the plane of the 2DEG, the in- plane kinetic energy of the electrons is quantized into Landau levels. This gives rise to magnetoquantum oscillations in the tunnel current when the field is swept at constant bias. The periodicity of the oscillations has been deduced by arguing that current minima occur when the Fermi level of the 2DEG lies in a gap between Landau levels [i]. The condition that the Fermi level lies midway between the (n-l)th and nth levels (n - I, 2, 3...) is n~ c = EF, where ~= - eB/m* is the cyclotron frequency, m* the effective mass and E F the Fermi energy of the 2DEG. This gives oscillations periodic in l/B, current minima occurring at fields B - B~/n, where B~ - m*EF/e~ , and has been used to determine the Fermi energy E F and sheet density n - eB~/~ of the 2DEG. This argument assumes a constant Fermi energy independent of B. However, if modulation of the current with increasing B is

* Present address: Department of Applied Science, City Polytechnic of Hong Kong, Kowloon, Hong Kong.

due to modulation of n, as Landau levels pass through the Fermi level, this charge modulation will change the subband wave function and energy, which will hence feed back to affect the Fermi energy and charge density. Thus magneto-oscillations in charge density and tunnel current must be determined self-conslstently.

EF I _ _ ~ ---" . . . . . . . .

eV

Z Z=O , n-(InGa)As InP n+(InGa)As

Fig. i Conduction-band profile of (InGa)As/InP tunnel structure under bias voltage V.

0749-6036/91/010023+03 $02.00/0 © 1991 Academic Press Limited

2 4 Super/attices and Microstructures, Vol. 9, No. 1, 1,997

To describe the bound state of the

inversion layer in a Si-MOSFET, the

one-parameter variational wave function introduced by Fang and Howard [4] has been

widely used. But for the accumulation layer in the n-(InGa)As layer of the structure in Fig. i, it leads to a misaligument between the Fermi level of the 2DEG and the Fermi level of the bulk n- layer (which is close to the band edge

since the donor binding energy is very small). To remedy this problem and retain the analytic simplicity of the Fang-Howard approach, we propose to use a similar two-parameter

variational wave function

*(z) ~a3/2 exp (z > -sa),

and ~ = 0 for z < -sa. Here the tunnel barrier interface is at z - 0 (Fig. i). The parameter a is related to the width of the charge

distribution and s is a shift parameter related to the penetration of the wave function into the barrier region (z < 0). We note that this wave function does not have an exponential tail inside the barrier. Since the probability of finding the electron in the barrier is small we do not expect this feature to significantly affect the results.

For this wave function we then solve Poisson's equation for the electric potential variation through the structure, assuming for simplicity that the depletion charge in the right hand n ÷ contact is a sheet of infinitesimal thickness. Hence we obtain the

total energy U of the system, which is the sum of the kinetic energy of the 2DEG electrons due to subband motion and in-plane motion and the

Coulomb potential energy. For a system under a given bias voltage V we must minimize the grand

potential ~ = U - n e(V - Vfb), where Vfb = -EFR/e is the fiat-band bias (EFR is the Fermi energy in the right hand n + contact as shown in Fig. I) with respect to variations in the parameters a, s and n . The minimization must be carried out subject to the conditions that the applied bias is held constant and the Fermi level of the 2DEG aligns with the Fermi level in bulk n-(InGa)As layer (taken to be at the band edge). This assumes that the tunnelling

rate through the barrier is small compared with

the diffusion rate of electrons through the n-(InGa)As layer, so that the thermodynamic state determined by the minimization procedure is a good description of the 2DEG under bias. The influence of the magnetic field occurs through the expression

n s = [IFg(E)dE,

where E is the in-plane kinetic energy of a 2DEG electron and g(E) is the 2D density of states (DOS) per unit area. For g(E) we use a Gaussian-broadened function for each Landau

level with width proportional to #B, together

with a constant background !5,6].

Detailed calculations were made for an (InGa)As/InP structure with barrier thickness 20 nm taking a conduction-band offset of 200

meV, effective mass m* = 0.045m e and dielectric permittivity E r = 12.5 (for both materials). As expected, the variation of n s with voltage was slightly nonlinear since the slope (differential capacitance) increased with voltage owing to the decrease in the mean offset distance of the 2DEG charge from the

barrier interface. We discuss the magneto-oscillations for a

fixed applied voltage of 150 mV corresponding to a sheet density n s = 2.9 x 1012 m -2. In

Fig. 2 the sheet density is plotted versus field B. It is also shown that the magneto-oscillation amplitude is much smaller than would be obtained if the Fermi energy were

fixed at the zero-field value. We can understand this in the following way. If E F is fixed then, when the Fermi level is near the centre of a Landau level (points marked A in Fig. 2), an increase in B forces a large number of electrons out of the 2DEG. But this decrease in charge density increases the width of the subband wave function (parameter a) and depresses the subband energy thereby increasing the Fermi energy, which tends to compensate for the decrease in electron density. Because of

this feedback mechanism, an increase in magnetic field reduces the electron density much less than would be the case for fixed

3 . 4

IE 3 . 0

o

2 . 6 t -

O

i I i i i ~ /

/ ! / t

l

I ,'

L ' i l l q V ' i

I / 2 I

/

"" ,t i I i i

10 2 0

~c (meV)

Fig. 2 Variation of sheet density n s with cyclotron energy ~c at fixed bias of 150 mV (~c = i0 meV corresponds to B = 3.9 T). Dashed curve gives result assuming fixed Fermi energy. Triangles denote fields when 2DEC Fermi level is in middle of Landau level, squares when Fermi level is midway between Landau levels.

Superlattices and Microstructures, Vol. 9, No. 1, 1991 25

r~

v --}

1 I l I

I I I i I

0 10 20

~ c (meV)

Fig. 3 Magneto-oscillations in tunnel current density J.

Fermi energy. This picture is confirmed by explicit calculations which show that the magneto-oscillations in E F are in antiphase with the oscillations in n .

To obtain the tunnel current we have calculated the transmission coefficient T in the WKB approximation and an attempt rate v from the average kinetic energy of the subband state. The current density is then given by J

envT. As n increases, v - h/m*a 2 increases also and so does T owing to the increased field across the barrier. Thus the magneto- oscillations in current are expected to follow closely those in electron density and this is indeed the case as shown in Fig. 3. Analysis of the current minima shows that they occur at fields given by B = Bf/(n + 4), with phase factor 4 - 0.i, whereas the empirical theory, which ignores modulation of the Fermi energy, gave ~ - 0. Nevertheless the fundamental periodicity field Bf can still be used to obtain the zero-field sheet density.

Recently measurements of the differential capacitance of modulation-doped GaAs/(AIGa)As heterostructures have been used to determine the DOS of a 2DEG in a magnetic field [3,5,6]. In the analysis of these experiments it was assumed that the charge density and width of the 2DEG were independent of magnetic field. As we have seen, for tunnelling structures this is not the case. Using our model we have derived the magnetocapacitance C = 6Q/6V by considering small changes 6Q = e6n and 6V in charge density and bias voltage. If we neglect penetration of the wave function into the barrier, the result may be written

i b + BZo 7 - - + - -

C ~r% e2g(EF )

'E , 3

7 o

O 2

Fig. 4 capacitance per

I I

1i

I [ p

i [ I 1 i

10 20

~ c (meV)

Magneto-oscillations in differential unit area C. Dashed curve

gives result assuming fixed sheet density.

where b is the barrier width, z o - 3a/2 is the mean offset distance of the 2DEG from the barrier interface and g(EF) is the DOS at the Fermi level. The parameters ~ and 7 are also charge dependent and for n, - 3 x l0 Is m -2 (a - i0 nm), ~ - 0.5 and 7 - 1.5. The constant charge model corresponds to 7 - i. Thus, in a tunnelling structure the contribution of the DOS to the capacitance is modified due to rearrangement of the charge distribution as the voltage (or magnetic field) is altered. The capacitance oscillations are shown in Fig. 4, in which comparison is also made with the results of a calculation assuming a fixed sheet density. At high fields the contribution due to modulation of the width of the 2DEG is comparable with the direct contribution due to the DOS.

Acknowledgement - This work is supported by the UK Science and Engineering Research Council.

[I] L. Eaves, B.R. Snell, D.K. Maude, P.S.S. Guimares, D.C. Taylor, F.W. Sheard and G.A. Toombs, Proc. Int. Conf. on Physics of Semiconductors, Stockholm (World Scientific) p. 1615 (1986).

[2] T.W. Hickmott, Phys. Rev. B32, 6531 (1985). [3] D. Weiss, K. von Klitzing and V. Mosser,

Two-dimensional Systems: Physics and New Devices (Eds. G. Bauer, F. Kuchar and H. Heinreich) Springer Series in Solid State Sciences 67, 204 (1986).

[4] F.F. Fang and W.E. Howard, Phys Rev. Lett. 16, 797 (1966).

[5] V. Mosser, D. Weiss, K. yon Klitzing, K. Ploog and G. Weimann, Solid St. Commun. 58, 5 (1986).

[6] T.P. Smith, B.B. Goldberg, P.J. Stiles and M. Heiblum, Phys. Rev. B32, 2696 (1985).