Communications in Physics, Vbl. 10, No. 1 (2000), pp. 8-14

DENSITY OF ELECTRON STATES IN MODULATION-DOPED SEMICONDUCTOR QUANTUM WIRES NGUYEN W E N TUNG AND TRAN DOAN HUAN Institute of Engineering Physics, Hanoi University of Technology

Abstract. We calculate the effect of impurity modulation doping on the one-dimentional election gas in a cylindrical semiconductor quantum wire, based on the semiclassical theory recently developedfor disordered low-dimensional elecfvonsystems. The calculation is carried out with the use of subband wave&nctions due to Gold-Ghazali describing a realistic (non-unform) distribution of electmns in the wire section. Plots illustrating the effect and comparison with the earlier resultsfor the uniform electmn distribution are given.

I. INTRODUCTION

There has been a great deal of recent interest in semiconductor quantum wire (QWR) structures, where the motion of electrons is essentially onedimensional [I] The quasione-dimensional electron gas, hereafter called for short a one-dimensional electron gas (IDEG), has been intensively studied both experimentally and theoretically The QWR structures have opened up the potential for different device applications In practice, intentional or unintentional doping is often inevitable The lDEG is generally strongly affected by disorder caused by a random field created by charged imvurities chaoticallv distributed in the sample . 12 - 41 The disorder has been shown to lead to considerable modifications in the energy spectrum of the lDEG, e g., smearing out the square-root singularity of the density of states (DOS) characteristic of the ideal lDEG [5, 61. Obviously, these in turn result in remarkable changes in many phenomena occurring in the wire, e.g., optical absorption. So far, only a few theoretical investigations have been made in order to understand the DOS of disordered IDEG's in QWR's, and merely numerical results have been available in the literature [ 5 , 6 ] . The reason is probably that the input function for disorder interaction was chosen to be the potential created by a single impurity. The single-impurity potential is screened by interacting electrons and is generally seen very complicated for realistic 1DEG's. Recently, an analytic theo~yof the DOS of disordered low-dimensional electron systems has been developed in [7 - 91. There, the disorder effect from impurity modulation doping on the ID DOS has been estimated, assuming the electrons uniformly distributed in the wire section, which is clearly irrelevant to realistic QWR's.

.

~=

DENSITY OF ELECTRON S X E S IN MODULATION-DOPED

9

Thus, the aim of the present paper is to calculate the disorder effect of impurity modulation doping on the DOS of the lDEG in a QWR, however, taking into account the realistic (non-uniform) electron distributior~ To this end, we will apply the 1D version of the semiclassical approach to disordered electron systems subjected to smooth random fields [9]

. .

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11. BASIC RELATIONS

To start with, we shall collect the formulae to be used for evaluation of the DOS of the disordered 1D EG in a semiconductor QWR. The electrons in the wire are considered to be confined in two dimensions, e.g., in they and z directions, and to move freely in the x direction (chosen as the wire axis). The disorder in the wire is normally caused by some random field U(x)affecting the motion of the electrons along the wire axis. In what follows, we will discuss the case when U(x)is a Gaussian field and, therefore, it may completely be described by an autocorrelation function, defined by [lo]

where the angular brackets stand for the averaging over all configurations of the disorder The disordered electron system is assumed ~nacroscopicallyhomogeneous, the autocorrelation functlon depending merely on the coordinate difference As usual, the strength of the random field is determined by the rms of its potential, y and of its force, F, given by

y2 =< u2>= W ( x - x'),=,I

(2)

and f?2

=< (VU)'>= vzV51W(3: - x')/z=ll.

(3)

The average potential is supposed small compared with the energy separation between neighboring subbands, so that the intersubhand scattering induced by disorder is negligible and the theory may be formulated in a one-subband approximation. Furthermore, we will assume that the random field in question obeys the following inequality:

%.

-

where m means the effective mass of the charge carriers with a parabolic subband. Under inequality (4) the field U ( x )and, hence, its autocorrelation function W (x- x') are varying slowly on the average in the x direction, so that a semiclassical approach is applicable to it It is well known [lo] that the DOS is the most adequate concept for describing the energy spectrum of disordered systems Recently, it has been pointed out [9] that in analogy

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NGUYEN HUYEN TLNG AND TRAN DOAN EUAN

with 2D and 3D cases [7, 8, 111, within the semiclassical approach to a smooth Gaussian field the DOS of disordered 1D electron systems may be, as expected, represented by an expansion with respect to the small quantity li2F2/4my3 In the lowest order, we obtain

with D,(x) being a parabolic cylinder function [12]. The rms of the potential and of the force of the random field in Eq. (5) are rewritten in terms of the Fourier transform of the autocorrelation function of the random field as

and

Thus, Eqs (5) - (7) set up a basis for studying the DOS of 1DEG's in semiconductor QWR's in the presence of Gaussian random fields In these equations, the input function describing the disorder interaction is the autocorrelation function, which may capture the microscopic details of both the origin of the disorder and geometry of the wire [2, 3, 91 111. MODULATION IMPURITY DOING IN CYLINDRICAL CEMICONDUCTOR QWR's For further calculation of the ID DOS in question, we have to find out the autocorrelation W ( k )for the system to be treated. So, we must specify our model by choosing a circular cylinder and confining the motion of the electrons in the cylinder by an infinite potential barrier at its surface [2, 13, 141 Accordingly, the wave functions of 1D subbands are proved to be given in terms of Bessel functions. The wave function of the lowest subband is of the form [2].

where R is the radius of the wire, Jo(x)is the zeroth-order Bessel function [12] Moreover, at zero temperature almost electrons are assumed to occupy this subband (extreme quantum limit), as evidenced experimentally [15] It is well known [lo] that the random field created by all ionized impurities is generally considered to be smooth at a high doping level In the case of modulation doping, the impurities are supposed to be randomly distributed in an infinitely thin tube coaxial with the wire The autocorrelation function of the total impurity field can then be given

DENSITY OF ELECTRON STATES IN MODULARON-DOPED.

in wave-vector space by the following expression [2,9]:

W ( k ) = ni--v:i(k)

t"k) '

( k = Ikl),

with n, being the 1D impurity density along the wire Here v,,(k) denotes the Fourier transform of the single-impurity potential, and t ( k ) the statistic dielectric function allowing for screening by interacting ID electrons It has been pointed out [16] that in contrast to the viewpoint that the random phase approximation QWA) becomes progressively worse in lower dimensions, it turns out to be a more accurate approximation in 1D than in 2D and 3D electron systems Within the RPA, the screening function for the 1DEG at zero temperature may be written as [2, 141

Here v,,(k) means the Fourier transform for the x direction of the electron-electron interaction potential, and the Fermi wave vector is fixed by the 1D carrier density n, via kl;. = (.1r/2)ne

It is well known [2, 91 that because of a finite extension of electron states in the wire cross section, the electron-impurity and electron-electron interactions figuring in Eqs (9) and (10) are to be weighted with the 1D-subband wave function After [9, 13, 141, the lowest-subband wave function was, for simplicity, chosen to be a constant in the directions y and z , which implies a uniform distribution of electrons in the wire section, i e ,

Instead of the above simplifying assumption, Gold and Ghazali [2] have proposed the following wave function for the lowest subband:

This function is proved to be a very good approximation to the exact wave function (a), however, in contrast to the latter, the former enables to get analytic results for the electronimpurity and electron-electron interaction potentials. Indeed, upon employing the wave function (ll), for the effective electron-impurity interaction we obtain [2]:

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NGUYEN HUYEN TUNG AND TRAN DOAN HUAN

Here Z denotes the charge of an ionized impurity in units of the electron charge, EL is the dielectric constant of the background lattice, and p the radius of the impurity tube. In what follows, I,(x) and K,(x)are the nth-order modified Bessel functions of the first and second kind, respectively [12] Furthermore, for the effective electron-electron interaction we have [2]

Thus, the autocorrelation for modulation doping is provided by Eqs. (9), (lo), (13), and (14).

I\! RESULTS AND DISCUSSIONS To illustrate the theory, we have carried out numerical calculations for cylindrical QWR's made of n-type GaAs at zero temperature, whose conduction subband is considered. The material parameters are the effective mass m = 0.067 m e and the dielectric constant EL = 12.9. our results are more general. This is because that the natural scales for the length, the energy, and the 1D DOS are atomic units: the effective Bohr radius a* = tLli2/me2, the effective rydberg Ry*=me4/2tili2, and p* = l/Ry*a*, respectively. For GaAs wires, we have a* = 100 A, Ry* = 5.6 m e y and p* = 1.79 x lo5 meV-lcm-l. The DOS of the disordered lDEG in a QWR is determined by Eq. (5), where the strength of the random field y,F are given by Eqs. (6), (7) in terms of its autocorrelation function (91.

ow ever,

Fig.1.

DOS p ( E ) in units of p* vs. energy for the lDEG in a modulation-doped QWR of a radius of the wire R = 3a* and of the impurity tube p = O (the impurities lying on the wire axis). The DOS is plotted under: (a) an electron density n, = 105 cm-' and different impurity densities ni = SO4, SO5, and lo6 cm-l, and (b) an impurity density ni = 106cm-' and various electron densities n, = lo4, SO5,and 106 cm-l. Dashed and solid lines refer to the uniform and non-uniform distribution of electrons, respectively. The dotted line represents the DOS of the ideal IDEG.

DENSITY OF ELECTRON SEXES IN MODULATION-DOPED...

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Figure 1 shows the DOS p ( E ) for a radius of the wire R = 3a* h d of the impurity tube p = 0 (the impurities lying on the wire axis) with the use of the electron distributions: uniform (dashed lines) and non-uniform (solid lines). In Fig. l a those are plotted under an electron density n, = lo5 cm-I and different impurity densities ni = lo4, lo5, lo6 cm-I, whereas in Fig. l b under a value of ni = lo6 cm-' and various values of n, = lo4, lo5, lo6 c m l . The DOS of the ideal lDEG is also given by a dotted line.

Fig.2. DOS p ( E ) in units of p* vs. energy for the lDEG in amodulation-doped QWR of a radius of the wire R = 3a' and of the impurity tube p = R (the impurities located on the wire surface). The interpretation is the same as in Fig. 1.

Figures 2 displays the DOS p ( E ) for a radius of the\Nire R = 3a* and of the impurity tube p = R (the impurities located on the wire surface). The interpretation is the same as in Fig 1 Figure 3 sketches the DOS p ( E ) for a radius of the wire R = 3a* and of the impurity tube p = 2R (the impurities located outside the wire) The interpretation is the same as in

Fig.3.

DOS p ( E ) in units of p' vs.energy forthe lDEG in amodulation-doped QWR of a radius of the wire R = 3a* and of the impurity tube p = 2R (the impurities located. outside the wire). The interpretation is the same as in Fig. 1

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NGUYEN HUYEN TUNG AND TRAN DOAN HUAN

From the lines thus obtained we may draw the following results. (i) Figures. 1, 2, and 3 indicate that the electronic energy spectrum is drastically changed due to disorder from impurity doping. The disorder gives rise to a band tail (of localized states) extending deep below the subband edge. (ii) A comparison of the solid (or dashed) lines of Figs. 1, 2, and 3 reveals the disorder effect depends considerably on the impurity position. The effect is larger when the impurities are more concentrated around the wire axis. (iii) An examination of the solid (or dashed) lines in Figs. la, 2a, and 3a reveals, as expected, the DOS tail is larger and more extended below the subband edge when elevating the impurity density. (iv) A comparison of Figs. lb, 2b, and 3b shows that the DOS depends remarkably on the electron density through the screening effect by 1D interacting electrons. (v) An inspection the solid and dashed lines in Fig. 1 (or Figs. 2 and 3) shows that at the used (high) electron densities theuniform electron distribution described by Eq. (11) turns out to be a rather good approximation to the non-uniform one described by Eq. (12).

ACKNOWLEDGMENT The author would like to thank Prof. DoanNhat Quang formany helpful discussions.

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Received 14 October I999