Steady-state polarization waves in photoexcited n-doped semiconductors

~Solid State Communications, Vol. 74, No. 9, pp. 8 8 9 - 8 9 2 , Printed in Great Britain.

STEADY-STATE

POLARIZATION

Departsunento

0038-1098/9053.00+.00 Pecgamon Press plc

1990.

WAVES IN PHOTOEXCITED

n-DOPED

SEMICONDUCTORS

Antonio Sez-gio EspePidi~o de Fisica, Universidade Federal da Bahia, 40021 - Salvador, BA, Brasil and

Ir~tltut~o

Aurea R. V a s c o n c e l l o s and Roberto Lugzi de Fisica, Urdversidade Estadual de Campina~, C.P. 6t65; 13081 - Campina~, SP, Brazil

(Recelved F e b .

I, 1990 by C.E.T. G0ncalves da Sllya)

We s h o w that, In n-doped l~u~er illumir~tion t h e h o m o g e n e o u s in the d e g e n e r a t e reglme, unstable order at the macroscopic level. equilibrium this nonequilibrium corresponding to the f o r m a t i o n of and holes. It i s the result of nonllnea~ d i ~ I p a t i v e e f f e c t s in t h e

semiconductors under continuous ete~dy-etete of c ~ i e ~ e becomes, agalr~t the foPms.~ion of spatial Beyond a critical distance from phase trar~Ition occurs, a polarization w a v e of electron~ the interplay of collective and far-from-equilibrium s ~ t e m .

Nonlinear e f f e c t s i n o p e n s y s t e n ~ are nowa~ a s ~ b J e c t of l~vge p h ~ i c a l and technical interest. Nonlinearity is the source of new and unexpected bel~evior in m a t t e r . (s'z) Decades ago, Tu~ir~ showed .t h a t morphological transitions may develop as a result of diffusion ir.~tabtlity in open swten~ governed by nonlinear kinetic laws. ~ We comideP h e r e a q u e s t i o n of t ~ i s tWe In the case of photoexcited ~emiconductors, aiming to put into evidence tha occurrence of s s p a t i a l l y self-organized structure of the carrier density. For that purpose we resort to the nonequilibrium statistical operator method ( ~ O H ) ~4~ in Zubarev's a p p r o a c h (~> which yields a nonlinear qu.~'~tum trar,~port theory, that can be co~idered a f a r - l, e a c h i n ~ : g e n e r a l i z a t i o n of the H i l b e ~ ' t - C h a p m a n - En,~kog ".ffi, and Ho~i's m e t h o c l ~ . (4> It provides the nordine~ equations of evolution of the nu~roscopic state of the ,cycles, ~nd then a way to evidence and describe at a microscopic level nonlinear. instabilities. We have applied Zubarev's N S O M t o the s t u d y of t h e evolution of a polar, semiconductor under continuous laser light lllumir~tion, (~ where a brief review of the method is given. Cort~ider a n- doped p~laP semiconductor wlth an electron concentration n high enough for them to

As shown in Ref. 6 after a sufficient delay time a homogeneous s~eady ste~e follows, chal-acterized by a statio~ry concentration n and quasi-temperature f#-s. A s s u m i n g a good therlmal contact of the system with the reservoir at temperature T , it follows that ~-t 0

kT . T h e

equations

0

6 remain semiconductor, kept in mind electror~ i8

for

n

~r~d

/~-i

of

Ref.

valid for the extrir~ic except, ths% tt must be that. the concentration of now n + n. T o te~t the O

stability of the homogeneotue s~eady stake ~alnst the formation of spatial ordering, we cor~ide~ an infinitesimal spatial fluctuation imposed on the s y , = b e m . F O r t,het p u x , p o s e we enl~ge t.IT,~, b~sic eel of N~OM- v~ri~bles with t.he inclusion of the n o n d i a g o n a l elements of the one- quasipar title der~ity matrix, namely n°

(t)

-

Tr(C t

n

(C)

= Tr

C p

h

(t))

,

(la)

,

(tb)

where CCC t ) and hCh t ) ~e ~ihilstion (creation) operators for electrons and holes in pl~e- wave states and p~ 1~

O

Zubarev's NSO. The NSOM-equatlor~ of evolutior, 1"olthe quantities of" Eqs. (i), iJ~ Zubarev's NSOM linear theory of relaxation and using the. random phase

behave as itAnerant carriers in the conduction band. Under the action of C.W. l ~ e e r Light, a concentration n of photoinJected electrous ~ holes is produced in p a i r s by photon absorption. 889

890

STEADY-STATE POLARIZATION NAVES IN PHOTOEXCITED n-DOPED SEMICONDUCTORS V o l . 74~ No. 9

approximation (RPA) interaction; a p e

ih

e

(t)IAE

to

n°

~

with

deal

Coulomb

r=(~) +

+2V(Q)Af "@

kQ

ko

L a n d a similar In ¢

one

2

holes

C2),

Eqs.

e(h)

fop

=h k /2me(h);

where

~

is

o

.~h)

V(Q)m4neZ/¢oVq~,

static

the

(backgro~d)

dielectric constant and V .the .volt,he of e(h) e(h) .e(h) --~ the '~ystem; b_.f.~., s f . , - , - f . ~ , wicn kQ

fe(h)

bein~

functions.

k÷O

Ferml-Dirac

k

distribution

Further,

n ( O ) = ~.Fn 0

If:+B

~

_ ~

~'~

,

ko

ko ,]L

ko,

Ta,

~) k~aJ

Bh ko

ko

(2b)

A E " ~ h ) _ . ( h)_

2

f'-A

_ i

D < ~ , 0 ) I | A E '~ +IB h | [ A E h + i ~ " l - B *

k~

kQ

-if

(2a)

+ i B e h n h - i B S e n e ÷N e kQ

wit, h

÷n h 1

(3)

(7b) It should be noted that c(~) is the static dielectric function of the nonequ/libx, ium stationary c~rier system. One solution of the secula~ eq~tion~ Eq. (5)~ i s n ( Q ) = O, i.e. t h e o n e cox, P e , m p o n d i n £ t o t h e uniform state, and another with n(Q) ~ 0 (spatial o ~ d e r i n ~ in t h e f o ~ m o f a nonequilib~ium stattonaz-y charge der~tty wave) is possible i f ~ ( ~ ) i O. I t s imagirk~ry part, is identically null, and then we must look f o r a z e r o o f its re~l p a r t , i.e. Ra ~=l-vr.

is, in u n i t s o f t h e e l e c t r o n c h a r g e , t h e Q- w a v e n u m b e r component of the charge density. Ne ~h~ ~ e term~ bilinear in t h e k~ quantities o f Ecl~. ( I ) , a n d coefficients B contain the effect of l~er light absorption, radiation recombination, and ca~rieP-phonon interactions. The latter produces relaxation effects much smaller than the others and is neglected. T h e n B*hm'Bhh='B e a n d B h v I B e e I B h ~ w i t h

~<~,O)A-'<~,O>=o

,

ca>

where

t,<~,~>ifAf" AE h +Af h ~E" ],SP.." A~"

.fA,.h _,: lf,"

÷e

*

If=

I, (ga)

ACE,~>=

~.

-t,E"

]

+Is' AE h -e" AE"

| (gb)

A and A depend on the matrix L m elements of the interaction potential o f cat-tiers with the laser and recombination fields respectively, whe~-e

ckx

=

and

hZkZ(m : i + m h S ) ~ ,

~ap. The point,) of

E°

is

the

energy

steady-state Eqs. (2)

solution corresponds

(fix to

e
= 0 (or n(Q) I 0), i.e. t h e kQ homo~eneotJ~ slate. Hence~ t h e lineax-ized equatlon~ around the steadv-state are in linealsta~illty ~naly~is of'?> EcI~. (2) putting N~ ( ~ . ) ! O. We explicitly look kQ for an instability ~/nst a static inhomogenelty, and thu,~ we need to analyse the existence of a null eigenvalue of the tineari~ed set of Eqs. (2). Lengthy and el~borate Algebra l e a d ~ u s tO the ~eculaP e q u a t i o n n

..e,4

n~(O> = 0

,



Numerical calculation~ ape performed u~in~ pa~Pameters cha~acterlst.ic of G~As, and a pl,mplng laser w i t h p h o t o n energy o f 2.4 e V a n d v =~Piable l~e~ power. Ou~ results show that there exists a critiaM l&~.'eP power, up.. in o t h e r woPd~, a critic ~I concentration n, f o r each value of n o

that satisfies Eq. (8) and therefore characterizes the instability of the homogeneou~ s t e a d y ~tate. F i g u r e I s h o w s the value o f t h e critical c o n c e n t r a t i o n , foP n m 10 ~? c m -~ a n d ~ x l O iv cm-S~ f o r o

several values of the wavenumber q. The calculation w.'m~ done taking the carriers in the very degenerate regime such that thei~ diatribution functions can be appPoxim~ted by step t'unctio~s~ centex'.ed at the qu~ei-ahem/cal potentiale. FoP O I O, t~ c~itic~l carrier photoconcentration i s g i v e n by

[

2mh

l'-tln

Lmh-m.J

j cr.

(tO)

in

o

where (6)

F<

It should be rioted no solution Lo Eq. ( 8 ) i n intrlr~ic semiconductor.

that thePe is t h e c a s e o f a~: In this case

Vol.

7 4 , No. 9

24

T

2O

016 n..

o i-

-

._.,

,

16

\

_,

qo.lxl O" cm-3 1~= 5X 1011'¢m"3

• ----

~

\

~,,

I , I , I 17 18 19 LOG CONCENTRATION(cm'~

Pig. 1: The crit`ical value of the photoinJect`ed concentrat.ion for. the diffel-ent values of the wavenumber of the emerging pol~ization wave, ~d t.wo values of n . The crit`ical concentration O

of phot`oinJect`ed c~rrier~ in the first c~e and t.he ~ e a o n d .

of 0 correspondirq; t,o points of high symmetry in the BriUouin zone. Sir~e it should be expected that. the polarization wave in the c~ler system would drive ~loD4~ t h e ions, a lor~ r.ar~e or.der in t,he whole mater.ial would follow. The situat.ion may then be cor-Bldered akin t.o the nent.her.nuld r . e c r . y s t . A l l l z a t i o n of ion Implant`ed semiconductors proposed by van Vecht`en. (~ We have also calculated the e l e c t ` l - o h i o P.aman s p e c t ` r u m t.o b e expeat~ed fr.om t h / s s y s t ` e m , d e p i a t . e d i n F i c u ~ e 2, As we have pr.evioualy ahow~ °) it, consist~ of fou~ bands due to scat`terin~ by: (t) opt.iaaI ple~morm, (2) quaegip~.icle excit.atior~, (3) upper, a c o u t t . i c plmsmor~, and (4) lower acoustic platoon8 the Lmst. t w o beir~ shown in P i g . 2. It` i s observed in t`hls caae of an n-doped semiconduct`or, that`, on approaching t`he c1"itical point., band (4) mer.ges and disappear.a int.o band (3). Since the for.met is a s s o c i a t . e d t`o t h e collective motion of holes inter.acting th~ouRh t,he scr.eened pa~t of Coulomb

is 9X~0 i 5 c m -9 4.5x1015 cm -s in I

what, i~ obt,alned is a met,~IHc t,o non-metaUlc transition in the extremely degenerat`e r . e g i m e . (°) There is no inst.abilit.y in t`he case of a p-doped material, and also in t,he n-doped semicorKluctor, at high temper.atur.es (cl~m'~ical Undt). T h i s t.ell,= u8 that. a n excess of light, p~r.tioles is neaess~r.y to er~ure t.he format.ion of t,he spatial structure but. such ordertrq; is destr.oyed by thermal agitation, i.e. formed in the degenerate l-egime n(Q) tends to ~el-o wit`h incr.ea~Inq; c~rier q,~-~ttemperat`u~e. The first instabiUty follows for O going to ~er.o (infinite wavelen~t.h) at, a m i n i m u m crit,ical la~er power I , and for subsequent, values of Q fox'

incr.e~i~

lL

above

I:.

It

is

worth

noticing that. t.his oh~acteristic impUes that the aar.r.ier ss~tem goes along a nonequIUbr.ium t.hermodynamic path fr.om t.her.m~l ohaos (at low IL) ~ t,o ol, der

891

STEADY-STATE POLARIZATION WAVES IN PHOTOEXCITED n-DOPED SEHICONDUCTORS

(spatial

structure

at. I ~ ) , i.

to

(o)

(b)

I

u

i

what,

-

ha~ been termed t u r b u l e n t nonequilibrium chaos (at. higher It), i.e. a b u n d a n c e of macroscopic length scales so that the syst`em 8ppe~"s ch~ot`ic. For t`he realist`it c~e of a finit`e system boundary oonditior~ mtmt be imposed; Q should admit` the values ka/L, k m 1,2,.-, if L is t`he dimension of the smnple in the direction of Q. Put~ther.mor.e, o u ~ s im a n i s o t , r . o p i c model and thu~ when crys~aUine symmet`ry is cor~ider.ed it would tnt.rodu(:e ~dditional remt`ri~t.ive cond/t.ior,~ and, aonJectur.ably, lead t.o ar.ltioal valuem

0.18

0.34

0.50 OJ8 0.54 FREQUENCY(CO/QVFe)

Fig. 2: The Roman sect.ion for the case C l o-cgk w i s e Cm_s; (b) am 0.5

is

,

from n -

I

c:=_s~

o

top left: (a) n ~ 10 s~ 5 x t O t
and (d) cm-i,~u~d the critical n ~ 9x10 i5 cm -s ¢r~t

scatteri~4~? n m i0

0.50

concentration

892

STEADY-STATE POLARIZATION WAVES IN PHOTOEXCITED n-DOPED SEMICONDUCTORS

int`er~,ct`lon a n d t`he lat`t`er t`o t`hat. ~',f elect`l-on~, t, h i s x,e s u J t , suf;~;e~t.s t.hat` a t . t`he t`]r-an~it`ion t ` h e ~ e i s f o z - n m t . l o n o f chal-~e den,=,It`ies of elect.ror~s~ and of" holes eloped t`o~et`he~, In what, may be t`ermed a polaPlzat`lon wave.

VOI. 74, NO. 9

A C K N O V / L E D G E H E N T . "~ • - O n e of ,l.~ ( A S E ) i~ ;~ Minist,ry of Educat.ion (CAPES) pre-docLoral fellow: t`he ot`her Lwo (ARV.RL) are Brasil Nat`ional Research CouJ~cil ( C N P q ) fellows.

i~ra~il

REFERENCES (3.

Ntcolis

and

Or g o n l z o t i o n

I.

in

P1-i~o~ine,

Spstem~ ( W i l e y , New Y o r k , 1 ~ 7 7 ) . H. Haken, Synergetics (Spl-tnl:er, Bel-Un- Hetdelbez-t;, t978). A.M. T~In~, Phil. T~ar~. Roy. Soc. B~7, 37 (t9~2). Fol ~ a l-evlew of the NSOM a n d i t , s applloa4,io~ see: R. L~zl and A.R. VasaonceUos, Fort`sch. Phys. No. i, Band 39 (in press); ~lso Rev. Bl-astl. Pts. |5, 106 (198~); ibld 16, 495 (1986); Ibid 17, 601 (1987). D.N. Zubapev, Ner'o~no~e~'noio

Stoti~tiche~rkoio (Izd,

Nauka..

[NonequillbritJm

Th~.rmodynomic~

Self-

Nonequilibrium

Termodinomik.o bioskwa.

19T1 )

Statistical

6 7

8

iO

(PlerltJm, New Y o r k . . 1074)]. T. Tom,~, A.R. V ~ c o n c e l l o s and R. L u z z i , P h y s i c s 1 4 4 B , 3?6 ( 1 9 8 7 ) . P. D a v i e s , Nonlinear Int~grol end Di f f ~ r e n d i o l Eclu,~fton~ ( D o v e r , , New York> 1962); also in Ref. (1), Pax't` II, S e c t i o n 6. A.S. E,=pe1-1dl~o, A.R. Vasconcellos and R. Lu~zl, Solid St`st`e C o m m u n .

(in pr-ess). J.A. v a n Vecht`en, J. P h y s . (P~a~Is) C 4 t , 4 ( 1 9 8 0 ) ; S o l i d St, a t . e Common. 3 g , 1835 (1981). A.S. Esperidi~o, A.R. Vasconcellos and

(in

R.

Luzzl~

press).

Solid

St`at`e

Commun.