Published by NATIONAL CENTRE FOR NATURAL SCIENCE AND TECHNOLOGY OF VIETNAM

Volume 18, Number 2

Jtrne 2000 Page

Contents

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Nguyen Vien Tho and Nguyen Van Thuan Motlcn of color charge in Schwarzschlld like gauge field. Huynh Thanh Dat, Le Phuong Thao, Duong Ai Phuong, and Nguyen Van Den The study of removal proc@ssof llgnin from wood by using FT-Ramanspectroscopy. Nguyen Huyen Tung cnd Tran Doan Huan - Effect of background doping on the density of states of 10 electron gases in quantum wires. Nguyen Ngoc Son, Tran Due Thiep, Nguyen Van Do, and Truong Thl An Systematics of mass distribution in low energy fission. Nguysn Ba An Multivalued steady state solutions It-, lasers coupled by an Intensity-dependent c;oss-loss mechanism. Tran Anh Vu Some conslderatlon in the design of a pohlarful TE-nitrogen laser.

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Phan Hong Lien

- l h e Abelian anomaly in the chiral current for nuclear slgma-

omega model. Tran Kim Anh, Pharn Thi Minh Chau, Nguyen Vu, Le Quoc Minh, Pham Hong Duong, Nguyen Trong Oanh, Tran Thu Huong, and Charles Barthoir Luminescence of ad\.nnced inorganic - organic - rare earth Ions materials and their applicution.

Communications inphysics, Vol. 10, No. 2 (2000), pp. 78-85

EFFECT OF BACKGROUND DOPING ON THE DENSITY OF STATES OF ID ELECTRON GASES IN QUANTUM WIRES N G W E N H W E N TUNG AND TRAN DOAN HUAN Institute qf Engmeering Physics,Hanoi University of Echnologq! Honor Abstract. We calculate the eflect arrs~ng~from background doprng on the densrty of states

oj the one-dimenszonal electron gns (IDEG) In a cylrndrrcal sernrconductor quantum wzre The calculatron u carrzed out wrthln a ID versron of the semrclassrcal approach to smooth randomjelds wrth the use ofsubband wave finctions due to Gold-Ghazalz descrrbrng a realrstrc (non-unlform) driistrrbutron of electrons rn the wzm cross sectron Plots rllustratrng the effect rn quantum wrres ofGaAs and comparzson wrth the earlrer msult are grven.

I. INTRODUCTION

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There has been a great deal of recent interest in semiconductor quantum wire (QWR) structures, where the motion of electrons is essentially restricted to be one dimensional [I] The quasi-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 IDEG is generally strongly affected by disorder caused by a random field created by charged impurities chaotically distributed in the sample 2-41, 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 1DEG [56 Evidently, these in turn result in remarkable changes in many phenomena occurring in the wire, e g , optical absorption Recently, an analytic theory of the DOS of disordered low-dimensional electron systems has been developed [7-101 There, the disorder effect from impurity doping on the 1D DOS in QWRs has been estimated, assuming the electrons uniformly distributed in the wire cross section. This is obviously irrelevant to realistic QWRs where the electrons are mainly concentrated around the wire axis [2]. Recently, we have involved the non-uniform electron distribution in the calculation of disorder effect from modulation doping on the 1D DOS when the impurities are placed on a cylindrical surface [ l l ] Thus, the aim of the present paper is to calculate the disorder #

EFFECT OF BACKGROUND DOPING ON THE...

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effect from background doping on the I D DOS when the impurities are distributed in a cylindrical tube of finite thickness, taking into account the realistic electron distribution. 11. ELECTRON DISTRIBUTION IN CYLINDRICAL QWRs

To start with, we must specify our model by choosing a circular cyIinder and confifiing the motion of the electrons in the cylinder by an infinite potential barrier at its surface [2,12]. Accordingly, the wave functions of I D subbands are proved to be given in terms of Bessel functions J,(x) [I:], e.g., for the lowest subband:

with r = jr/ This function describes a non-uniform electron distribution in which the particles are mainly concentrated around the wire axis [2] Moreover, at zero temperature almost all electrons are assumed to occupy this subband (extreme quantum limit), as shown experimentally [I?] For similicity, Lee and Spector [12] have chosen the lowest-subband wave function to be a constant in the directions y and z, viz

This implies a uniform distribution of the electrons in the wire cross section, which is clearly irrelevant to the above-mentioned real distribution. However, a merit of the LeeSpector wave function is that it enables one to get analytic results for the electron-impurity and electron-electron interaction potentials Consequently, w e may have developed an analytic theory both of the mobility [12,15] and the DOS [9,lO] in cylindrical QWRs Instead of the above simplifying assumption, Gold and Ghazali [2] have proposed the lowest-subband wave function as follows.

This function is proved to be a very good approximation to the exact wave function given by Eq. (I), on the other hand, in contrast to the latter, the former has the merit of the Lee-Spector wave function, i.e , it enables one to get also analytic results for the electronimpurity and electron-electron interaction potentials.

In. BACKGROUND DOPING IN CYLINDRICAL SEMICONDUCTOR QW& Next, we arqconcerned with a background-doped sample in which the impurities are distributed absoIutely randomly but uniformly on the average in a tube o f finite thick-

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NGUYEN HUYEN TUNG AND TRAN DOAN H U M .

ness and coaxial with the wire. Then, the autocorrelation function of the total impurity field can be cast in wave-vector space into the following form [16,17] :

where v ( k ,pi) denotes the Fourier transform in the x direction of the one-impurity potential, p, and p, are the inner and outer radius of the impurity tube, and n~is the 3D impurity density. The impurity potential is to be screened by interacting electrons in the 1DEG. This can be quantified by introducing a static dielectric function as .

where v,: ( k ,p,) is the unscreened one-impurity potential. It is well known [2] that the impurity potential is to be modified by a finite transverse extension of the subband wave function 4 ( r )in the y and z directions, i.e., weighted as

with Z being the charge of an ionized impurity in units of the electron charge e . The Coulomb potential figuring in Eq. (6) describes bare interaction between an electron at r and an electron at r', given by

with E L being the dielectric constant of the background lattice. In what follows, In(x)and Kn(x)are the nth-order modified Bessel functions of the first and second kind, respectively [13]. It has been pointed out [18] that in contrast to the viewpoint that the random phase approximation (RPA) becomes progressively worse in lower dimensions, it turns out to be a more accurate approximation in ID than in 2D and 3D electron systems. Within the RPA, the screening function for the lDEG at zero temperature may be written as [12,15]

Here vee(k)is the Fourier t r a n i f o ~in the a: direction of the electron-electron interace is fixed by the 1D carrier density ne via kF = tion potential, and the ~ ~ r @ i h q a vvector (7r/2)ne.The electron-elecbn interaction is also to be weighted with the lowest-subband

EFFECT OF BACKGIlOUND DOPING ON 'IXE...

wave function as [2]

Now, we have to insert Eq (3) for the Gold-Ghazali wave function and Eq. (7) for the bare electron-impurity interaction into Eq. (6), and expand Ko(klpz- r / )in products of Bessel functions with the help of a summation theorem [I31 For doing the resulting integrals, the following mathematical formulae turn out to be useful [13]

with Rev > -1. As a consequence, for the effective electron-impurity interaction we obtain:

where n = kR and 6, = kp,. Analogously. inserting Eq. (3) into Eq (9) yields the effective electron-electron interaction

According to Eq. (4), the autocorrelation function far doping depends on the geometry of the impurity system In what follows, we distinguish between two limiting cases of interest inside uniform background doping with p, = 0 , p , = I< (model U1) and outside uniform background doping with p, = R (model U2) Next, we must put Eqs (5) and (1 2) into Eq (4) and do the appearing integrals with the aid of Eqs. (lo), (11) and the following formulae [19]

with Rev > -1, and

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NGUYEN HUYEN TUNG AND TRAN DOAN HUAN.

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with a > 0 Consequently, we may present the autocorrelation for inside uniform doping in an analytic form:

The autocorrelation for outside uniform doping reads as

C(Q){a2[K?(Q)- K;(Q)I - & [ K ; ( s M )

-K;(SM)]), (17)

where 6M = kp,,

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IV EFFECT OF BACKGROUND DOPING ON THE 1D DOS OF QWRs

It is well known 116,171 that the random field created by all ionized impurities is generally considered to be smooth at a high doping level. Then, the total impurity field is a Gaussian one. Hereafter, we will assume that the field in question obeys the following inequality:

where rn means the effective mass of the charge carriers with a parabolic subband, y and F are the rms of the random potential and of the random force, respectively. It has been pointed out [7,9,17] that under the smoothness condition (18) a semiclassical approach to the field is applicable. As a result, the DOS of disordered 1DEG's may be represented in terms of an expansion with respect to the small quantity ti2F2/4my3.In the lowest order, we obtain

with U,(x) being a parabolic cylinder function [I31 . The rms of the potential and of the force figuring in [9] are rewritten in terms of the Fourier transform W (k) of the autocorreiation function of the random field as

EFFECT OF BACKGROUND DOPNG ON TI--E

and

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Thus, in the present scheme for the DOS calculation the input function describing the disorder effect is the autocorrelation function For background doping, this is given by Eqs. (16) and (17). To illustrate the forgoing theory, we ha\: carried out numerical calculations for 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 meand the dielectric constant E L = 12.9. The natural scales for the length, the energy, and the I D DOS are atomic units: the effective Bohr radius a* = c Lh2/mc" the effective rydberg ~ ~ * = r n e ~ / 2 c ~ f i and p* = l/Ry*a', respectively For GaAs wires, we have a* = 100 A, Ry* = 5.6 me\! and p* = 1.79 x lo5 r n e ~ - ' c m ~ ' Figure 1 shows the DOS p ( E ) in units of p* for the IDEG in a wire of radius R = a* under model U1 (inside background doping) and different electron distributions: uniform (dashed lines) and non-uniform (solid lines) In Fig. l a the DOS is plotted under an electron density n, = lo4 cm-I and different I D impurity density n, = x R 2 n I = lo1', lo5, lo6 cm-', whereas in Fig. l b under a value of r ~ ,= lo6 cm-' and various values of n, = lo4, lo5, l o 6 cm-' The DOS of the ideal lDEG is also given by a dotted line.

Fig.1. DOS p(E) in units of p* vs energy for the lDEG in a QWR of radius R = a' under model U l (inside background doping. The DOS IS plotted under: (a) an electron density n, = lo4 cm- and different 1D impurity densibes n, = lo4, lo5, and 10" cm-l, and (b) an impunty density n, = lo6 cm-I and vanous electron densties n, = lo4, 1G5, and lo6 cm-I. Dashed and solid lines,r t - h to the uniform and nonuniform distfibut~onof elec%fons,respci:;vz!y The dotted line represents the DOS of the ideal 1DEG. +

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.NGUYEN I - I U m TUNG AND TRAN DOAN W A N

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Fig.2. DOS p(E) In units of p* vs energy for the lDEG in a QWR of radius R = a* under model U2 (outside background doping) wlth outer rad~usp, = 6R The interpretation is the same as In Fig. 1.

Figures 2 displays the DOS p ( E ) in units of p* for the IDEG in a wire of radius a* under model U2 (outside background doping) with outer radius p, = 6R. The interpretation is the same as in Fig. 1. From the curves thus obtained we may draw the following results (i) Figures. 1 and 2 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) An examination of the solid (or dashed) lines in Figs l a and 2a reveals, as expected, the DOS tail is larger and more extended below the subband edge when elevating the impurity density. (iii) An inspection of Figs l b and 2b shows that the DOS depends appreciably on the electron density through the screening effect by 1D interacting electrons (iv) A comparison of the solid and dashed lines in Fig 1 (or Figs 2) shows that at a high electron density the uniform electron distribution, described by Eq (3) turns out to be a rather good approximation to the non-uniform one, Eq (1)

I?

=

ACKNOWLEDGMENT The authors would like to thank Prof. Doan Nhat Quang for many helpful discussions.

REFERENCES 1 C. Wclsbuch and B. Vinter, Quantum Semzconductor Structures, Academ~c.San Diego, 1391. 2. A. Gold and A. Ghazali, Phys. Rev. B 41 (1990) 7626. 3 J. S. Thakur and D. Neilson, Phys. Rev. B 56 (1997) 4679, ~bzd.56 (1997) 7485.

EFFECT OF BACKGROUND DOPING ON THE...

4. J. Motohisa and H. Sakaki, Appl. Phys. Lett. 60 (1992) 1315. 5. M. Takeshima, Phys. Rev. B 33 (1986) 7047. 6. A. Ghazali, A. Gold, and J. Sene, Semicond. Sci..Echnol. 8 (1993) 1912. . 7. D. N. Quang and N. H. Tung, Phys. Status Solidi B 207 (1998)-111. 8. D. N. Quang and N. H. Tung, Phys. Status Solidi B 209 (1998) 375. 9. D. N. Quang and N. H. Tung, Phys. Rev. B 60 (1999) 13648. 10. D. N. Quang and N. H. Tung, Phys. Rev. B (submitted 1999). 11. N. H. Tung and T D. Huan, Commun. in Phys. 10, No. 1 (2000), 8. 12. L. Lee and H. E Spector, J. Appl. Phys. 57 (1985) 366. 13.1. S. Gradshteyn and I. M. Ryzhik, Table oflntegrals, Series, andpmducts, Academic, New York, 1980. 14. A. R. G8ni, A. Pinczuk, J. S. Weiner, J. M. Calleja, B. S. Dennis, L. N. Pfeiffer, and K. W. West, Phys. Rev Lett. 67 (1991) 3298. 15. G. Fishrnan, Phys. Rev. B 34 (1986) 2394. 16. E Van Mieghem, Rev Mod. Phys. 64 (1992) 755. 17. D. N. Quang, N. N. Dat, anbD. V. An, J. Phys. Sbc. Japan. 66 (1997) 140. 18. Q. I! Li and S. Das Saxma, Phys. Rev Lett. 68 (1992) 1750; Phys. Rev. B 48 (1993) 5469. 19.. A. E Prudnikov, Yu. A. Brychkov, and 0. I. Marichev, Integrals and Series, Vol. 2, Gordon & Breach Science Publishers, New York, 1986.

Received I0 December 1999