Electron diffusion and harmonic generation in super-lattices Текст научной статьи по специальности «Физика»

ФИЗИКА И МАТЕМАТИКА

G. M. SHMELEV (Volgograd), E. M. EPSHTEIN (Moscow), E. N. VALGUTSKOVA (Volgograd)

ELECTRON DIFFUSION AND HARMONIC GENERATION IN SUPERLATTICES

There are many works devoted to current harmonic generation in one-dimensional superlattices (SL) (Bass, Bulgakov, Tetervov, 1989; Feise, Citrin, 1999; Pavlovich, Epshtein, 1976; Romanov, Romanova, 2001; Romanov, Romanova, 2004).The second harmonic generation (SHG) occurs in presence of DC electric field or other cause breaking initial inversion symmetry. In present work, we investigate SHG in SL under presence of the carrier density gradient (V n )■ We show that the SH intensity can be comparable with that of the current fundamental harmonic in that case.

We calculate the current along the SL axis driven by DC and AC electric fields

and gradient Vn - All the vectors are directed along the SL axis, so that we have a one-dimensional quasi-classical problem (the fields are assumed to be non-quantizing ones). With an additive energy spectrum, the electron distribution function is f(p,r,t) = /(p1,pi,xj), where p is electron quasi-momentum, p± is its component perpendicular to the SL axis. For brevity, we omit p± and write px as p below. The Boltzmann kinetic equation takes the form

approximation (Bass, Bulgakov, Tetervov, 1989). In our case, such an approach meets with some principal difficulties. Therefore, we use a model Bhatnagar — Gross — Crooke integral

E(/) = Ecosco/4 E,, Ej = const

0)

where u (р) = дг{р)/др, є(р) is electron energy, St /(p,r, t) is collision integral. In many works on the electron transport in SLs, that integral is taken within the simplest т -

, (т = const), (3)

і

where f0 ~ /0 (p) is equilibrium distribution function, f0 (p) = /?0 is equilibrium electron density and /(p,r./) = n(r,t) is nonequilibrium one. This integral obeys necessary P 1 f n ]

condition fQ—— f = 0 . A continuity equation follows from Eqs. (2) and (3)

i> i> ^ L no

(4)

where

is current density.

We seek solution of Eq. (2) with collision integral (3) as

j(r>0=e'Ev(p)f(p>r’t)

p

(5)

Here the function

f{0)(P’t)= I exP^“^~j/o — 00

p-e\E(t")dl" v t'

d t' X

(7)

obeys equation

<3/(o)

dt

eE{t)

(0)

(8)

and =n0 condition, while (p is unknown function satisfying ^9 = 0 condition,

p p Substituting Eq. (6) into Eq. (2) with Eqs. (4) and (8) taking into account, we obtain an equation for cp function:

дф(/УУ) + c£^atpQ?,x,f) + u dg>(p,x,l) + Ф(p.x,t) = J_ ^,(0)^i)——

dt dp дх т nn' dx

-j{x,i)-v(p)n(xj)

.e

0 = ./,)•

(9)

It can be found by direct substitution that the solution of Eq. (9) is

— GO

ip)

eE(t’)

-,t

(ft \ ^

p-e —s(p)

V t' )

x+

eE(t')

-j

t

\

p-e \E(t")dt"

\ t' J

dt', (/ » x).

(10)

In general, the solution (10) has a formal character, because the problem in consideration is to be solved by a self-consistent way together with the Poisson equation. Besides, the inhomogeneity source should be indicated. However, such a procedure is not necessary to evaluate the SHG efficiency.

We take the electron density gradient within linear approximation below. In that case, we obtain from Eq. (10)

\

-00

>i(x,t')

. t' J

Г t

etln

p~e

К (

j

At’,

01)

where j^'\t) is the cuixent density calculated with substitution /l0) function into Eq. (5). It can be seen that Eq. (11) satisfy ^ cp = 0 condition.

It has been shown in (Balkarey, Epshtein, 1972; Epshtein, 1978) devoted to one-dimensional diffusion from a plane source in bulk materials in presence of a high-frequency electric field, that n(xj) ~ rt(li(x) « nt, approximation can be assumed in real situations with /7l0|(x) being DC component of the electron density. In that approximation, we obtain from Eq. (11) at constant temperature (T).

I*expf— |/,") p-e}E{t’)dt"j' i ■—p-e | /") dr'

«„ 3.x J { т ) I f Jj ец, I f

df

(12)

Therefore, the function (6) with cp function defined by Eq. (12) is the desired solution of Eq. (2). Substituting the /’function into Eq. (5), we obtain j(x,t)= j0(xj)+ j ^(xj), where j0(xj) = j^(t)n^(x)/nQ , a jd being the diffusion current density

j,i (x,t)=-eD(t)-

OX

(13)

]+t2

■(!>)

0 0

-/■>(,w,)-U

Щ,

p-e

V

t-t.

dttdt2

(14)

Jj

p-e j*£(r’)d?'’

, t-h-h

is electron diffusion coefficient. Drift current density has been calculated in (Pavlovich,

Epshtein, 1976).

The electron dispersion law in SL under tight-binding approximation takes the form

-ii. N

. Pxd

2 m

1-cos-

h

(15)

where2A is lowest miniband, d is SL spatial period and m is in-plane electron effective mass. In this case, the quasi-classical situation with the Boltzmann equation being valid is determined by following conditions: 2A»ftco, fi/x, j<?£T|<i , Ag?|Va. n^^jriQ . The diffusion coefficient takes the form

COCO

00

sin

pd ed ft ft

V

V ' 'I

/■

t-t,

d?, dt1

000

V

t-t-I,

V

(16)

cococo I 'If ) -I

-c- 1/Ap) JJJexf^7A)sin f+f K)d/' sin j J-

t t.

where Z)0=Uqt/2, u0 = A djti is the maximal electron velocity along SL

axis,

C, = (cos

I pd\

^ j, angular brackets denote averaging over the equilibrium carrier

r distribution.

For nondegenerate electron gas C, = I, (A / k'l)f lu (A / kT), 11 (z) is modified Bessel function, k is Boltzmann constant. By substituting Eq. (1) into Eq. (6) we have

00 00 Д(/Н4,[ J

0 0 00 00 - ft-

cos | asinc; t-asmco (r-/,)-2osinft> (/,+2t)

Й

d ttd t7

-2Q

0 0

00 00 00

Ш

0 0 0

(17)

|exp| —-—-—- jsinf asinffl t-asmco[t-ti-tn)+^~{tl+t2) |x

I \ i \ deE. \dt..d udt. xsm <7simy [t-tj-asmoy-t,-/,)+■---------1, —:---=---

V ' й J r'

\

deE E, 1

where a----------- =----------• By expanding the integrand in Eq. (17) into Fourier series, the

Tico E0 cot

diffusion coefficient can be written as

D(?)=D(0)+j^#v)cos(siof-ps); (is)

s—I

where D'0- is DC component. At Q = 0 (18) does not contain even harmonics, in accordance with symmetry arguments. The analytical expressions for and coefficients, which depend on fix , cox , EjE0 and AjkT , are rather unwieldy. So we consider some special cases.

At Qx = 0 ,

D(0)

■ = JS (a) - С A {a)J0{2a) + 2 J jf M) s=i 1 + («ог]'

-c, £ vl-

/=—•/. [l -\-\scot) j [1 + ((.V + l)co r) J

0. /*o

<±± ~J]tf rr

.s'ir.—-f_. [l + (/ft>rj J[1 + {mcoT) j[i + (.sft>r) J

- / It) I 1 ~ s(l+mXajTf - X

[l + (/ corf ] [l + (m cot)2 ] [l + (so r)2 ] where Js (b) is Bessel function of the first kind.

At (ox » 1,

(19)

1 + qV-

1 + С, Ja (2 о - 4 С г J l(a )-

1 + 4Q~x~ 1

'(i + nV )2 j ’ <20>

If AC field is absent (a = 0), then the result (Ignatov, Shashkin, 1984) follows from Eq. (20) or (19):

Fig. 1. a) DC component of the diffusion coefficient and b) function A as functions of AC field ampli-

tude EjEu at various values of Qx : 1 — 0; 2 — 0.5; 3 — 2; (Aj kl — 0.5 , OX — 5)

Fig.2.

tude

a) DC component of the diffusion coefficient and b) function A as functions of AC field ampli-EjEu at various values of Qx : 1 - 0; 2 — 0.5; 3 - 2; (Aj kT = 0.5 , 00X =0.5)

e/e,

Fig.3. The second harmonic amplitude of the diffusion coefficient D (2,/D() as functions of AC field amplitude E/E0 at various values of Q r : 1 — 0; 2 — 0.5; 3-2; AjkT = 0.5, a) cot =5, b) cot =0.5

D(n) = D,. 1

Г

l + QV

l + C,2«X^i-4C7 042

(21)

1 + 4Q:t2 1 (l + Q-t2)2 _

where Q = E,/E0 t is the Stark frequency. At Q = 0 , we obtain from Eq. (20)

Z)(,’>= Dl,J(;(a)[l-C2Jil(2a)]. (22)

The typical behavior of the diffusion coefficient DC component describing with Eq. (19) is shown in Figs. 1(a) and 2(a).

It follows from Eq. (17) for second harmonic at Q - 0

D(2) = D0 -Jgf+gf , (tg/?, =g,/gi), gi s v2 - c2 V6 - c,2 (vw -vj, g2 c3 vg - - Cf (K)2 ~ vj,

V2 = 2Z (a) + ^,-2 (o)]-^/ (<?) ""' V ’ V4 35 2Z (°) “ J'-2 (°)]'Jl (a)--’

k - i r1 •

[l + ((s-/)fflr) |l + (/®rf J

vs = Z Z['/.-/-2W--/.v-,+2(a)k(«)'7/(2«)f ^2/

(23)

[l+((,s-/)j;r)2j[l + (/f<jr):!]

[1 +((/+p-.v)fyr) J[ 1 + (a <2Jt) j[l+(/fyr) J

II#.......................

..

The results describing by Eq. (23) are shown in Fig. 3 for some special cases.

The fundamental harmonic of current j0 (xj) takes the form

7o)(-1c>'‘)=[/>(0)(x)Ao]^cos(®/-ari)’ (tga I =a2/fliK (24)

where

A = C, yj+ a^_ ,

«, =2f[jM(a)+J, ,(fl)]/,(a)T-----MilWLlM]-------------------, (25)

mV [l + (/an-)2 H-^Qr)2] -4(Qr)2(/«r)2

- 2 (*).,, (a)-* + 2± [jM (a)-J„ (a)]j, («U-.. v

1 + (Qr) [l+ (/corY +(flr) J -4(fir)2(/or)2

and jc=ev0n0.

Using Eq. (25), A as a function of E/E0 ratio is shown in Figs. 1(b) and 2(b) at the parameter values corresponding to Figs. 1(a) and 2(a). The ratio of the current second and

fundamental harmonic amplitudes is u0x|c5 n^jd ^{n^D^A). At

dn^/dx^j « ZT1 (£ is diffusion length) that ratio is of order of £>^ \_>0t / LD0A ■

Let us some estimates. At A = 10 2eV, d -- l(r6cm we have u0«107cm/s. At x = 10'12s . L = 10~4cm, kT « 2 J , cot = 0.5, £ « £0, (£0 « 600 V/cm) the amplitudes of the fundamental and second harmonic become comparable (see Figs. 2(b) and 3(b), curves 1). At temperature kT>2A situation for SHG becomes more optimal.

Similar results are obtained with the non-uniformity due to a temperature gradient.

The work was supported by the Russian Foundation of Fundamental Investigations (Project No. 02—02—16238).

References

Balkarey Yu. I., Epshtein E. M. // Fiz. Tekhn. Polupr. 1972. V.6 (4). P. 762-763.

Bass F. G., Bulgakov A. A., Tetervov A. P. High frequency properties of semiconductors with superlattices. M.: Nauka, 1989. 288 p.

Epshtein E. M. // Fiz. Tekhn. Polupr. 1978. V.12 (1). P. 182-184.

Feise M. W„ Citrin D. S. // Appl. Phys. Lett. 1999. V.75 (22). P. 3536-3538.

Ignatov A. A., Shashkin V. I. // Fiz. Tekhn. Polupr. 1984. V.18 (4). P. 721-724.

Pavlovich V. V., Epshtein E. M. // Fiz. Tekhn. Polupr. 1976. V.10 (10). P. 2001-2003.

Romanov Yu. A., Romanova Yu. Yu. // Fiz. Tekhn. Polupr. 2001. V.35 (2). P. 211-215.

Romanov Yu. A., Romanova Yu. Yu. // Fiz. Tverd.Tela. 2004. V.46 (1). P. 156-161.

В. К. ИГНАТЬЕВ (Волгоград)

НЕЛОКАЛЬНАЯ ЭЛЕКТРОДИНАМИКА СВЕРХПРОВОДНИКА ВТОРОГО РОДА

Введение

Полвека, прошедшие со времени пионерской работы В.Л.Гинзбурга и Л.Д.Ландау [1], не снизили интерес к электродинамике сверхпроводников второго рода. Магнитный поток в них переносится вихрями Абрикосова [2], образующими как сравнительно регулярную структуру (решетку), обладающую ближним порядком, так и своеобразную вихревую жидкость, характеризующуюся дальним порядком, и даже «вихревую плазму» [3]. При этом уже в диапазоне радиочастот проявляются существенная пространственная дисперсия и наведенная анизотропия, наблюдаемые в диэлектриках лишь в оптическом диапазоне.

Характерной особенностью сверхпроводников второго рода, затрудняющей анализ электромагнитных процессов, является невозможность разделить заряды на связанные, свободные и намагниченности. В диэлектриках и магнетиках такое разделение проводится по характерному масштабу перемещений. Связанные заряды совершают колебания порядка межатомных расстояний, а без внешнего поля покоятся; токи намагничивания без внешних воздействий создаются движением связанных зарядов по замкнутым траекториям, свободные заряды под действием внешнего поля перемещаются на макроскопические расстояния, а без поля покоятся. Соответственно вводятся макроскопические (средние по объему) векторы поляризации, намагниченности и тока проводимости (транспортного) [4].

В диапазоне оптических частот из-за значительного тока смещения нельзя однозначно выделить замкнутые траектории связанных зарядов, а перемещение свободных зарядов сравнимо с перемещением связанных. Поэтому состояние среды характеризу-