semiclassical approach to the density of states of the disordered electron gas in a quantum wire

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PHYSICAL REVIEW B 15 NOVEMBER 1999-IVOLUME 60, NUMBER 19

Semiclassical approach to the density of states of the disordered electron gas in a quantum wir

Doan Nhat QuangCenter for Theoretical Physics, National Center for Natural Science and Technology, P.O. Box 429 Boho, Hanoi 10000, Viet

Nguyen Huyen TungInstitute of Engineering Physics, Hanoi University of Technology, 1 Dai Co Viet Road, Hanoi, Vietnam

~Received 23 December 1998!

A theory is given of the density of states~DOS! of the quasi-one-dimensional electron gas~1DEG! in asemiconductor quantum wire in the presence of some random field. For a smooth random field, the derivationis carried out within Gaussian statistics and a semiclassical model. The DOS is then obtained in a simpleanalytic form, where the input function for disorder interaction is the autocorrelation function of the randomfield. This allows one to take completely into account the geometry of the wire, the origin of disorder, and themany-body screening by 1D electrons. The DOS is demonstrated to be composed of the classical DOS and thequantum correction, which are connected with fluctuations in the random potential and in the random force,respectively. The disorder is found to smear out the square-root singularity of the DOS of the ideal 1DEG intoa finite peak tailing below the subband edge. The disorder effects from impurity doping and surface roughnesson the DOS of the 1DEG in a cylindrical GaAs wire of radiusR are thoroughly examined. It is shown that forR<a* /2 ~with a* as the effective Bohr radius! surface roughness with a radius fluctuation equal to 10% ofRoverwhelms impurity doping with a densityni5106 cm21, but for R>2a* the latter is dominant.@S0163-1829~99!02940-9#

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I. INTRODUCTION

In recent years, quantum wire~QWR! structures have attracted much attention, and promising advances have bobtained in fabrication and application of QWR’s, e.g., ladevices.1 This is primarily motivated by their unique transport properties, viz., high electronic mobility2 as well as theexpected enhanced optical properties such as high diffetial optical gain3 and optical nonlinearities.4 Thus, from bothfundamental and applied physics viewpoints, there has ba growing interest in understanding the electronic properof QWR’s.

It is well known that various QWR’s may be adequatedescribed in terms of a quasi-one-dimensional electronhereafter called for short a one-dimensional electron~1DEG!, where the electron dynamics is essentially restricto be one dimensional. In practice, 1DEG’s are alwaysfected by disorder caused by some random field presenQWR’s. The field is of different origins, e.g., impuritdoping,5–8 surface roughness,5–10 and alloy disorder.5,7,11

The disorder has been shown to lead to considerable chain the electronic energy spectrum of the 1DEG, e.g., desting the square-root singularity of the density of states~DOS!characteristic of an ideal 1DEG.12,13 Moreover, the disorderalso gives rise to an appreciable modification in the elemtary collective-excitation spectrum of the 1DEG, e.g., tdispersion for long-wavelength plasmons.14 Obviously, thesein turn result in remarkable changes in many phenomoccurring in the wire, e.g., optical absorption. Therefohaving useful~especially analytic! expressions for the DOS’of disordered 1DEG’s is of fundamental importance in boexplaining observable properties of QWR’s and analyzthe performance of modern semiconductor devices basethem.

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So far, only a few theoretical investigations12,13,15,16havebeen made in order to understand the electronic energy strum of disordered 1DEG’s in QWR’s, and only numericresults have been available in the literature. It should beprisingly noted that the existing theories of the effect of dorder on the 1D DOS have been established only fordisorder arising from charged impurities chaotically distruted in the sample. The other sources of disorder, especsurface roughness, have recently been confirmed experimtally to be of importance in very thin QWR’s.17–19Neverthe-less, the consideration of their influence on the 1D DOSseemingly been quite scarce. The reason for this is probthat in the earlier theories the input function for disordinteraction has been chosen to be the potential created bindividual center of force, e.g., a single ionized impuritydoped QWR’s. The one-center potential is clearly inadequfor describing disorder interaction connected, e.g., with sface roughness.

Furthermore, existing theories of the impurity doping efect turn out to be partly unsatisfactory. Indeed, Das Sarand Xie15 calculated the DOS for a 1DEG in a doped reangular QWR. The calculation is based on a self-consisBorn approximation and, hence, is adequate mainly for adoping level. Takeshima12 evaluated the 1D DOS for aheavily doped square wire, employing the semiclassmodel suggested by Kane20 and Bonch-Bruevich21 for 3Delectron systems. However, a serious shortcoming oftheory is the use of a 3D~bulklike! screened Coulomb impurity potential, which is evidently seen to dramaticaoverestimate the screening by 1D electrons since theelectron system may be polarized merely along the waxis.22 Therefore, the authors of Ref. 13 applied Klaudebest multiple-scattering approach23 to evaluating the 1DDOS in a doped cylindrical wire. This enables us to allow f

13 648 ©1999 The American Physical Society

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PRB 60 13 649SEMICLASSICAL APPROACH TO THE DENSITY OF . . .

the geometry of the wire and the features of the mabody interaction in the 1DEG, e.g., 1D screening of the ipurity field. However, the method must invoke drastic aproximations, and despite these it is computationally svery complicated for realistic 1DEG’s. Further, the authoof Ref. 16 used a 1D version of the multiple-scatteritheory of Matsubara and Toyozawa.24 Nevertheless, this isessentially a single-band scheme and, hence, can describimpurity band formation rather than the tailing of th1D DOS.

Thus the goal of the present paper is to develop a theof the electronic energy spectrum of disordered 1DEGin QWR’s, which is to be applicable to disorder nonly arising from impurity doping but also of any originrelevant to realistic QWR structures, and which must elimnate the above-mentioned difficulties of the existing theorTo this end, we will propose a version of the semiclassiapproach to 1DEG’s subjected to smooth random fiebased on modifying the method suggested by Quand Tung25 for 2D electron systems. This version allowtaking complete account of the microscopic details ofgeometry of the wire and the origin of the disorder, aproviding the DOS of disordered 1DEG’s in a simple anlytic form.

In Sec. II, we start with a collection of the formulas to bused for calculating the DOS of the 1DEG in the presencesome Gaussian random field. The derivation of the DOSdisordered 1DEG’s in smooth Gaussian fields proceedSec. III within a semiclassical approach. This accountsthe classical effect due to fluctuations in the random potenas well as the quantum effect due to fluctuations in the rdom force. In Sec. IV, the theory is applied to a cylindricwire where the disorder is produced by both impurity dopand surface roughness. In Sec. V, illustrating plots and cclusions are presented. Finally, Sec. VI is devoted to a smary.

II. BASIC RELATIONS

Two striking aspects of strictly 1D electron systems habeen shown theoretically. For noninteracting electrons inpresence of any disorder, all electronic states are expotially Anderson localized.26 On the other hand, within theTomonaga-Luttinger model for interacting electrons~linearelectron energy dispersion, infinite density of negativenergy electrons, and short-range interaction!, the electronsin a single-subband QWR in the absence of disorder maka singular strongly correlated liquid.27 Thus the disorderedinteracting strictly 1DEG was believed, in principle, to bhave as a localized, strongly correlated Luttinger liquid.

Contrary to the above long-standing theoretical claDas Sarma and co-workers28–30 recently proved, based onmore realistic model~parabolic dispersion, finite electrodensity, and Coulomb interaction!, that even a small amounof disorder in an interacting 1DEG would restore the Fersurface if the localization length becomes larger thanphysical length of the wire. The typical localization lengtturn out to be very long~larger than 10–100mm!.31 There-fore, 1DEG’s in QWR’s with both disorder and electroelectron interactions behave, for all practical purposes,sentially as delocalized Fermi liquids.28–31There has been a

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abundance of recent experiments which offer clear evidesupporting the standard Fermi-liquid model for actu1DEG’s.32–37 Until now, all theories of disordered interacing 1DEG’s, e.g., in semiconductor QWR’s5–8,12–16,28–31,38,39

and graphene nanotubules,40–42have seemed to be developewithin this model. These were found to be quantitativehighly successful in explaining the experimental data onelectronic properties of realistic QWR’s, e.g., Raman scating, photoluminescence, and band-gap renormalization.

Thus in what follows we may ignore all aspects of locaization and Luttinger-liquid physics uncritically, startinfrom the standard Fermi-liquid model. The electrons in twire are considered to be confined in two dimensions, e.gthe y andz directions, and to move freely in thex direction~chosen as the wire axis!. The disorder in the wire is usuallycaused by some random field affecting the motion ofelectrons along the wire axis.

Hereafter, we shall restrict the discussion to the case wthe random fieldU(x) obeys Gaussian statistics. Therefoit may completely be described by the autocorrelation fution of the disorder potential

W~x2x8!5^U~x!U~x8!&, ~1!

where the angular brackets stand for averaging over all cfigurations of the random field. The disordered electron stem is assumed macroscopically homogeneous, the autorelation function depending merely on the coordinadifference.

As usual, the strength of the field is determined by the rof its potentialg and of its forceF, defined by

g25^U2&5W~x2x8!ux5x8 ~2!

and

F25^~¹U !2&5¹x¹x8W~x2x8!ux5x8 . ~3!

The average potential is supposed small compared withenergy separation between neighboring subbands, so thaintersubband scattering induced by disorder is negligiblethe theory may be formulated in a one-subband approxition.

It is well known43 that the DOS is the most adequaconcept describing the energy spectrum of disorderedtems. The electronic DOS per unit length can be represein terms of a Fourier transform of the Green’s function a

r~E!51

p\ E2`

`

dt exp~ iEt/\!^G~ t !&, ~4!

where a spin degeneracy of 2 is included. Here^G(t)& de-notes the diagonal part of the configuration-averaged Grefunction, which describes the one-particle properties of1DEG in the presence of a random field and is clearly inpendent of the spatial coordinate in virtue of the macroscohomogeneity of the electron system.

The Green’s function may be written within the onsubband approximation in terms of a Feynman path inteas43

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13 650 PRB 60DOAN NHAT QUANG AND NGUYEN HUYEN TUNG

G~ t !5S m

2p i\t D1/2 1

N R Dx~t!expH i

\ E0

t

dt

3Fm

2x2~t!2U@x~t!#G J . ~5!

Here m is the effective mass of the charge carriers withparabolic 1D subband reckoned from the unperturbed sband edge,Dx(t) means the Feynman measure, andN is thenormalization factor:

N5 R Dx~t!expH i

\ E0

t

dtm

2x2~t!J . ~6!

The path integrals in Eqs.~5! and ~6! are taken over closedpaths:x(t)5x(0).

For a Gaussian random field, it holds for the averagGreen’s function43 that

^G~ t !&5S m

2p i\t D1/2 1

N R Dx~t!expH i

\ E0

t

dtm

2x2~t!

21

2\2 E0

t

dtE0

t

dt8W@x~t!2x~t8!#J . ~7!

Thus Eqs.~4! and~7! set up a basis for studying the DOof the 1DEG in a QWR in the presence of a Gaussian rdom field. The disorder effect is then allowed for via tautocorrelation function, whose form is specified by theometry of the wire and the nature of the disorder.5–8

III. DOS OF DISORDERED 1DEG’S IN A SEMICLASSICALMODEL

For further evaluation of the DOS of disordered 1DEGwe have to calculate the averaged Green’s function proviby Eq. ~7!. Hereafter we will assume that the random fieU(x) producing disorder in the electron system obeysinequality

\2F2

4mg3 !1, ~8!

whereg and F are, as before, the averages of the randpotential and of the random force, given by Eqs.~2! and~3!.

It has been pointed out44,45 that if inequality ~8! is ful-filled, the fieldU(x) and, hence, its autocorrelation functioW(x2x8), are varying slowly on the average along the waxis, so that a semiclassical approach to it may be applicaAs a consequence, the Green’s function~7! is to be approxi-mated in the lowest order by25,46

^G~ t !&5S m

2p i\t D1/2

expS 2g2t2

2\2 D F11g2t22J~ t !

2\2 G ,~9!

whereJ(t) is a path integral, defined by

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N R Dx~t!expH i

\ E0

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dtm

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3E0

t

dtE0

t

dt8W@x~t!2x~t8!#. ~10!

It is to be noticed that the first and second terms insidesquare brackets on the right-hand side of Eq.~9! are respon-sible for the classical DOS in the subband-bending moand the quantum correction relative to fluctuations inrandom force, respectively.

Now, we must evaluate the functionJ(t) entering Eq.~9!.The evaluation of the integrals present in Eq.~10! is outlinedas follows. First the autocorrelation function in Eq.~10! is tobe replaced by its Fourier transform withk as a 1D wavevector, defined by

W~x!5E2`

` dk

2pW~k!exp~ ikx!. ~11!

Then the path integral is straightforward, yielding25

J~ t !5E2`

` dk

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t

dtE0

t

dt8 expF2 i\k2t

2ms~12s!G ,

~12!

wheres5ut2t8u/t<1 andk5uku.Next, it is evidently seen from Eqs.~4! and ~9! that the

main contribution to the semiclassical DOS results from sua time region that

gt/\&1, ~13!

with g being the average random potential.Furthermore, thek integral in the Fourier representatio

~11! of the autocorrelation function is primarily extendeover such a wave-vector region that

k&1/L. ~14!

Here L is a correlation length of the random field that thfunction W(x) becomes small underuxu*L, the field fluc-tuations with a spacing larger thanL being statistically inde-pendent.

Upon combining inequalities~13! and ~14!, we are in aposition to estimate the upper limit of the variable of texponential in Eq.~12!:

\k2t

m&

\2/mL2

g. ~15!

In accordance with the semiclassical nature of the randfield, we may, as usual, adopt the following inequality:

\2/mL2

g!1. ~16!

Then expanding the exponential in Eq.~12! into a series inpowers of the small quantity\k2t/m, the functionJ(t) is tobe approximated in the lowest order by

J~ t !5g2t21\F2

12m~ i t !3. ~17!

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The rms of the potential and of the force of the random fifiguring in Eq. ~17! can now be rewritten in terms of thFourier transform of the autocorrelation function as

g25E2`

` dk

2pW~k! ~18!

and

F25E2`

` dk

2pk2W~k!. ~19!

Finally, on substitution of the functionJ(t) in Eq. ~9! byEq. ~17!, we may immediately arrive at the following aveaged Green’s function describing the 1DEG moving insmooth Gaussian field:

^G~ t !&5S m

2p i\t D1/2

expS 2g2t2

2\2 D F12F2

24\m~ i t !3G ,

~20!

Let us now return to the derivation of the DOS for th1DEG in question. We need to insert the Green’s funct~20! into Eq. ~4!, and perform easily the appearingt integralby means of47

E2`

`

dx~ ix !n exp~2b2x22 iqx!

5Ap

2n/2bn11 expS 2q2

8b2DDnS q

b&D , ~21!

with Dn(x) being a parabolic cylinder function. As a consquence, we obtain

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p S 2m

\2 D 1/2 1

~2g!1/2expS 2E2

4g2D3FD21/2S 2

E

g D2\2F2

24mg3 D5/2S 2E

g D G . ~22!

Thus, within the semiclassical model, the DOS of disorde1DEG’s in QWR’s is provided by Eqs.~22!, ~18!, and~19!.This general analytic form of the 1D DOS is the centresult of the present paper.

With the aid of the asymptotic expansion of the parabocylinder function,47 it follows immediately from Eq.~22! thatif the disorder is absent the semiclassical DOS can reprodthe square-root singularity at the subband edge, characteof the ideal 1DEG,1

r0~E!51

pS 2m

\2 D 1/2u~E!

E1/2 , ~23!

whereu(t) is the Heaviside step function.In analogy with disordered 2D electron systems,25 within

the semiclassical approach to a smooth Gaussian fieldDOS of disordered 1D electron systems may, as expectedrepresented in terms of an expansion with respect tosmall quantity\2F2/4mg3. It is obvious that the first terminside the square brackets on the right-hand side of Eq.~22!refers to the classical DOS, while the second one to the qutum correction. The former depends merely on the rms

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fluctuations in the potential of the random field,g, whereasthe latter depends not only on the rms of fluctuations inpotentialg but in its forceF as well.

It is interesting to note that our approach to a random fiis capable of including the disorder effect from the potenand its first spatial derivative into the DOS calculation, whthe purely classical approximation of Kane20 includes thatfrom the potential only, and the quantum method of Halpeand Lax43 that from the potential as well as its first ansecond derivatives.

It should be stressed that the semiclassical approachassumed the random field to be smooth, which implies,seen below a large width of the wire, a high doping levand small fluctuations in the wire width. The smoothnecondition ~8! involves only long-range fluctuations in thdisorder potential, and corresponds to the short-time cspecified by inequality~13!. Consequently, the theory madescribe properly the high-energy region above and nearsubband edge~the latter is of the order of the rms potentia!.This may be understood physically by considering tHeisenberg energy-time uncertainty relation in combinatwith inequality ~13!.

It is worthy to note that because of its simple form, oapproach is a flexible tool for the theoretical investigationdisorder effects on the DOS of 1DEG’s in semiconducQWR’s. It follows from Eqs.~22!, ~18!, and ~19! that indistinction from the existing theories,12–16rather than the po-tential due to a single center of force but the autocorrelatfunction of the random field plays the key role as the inpfunction for disorder interaction. In the most discussed cof impurity doping, the random field and, hence, this funtion are connected with all ionized impurities present in tsample.43 It has been shown5–8 that the autocorrelation function may capture the microscopic details not only of the gometry of the wire and the nature of the disorder but alsothe many-body interaction in the 1DEG. On the other haour approach clearly works quite well with an arbitrary forof that function. Therefore, this enables us, to calculatedisorder effect on the energy spectrum of the interact1DEG in a QWR of any geometry and subjected to a smodisorder of any origin.

IV. AUTOCORRELATION FUNCTIONS FOR ACYLINDRICAL SEMICONDUCTOR QWR

We shall now apply the foregoing theory to assessinfluence on the DOS of 1DEG’s in QWR’s due to disordarising from impurity doping and surface roughness, whare, in general, almost unavoidable in realistic wires. Asmodel of QWR’s, we choose a circular cylinder and confithe motion of the electrons in the cylinder by an infinipotential barrier at its surface.5–8,48,49Accordingly, the wavefunctions of 1D subbands are proved to be expressedterms of Bessel functions.5 Moreover, at zero temperaturalmost all electrons are assumed to occupy the lowestsubband ~extreme quantum limit!, as shownexperimentally.33,37 As indicated just above, the theoreticanalysis of the disorder effect is simply reduced to findithe autocorrelation function for the electron system totreated.

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A. Impurity doping

We begin by examining random fluctuations in the imprity density in the sample. It is well known20,21 that the ran-dom field created by all charged impurities is generally csidered smooth at a high doping level. Moreover,autocorrelation function of the total impurity field can be cainto the form43

WID~x2x8!5nIE d3r iv~x2r i !v~x82r i !, ~24!

with nI the 3D density of ionized impurities. Herev(x2r i)is the potential energy of an electron atr5(x,0,0) due to asingle impurity atr i5(xi ,ri).

In the case in question, this potential obviously possesthe symmetry described byv(x2r i)5v(ux2xi u,ri). There-fore, the Fourier transform of Eq.~24! reads

WID~k!5nIE d2riv2~k,ri !, ~25!

wherev(k,ri) is the Fourier transform in thex direction ofthe one-impurity potential.

The impurity potential is to be screened by interactielectrons in the 1DEG. This can be quantified by introduca static dielectric function as

v~k,ri !5vei~k,ri !

e~k!. ~26!

Here vei(k,ri) is the unscreened one-impurity potentiawhich is to be modified by a finite extension of the electrwave functionf(r ) in the y andz directions, i.e., weightedas5,14

vei~k,ri !52ZE d2r uf~r !u2v~ri ,r ,k!, ~27!

with Z the charge of an ionized impurity in units of thelectron chargee. The Coulomb potential figuring in Eq.~27!describes bare interaction between an electron atr and anelectron atr 8, given by14

v~r ,r 8,k!52e2

eLK0~kur2r 8u!, ~28!

with eL the dielectric constant of the background lattice.what follows, I n(x) and Kn(x) are the nth-order modifiedBessel functions of the first and second kind, respectivel47

As previously,48,49 the electrons are, for simplicity, considered to be distributed with a constant density in the wcross section. This implies that for the lowest subbandmay take

uf~r !u25~pR2!21u~R2ur u!, ~29!

whereR is the wire radius. The assumption of the unifordistribution of electrons may be a somewhat good appromation in the high-electron density limit because they reeach other.

Upon putting Eqs.~28! and ~29! into Eq. ~27!, we obtainthe following analytic form for the effective impuritypotential:49

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eL

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a2 H12aK1~a!I 0~d i ! for d i<aaI 1~a!K0~d i ! for d i>a,

~30!

wherea5kR andd i5kuri u.According to Eq.~25!, the autocorrelation function fo

impurity doping depends on the geometry of the impursystem. Hereafter, the charged impurities are supposed trandomly distributed in a cylindrical tube coaxial with thwire.

First, we are concerned with an infinitely thin impurittube of radiusr, integral~25! being trivial. The autocorrela-tion function depends on the position of the impurity systeas follows.

~1! r<R and WID5WMI ~modulation doping inside thewire!:

WMI~k!5S 4Ze2

eLD 2 ni

e2~k!

1

a4 @12aK1~a!I 0~d!#2,

~31!

whereni is the 1D impurity density along the tube axis, and5kr.

~2! r.R andWID5WMO ~modulation doping outside thewire!:

WMO~k!5S 4Ze2

eLD 2 ni

e2~k!

1

a2 I 12~a!K0

2~d!. ~32!

For an impurity tube of finite thickness with radiirm andrM , it is useful to distinguish between two limiting casesinterest.

~3! rm50, rM5R, andWID5WUI ~uniform doping insidethe wire!:

WUI~k!5S 4Ze2

eLD 2 ni

e2~k!

1

a4 $124I 1~a!K1~a!

1a2K12~a!@ I 0

2~a!2I 12~a!#%. ~33!

~4! rm5R, rM.R, andWID5WUO ~uniform doping out-side the wire!:

WUO~k!5S 4Ze2

eLD 2 ni

e2~k!

1

a4 I 12~a!$a2@K1

2~a!2K02~a!#

2dM2 @K1

2~dM !2K02~dM !#%, ~34!

wheredM5krM . The 1D impurity density in Eqs.~33! and~34! is fixed by the 3D one viani5pR2nI .

If there exist several impurity species with chargesZi anddensitiesni in the sample, the productZ2ni in Eqs.~31!–~34!is to be replaced with an effective impurity density, definby

ni* 5(i

Zi2ni . ~35!

B. Surface roughness

Next we deal with random fluctuations in the wire radithat have been found to be important in very thin wires, emade from AlxGa12xAs/GaAs.17–19 It is well known50 that

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the surface roughness in a QWR is characterized by theerage size fluctuationD and the correlation lengthL alongthe wire axis. Naively, one would argue that a decreasinDor an increasingL corresponds to a smoother interface.Gaussian-like decay of the wire radius fluctuations is usuassumed.

For the lowest subband, the autocorrelation functionsurface roughness in a cylindrical QWR of radiusR is thensupplied by5–8

WSR~k!5~2.4!4Ap\4

m2

D2L

R6

exp~2L2k2/4!

e2~k!, ~36!

where e(k) allows for screening the surface roughneinduced field by the interacting 1DEG.

It is to be noted51 that in case the 1DEG is affected simutaneously by both sources of disorder, the autocorrelafunction is obviously additive if correlations between theare neglected, i.e.,

W~k!5WID~k!1WSR~k!. ~37!

C. RPA screening

It has been pointed out52 that the disorder and screenineffects are to self-consistently determine each other. Scring by interacting 1DEG’s in QWR’s is of great importancin determining disorder. In what follows, for the sakenumerical simplicity we neglect the influence of disorderscreening.13,49Then, within the random-phase approximatiof the standard Fermi-liquid model, the static dielectric funtion for a 1DEG at zero temperature may be written as14,28–30

e~k!5112m

p\2

vee~k!

klnUk12kF

k22kFU. ~38!

Herevee(k) denotes the Fourier transform in thex directionof the bare electron-electron interaction potential, andkF isthe Fermi wave vector fixed by the 1D carrier densityne viakF5(p/2)ne .

The screening function~38! clearly exhibits a logarithmicsingularity atk52kF . It is well known that within the self-consistent Born approximation, the 2kF singularity of thefunction~38! leads to meaningless results at zero temperafor the 1D transport property52 and the electronic 1D DOS,6

so that the inclusion of the disorder effect in the calculatof screening is very important. However, this singularity psents no difficulty in our DOS calculation since all quantitiof interest—the average potentialg and forceF—are givenin terms of the convergentk integrals as seen directly fromEqs.~18! and~19!. The influence of disorder on screeninglikely of less importance in the estimation of the fiestrength since this is significant mainly around the integrasingular point.14

The electron-electron interaction given by Eq.~28! is tobe weighted with the wave function of the subband as5

vee~k!5E d2rE d2r 8uf~r !u2uf~r 8!u2v~r ,r 8,k!. ~39!

Inserting Eqs.~28! and~29! into Eq.~39! yields the effectiveelectron-electron interaction in analytic form:49

v-

ly

r

-

n

n-

-

re

n-

y

vee~k!54e2

eL

1

a2@122I 1~a!K1~a!# ~40!

(a5kR).

V. RESULTS AND DISCUSSIONS

To illustrate the theory developed in the preceding stions, we have carried out numerical calculations for cyldrical QWR’s made ofn-type GaAs at zero temperaturewhose conduction subband is considered. The materialrameters are the effective massm50.067me and the dielec-tric constanteL512.9. However, our results are more geeral. This is because that the natural scales for the lengthenergy, and the 1D DOS are atomic units: the effective Bradius a* 5eL\2/me2, the effective Rydberg Ry*5me4/2eL

2\2, and r* 51/Ry* a* , respectively. For GaAswires, we havea* 5100 Å, Ry* 55.6 meV, andr* 51.793105 meV21 cm21. The DOS of the disordered 1DEG inQWR is determined by Eq.~22!, where the average potentia

FIG. 1. rms of the~a! potentialg and ~b! force F for impuritydoping vs wire radiusR under densityni5ne5106 cm21. The la-bels M1, M2, and M3 refer to modulation doping with variouimpurity positionsr50, R, and 2R. U1 andU2 refer to inside andoutside uniform doping withrm50, rM5R and rm5R, rM

56R, respectively.

p

m

-

g

d

id

ide

ess

ation

et

in

deal


g and forceF of the random field are given by Eqs.~18! and~19! in terms of its autocorrelation function.

A. Impurity doping

First, we evaluate the disorder effect due to impurity doing with the autocorrelation function given by Eqs.~31! and~32! for modulation doping, and Eqs.~33! and ~34! for uni-form doping. From these it follows that the random paraetersg andF depend on the radius of the electron wireR andthose of the impurity tuberm andrM as well as their densities ni andne . Hereafter, the case ofni5ne is discussed.

In Fig. 1 the rms of the potentialg and forceF for impu-rity doping are plotted vs wire radiusR under impurity den-sity ni5106 cm21: modulation doping with various impuritypositionsr50, R, and 2R; and inside and outside dopinwith rm50, rM5R and rm5R, rM56R, respectively. InFig. 2 the random parametersg andF are plotted vs impuritydensityni for different wire radiiR5a* /2 anda* . In Fig. 3the DOSr(E) is plotted vs energy for the 1DEG subjecteto impurity doping with densityni5106 cm21 under differ-ent wire radiiR5a* /2, a* , and 2a* . The DOS of the ideal1DEG is depicted according to Eq.~23!. In Fig. 4 the DOSr(E) is plotted vs energy for the 1DEG subjected to outs

FIG. 2. rms of the~a! potentialg and ~b! force F for impuritydoping vs densityni (ni5ne) under different wire radiiR5a* /2~dashed lines! and a* ~solid lines!. The labels are the same asFig. 1.

-

-

e

doping with wire radiusR54a* , under various impuritydensitiesni5105, 53105, and 106 cm21.

In should be mentioned that we cannot employ Eq.~22! tocalculate the disorder effects on the 1D DOS due to insand outside doping withni<53105 cm21 andR<2a* , andinside doping withni<106 cm21 andR<a* /2, since in thesecases the semiclassical condition~8! is broken.

B. Surface roughness

Next we assess the disorder effect from surface roughnwith the autocorrelation function~36!. It is obvious that thesurface roughness parameters depend on the wire fabrictechnique. For GaAs the average size fluctuationD is as-sumed ranged from 3 to 20 Å, and the correlation lengthLfrom 20 to 200 Å.10 Small values ofD correspond to fabri-cation by molecular-beam epitaxy53,54 and the large ones tofabrication, e.g., by electron-beam lithography and wchemical etching.55

In Fig. 5 the rms of the potentialg and forceF are plottedvs wire radiusR under correlation lengthL5a* , various

FIG. 3. DOS r(E) in units of r* vs energy for the 1DEGsubjected to impurity doping with densityni5ne5106 cm21 underdifferent wire radiiR5a* /2 ~dashed lines!, a* ~dash-dotted lines!,and 2a* ~solid lines!. The DOS is plotted for~a! modulation dopingM1 andM3, and~b! uniform dopingU1 andU2. The labels are thesame as in Fig. 1. The dotted line represents the DOS of the i1DEG.

n-

na

ghg

ef

bfir

ing

omessire

es

ofing

ceas-

tion


radius fluctuationsD53, 10, and 20 Å, and different electrodensitiesne5105 and 106 cm21. In Fig. 6 the random parametersg andF are plotted vs correlation lengthL under wireradiusR5a* , various radius fluctuationsD53, 10, and 20Å, and different electron densitiesne5105 and 106 cm21. InFig. 7 the DOSr(E) is plotted vs energy for the 1DEGsubjected to surface roughness with correlation lengthL5a* and various radius fluctuationsD53, 10, and 20 Åunder electron densityne5106 cm21 and different wire radiiR5a* /2 anda* .

Finally, Fig. 8 displays the DOSr(E) of the 1DEG sub-jected to both sources of disorder: impurity doping with desity ni5106 cm21 and surface roughness with radius fluctution D510 Å and correlation lengthL5a* . The DOS usingEq. ~37!, is plotted under different wire radii:R5a* /2 forcombining the effects of outside doping and surface rouness, andR5a* for combining the ones of inside dopinand surface roughness. As in the 2D case,56 the resultingeffect is found not equal to merely the sum of the twofects, each one being treated separately.

It is worth noting that the average potential is seen tosmall compared with the energy separation between the

FIG. 4. DOS r(E) in units of r* vs energy for the 1DEGsubjected to outside impurity doping with wire radiusR54a* un-der various densitiesni5ne5105 cm21 ~solid lines!, 53105 cm21

~dash-dotted lines!, and 106 cm21 ~dashed lines!. The DOS is plot-ted for ~a! modulation dopingM3 and~b! uniform dopingU2. Thelabels are the same as in Fig. 1.

--

-

-

est

and second subbands given by57 E12E058.9(a* /R)2Ry* ,which warrants the one-subband approximation used.

C. Conclusions

From the curves thus obtained we may draw the followresults.

~i! Figures 1 and 5 indicate the strength of the randfields connected with impurity doping and surface roughnin thin QWR’s is enhanced rapidly when reducing the wsize. ForR<a* /2 surface roughness~especially with largeD! is observed to overwhelm impurity doping. But forR>2a* the former is negligible, whereas the latter becomdominant. These disorder sources compete forR'a* .

~ii ! It follows, as expected, from Fig. 2 that the strengththe total impurity field increases when elevating the doplevel ni .

~iii ! Figures 5 and 6 reveal the strength of the surfaroughness field exhibits a linear enhancement with increing the radius fluctuationD; see Eq.~36!. Moreover, therandom field is weakened when increasing the correla

FIG. 5. rms of the~a! potentialg and ~b! force F for surfaceroughness vs wire radiusR under correlation lengthL5a* , anddifferent electron densitiesne5105 cm21 ~solid lines! and 106 cm21

~dashed lines!. The labelsS1, S2, and S3 correspond to surfaceroughness with various size fluctuationsD53, 10, and 20 Å, re-spectively.

heis

nothestaile.bedge

reeg-

tofDb-t in5.

i:s

e


FIG. 7. DOS r(E) in units of r* vs energy for the 1DEGsubjected to surface roughness with correlation lengthL5a* underdifferent wire radiiR5a* /2 ~dashed lines! anda* ~solid lines!, andelectron densityne5106 cm21. The labels are the same as in Fig.

FIG. 6. rms of the~a! potentialg and ~b! force F for surfaceroughness vs correlation lengthL for wire radiusR5a* , underdifferent electron densitiesne5105 cm21 ~solid lines! and 106 cm21

~dashed lines!. The labels are the same as in Fig. 5.

length L. The surface roughness potential drops withLrather rapidly for smallR and largeD, but slowly for largeR.

~iv! An examination of Figs. 3, 4, 7, and 8 shows tsquared-root singularity of the DOS of the ideal 1DEGdestroyed by disorder into a finite peak. The disorderonly dramatically reduces the height of the peak and pusit up toward higher energies, but also gives rise to a band~of localized states! extending deep below the subband edgThe electronic energy spectrum of the 1DEG is found toconsiderably modified in a region around the subband eand of the order of the average potentialg, as quoted previ-ously. The DOS of the ideal and disordered 1DEG’s aasymptotically equal. This means the disorder effect is nligible at very high energies.

It is worthy to recall that the DOS tail (E,0) cannot beprovided within the 1D semiclassical theory dueTakeshima,12 nor within the 1D multiple scattering theory oRef. 16. Further, in comparison with the 2D and 3cases,25,43 the modification of the 1D DOS around the suband edge is much more drastic. The reason for this is tha

FIG. 8. DOS r(E) in units of r* vs energy for the 1DEGsubjected to both impurity doping with densityni5ne5106 cm21

and surface roughness with radius fluctuationD510 Å and corre-lation lengthL5a* . The DOS is plotted under different wire radi~a! R5a* /2 for outside dopingM3 andU2 and surface roughnesS2, and the combined effect is marked byM3S2 andU2S2. ~b!R5a* for inside dopingM1 andU1 and surface roughnessS2,and the combined effectM1S2 andU1S2. The labels are the samas in Figs. 1 and 5.

dni

thnt

on

reth

hiis

toin

h

daba

an

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ityodyu-

ma-

-edof

bere-hehe

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n-o-cee-h

th-the

ap-by

a-redes-ateri-


the 1D case the singularity of the DOS at the subband eand the disorder interaction are both stronger, the screeby the 1DEG being weaker.

~v! The characteristics of the DOS~its peak and tail! areseen to depend on the wire size. Figures 3 and 7 indicatethe DOS peak is lowered, and the DOS tail is larger amore extended below the subband edge when reducingwire radius. This means that the thinner the wire, the strger the disorder effect becomes.

~vi! Figures 4 and 7 reveal that the DOS peak is loweand the DOS tail is larger and more extended far belowsubband edge when increasing the doping levelni or the sizefluctuationD.

~vii ! As in the 2D and 3D cases, the disorder-induced sof the Fermi level of the 1DEG is found to be small, whichin contrast to the earlier result.12 Indeed, at densityne5ni5106 cm21 the Fermi level of the ideal 1DEG is equalEF

(0)52.50 Ry* . The Fermi level of the disordered 1DEGa wire of radiusR5a* , using Eq.~22!, is estimated to beEF52.70 and 2.94 Ry* under inside modulation doping witr50 and outside uniform doping withrM56R, respec-tively; and EF52.54 Ry* under surface roughness withL5a* andD510 Å. This is interpreted as follows. As quoteabove, the 1D DOS, on the one hand, is reduced appreciin the intrasubband near the subband edge; on the other hit is remarkably enhanced on the tail far below the subbedge, the Fermi level being slightly shifted.

VI. SUMMARY

In this paper we have achieved a simple analytic expsion @Eq. ~22!# for the DOS of disordered quasi-1DEG’ssemiconductor QWR’s, in which the input function for diorder interaction is the autocorrelation function in wavvector space. This enables us, to calculate the effect on

s

v.

.

d

geng

atdhe-

de

ft

lynd,d

s-

-he

1D DOS due to a source of disorder other than impurdoping, e.g., surface roughness, and that due to many-bscreening caused by realistic 1DEG’s in QWR’s. In particlar, it is possible to go beyond the random-phase approxition, including the local-field correction~electron exchangeand correlation! into the DOS calculation.

In our model of cylindrical QWR’s with an infinite potential barrier, the cylindrical symmetry is used. As indicatpreviously,5,49 this could not alter the order of magnitudethe disorder effects, but does enable the input function togiven in analytic form for various disorder sources. Thefore, a numerical calculation of the 1D DOS as well as tobservable properties of realistic QWR’s relative to tDOS, e.g., optical absorption, presents no difficulty.

It should be kept in mind that our semiclassical approadescribes a smooth random field which obeys inequality~8!valid for disorder of any origin. The disorder is clearly conected only with long-range fluctuations in the disorder ptential ~e.g., due to heavy doping and slowly varying surfaroughness! and modifies the 1D DOS in the high-energy rgion ~around the subband edge!. For disorder associated witshort-range potential fluctuations~e.g., doping inside a verythin QWR or at a lower doping level, and alloy disorder wia d potential!, the theory is inapplicable. Then, also to include the short-range potential fluctuations and, hence,low-energy region~deep tail!, in a forthcoming paper we willsupply a path-integral approach based on a cumulantproximation, modifying the method developed recentlyQuang and Tung58 for 2D electron systems.

Owing to the absence of detailed experimental informtion about the electronic energy spectrum of disordeQWR’s, a quantitative comparison with experiments is prently impossible. We hope that our analytic results stimultheoretical investigations and help to clarify future expemental results.

ag,

nd

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semiclassical approach to the density of states of the disordered electron gas in a quantum wire

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