Photoinduced amplification of hypersound in superlattices

PERGAMON

Solid State Communications 120 (2001) 47±51

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Photoinduced ampli®cation of hypersound in superlattices O.A.C. Nunes*, A.L.A. Fonseca, D.A. Agrello Institute of Physics, University of BrasõÂlia, P.O. Box 04455, 70919-970 Brasilia D.F., Brazil Received 22 February 2001; accepted 16 April 2001 by C.N.R. Rao; received in ®nal form by the Publisher 17 July 2001

Abstract The high-frequency acoustic-phonon scattering by electrons in superlattices in the presence of an intense radiation ®eld is discussed. It is shown that in the intense ®eld regime of the electromagnetic wave the hypersound attenuation coef®cient may revert its sign and hypersound ampli®cation can be obtained. An application is made for GaAs based superlattices. q 2001 Elsevier Science Ltd. All rights reserved. PACS: 72.10.Di; 73.50.Rb; 73.20.Di; 78.66.2w Keywords: A. Semiconductors; D. Electron±phonon interaction; D. Phonons

1. Introduction The ampli®cation of acoustic waves by external ®elds in homogeneous semiconductors has been considered on several occasions [1±4] and the advent of semiconductor superlattices (SL) with interesting properties [5] has renewed the interest on the subject [6]. Superlattices are periodic semiconductor structures characterized by the presence of an additional periodic potential which splits the energy bands of the bulk semiconductor into a set of narrow (less than 0.1 eV) allowed and forbidden sub bands (mini bands). Among the works devoted to the interaction of sound with conduction electron of the SL one can mention those [7,8] where acoustic-wave ampli®cation in a SL in the presence of static (nonquantizing) electric ®eld has been investigated in complete analogy to the sound ampli®cation in homogeneous semiconductors [9]. We can also mention the works [10±12] in which hypersound attenuation in the presence of a radiation ®eld has been investigated. In particular, it has been shown [11,12] that the attenuation coef®cient depends on the phonon wave vector q~ in an oscillatory manner. However, in Refs. [11,12] only the effect of the strong laser amplitude on the attenuation coef®cient has been considered, i.e. no photon absorption …` ˆ 0† (` is the number of photons) by electrons in the SL has been * Corresponding author. Fax: 161-3072363. E-mail address: oacn@helium.®s.unb.br (O.A.C. Nunes).

investigated. Here in this paper we investigate the possibility of hypersound ampli®cation (negative attenuation) as a consequence of intraminiband photon absorption …` ± 0† by electrons in SL. Intraminiband absorption of electromagnetic waves by the semiconductor superlattices has been a subject extensively investigated [13±15]. The intraband absorption of light by charge carriers in semiconductors is always followed by emission or absorption of phonons (or via other scattering mechanisms) so that the electrons can gain the necessary momentum for the transition. The motivation for such a study of hypersound ampli®cation, is the possibility of a high-frequency sound (hypersound) apparatus similar to a long wave sound generator. Also, the ®ltering action of a SL as GaAsAl0.5/Ga0.5As has been observed experimentally and it can be exploited for the construction of a high-frequency (,2.2 £ 10 11 s 21) phonon spectrometer [16]. It is therefore important to consider in this paper the negative attenuation (ampli®cation) of hypersound by the multiphoton absorption of an intense radiation ®eld in superlattices.

2. Formalism In solving this problem of phonon ampli®cation in SLs we shall use the Hamiltonian of the electron±phonon system in a SL in second quantized notation …" ˆ 1†;

0038-1098/01/$ - see front matter q 2001 Elsevier Science Ltd. All rights reserved. PII: S 0038-109 8(01)00315-5

48

O.A.C. Nunes et al. / Solid State Communications 120 (2001) 47±51

namely Hˆ

X



e…s† p~ 2

p~;s



X e~ vk~ b1 b A…t† a1 p~;s ap~ ;s 1 k~ k~ c ~ k

1 X X 1 1 p C~ Ms;s 0 …kz †a1 ~ g;s 0 £ …bk~ 1 b2k~ † p~;s ap~ 2k1n~ N p~;k~ s:s 0 ;n k …1† where

e…s† …~p† ˆ

p~2' 1 Ds …1 2 cos pz d† 2m

s ˆ 1; 2; 3; ¼

…2†

Here d is the superlattice period, m the effective electron mass, Ds the half width of the s-th allowed miniband, p' and pz are the quasi-momentum components across and along ~ the SL axis, A…t†; which is directed along the SL axis, is the vector potential related to the electric ®eld of the electro~ ˆE ~ 0 sin Vt by the relation E ~ˆ magnetic wave E…t† ~ 2…1=c†…2A=2t†; a22 p~s and ap~s are the creation and annihilation operators of an electron with quasi-momentum p~ in the s-th and bk~ are the creation and annihilation operaminiband, b1 k~ tors for phonons, N is the number of SL periods; g ˆ …0; 0; 2p=d† is the reciprocal SL vector, Ms;s 0 …kz † is given by Ms;s 0 …kz † ˆ

ZNd 0

wps 0 …z†ws …z† eikz z dz

…3†

where ws …z† is the wave function of the s-th state in one of the one-dimensional potential wells from which the SL potential is formed; Ck~ is the matrix element of the electron±phonon interaction in homogeneous material which is given by 8 4p e b > < …2r0 vk~ †21=2 …piezoelectrical interaction† e0 Ck~ ˆ > : …2r0 v~ †21=2 Lk~ …deformation potential†

condition k` q 1 means that the hypersound wavelength is far smaller than the mean free path of the electron. In order to determine the attenuation (ampli®cation) coef®cient we shall use the equation of motion method [17], starting from the Hamiltonian (1). We shall assume that for t ˆ 21 the external ®eld is absent and we have a system of noninteracting electron and phonons in thermal equilibrium, so that for all k~ we have kbk~ l21 ˆ 0: Here we use the notation kXlt ˆ Tr{Xr…t†}

where r…t† is the density matrix of the system de®ned by the equation i

2r ˆ ‰H; rŠ 2t

…5†

Then we consider that the electron±phonon interaction and the external laser ®eld are set in operation in an adiabatic manner, and by means of some external source nonequilibrium phonons with a wave vector k~ are excited, so that starting from this instant, kbk~ lt ± 0: It is the subsequent fate of these phonons which we intend to determine. It follows from Eqs. (1), (4) and (5) that i

2 kb~ l ˆ k‰bk~ ; HŠlt 2t k t X X 1 ˆ vk~ kbk~ lt 1 p C2k~ Mss 0 …2kz † N p~ s;s 0 ;n £ ka1 ~ g;s 0 lt p~;s ap~ 1k1n~

…6†

On the other hand, the equation of motion for ka1 ~ g;s 0 l is p~;s ap~ 1k1n~

k

where r0 is the crystal density; vk~ the phonon frequency ~ L the deformation potential constant; with wave vector k; b the piezoelectric modulus; and e 0 is the lattice dielectric constant. Some approximations underlying the starting Hamiltonian (1) have been assumed. The electromagnetic wave frequency is assumed to be large compared with the reciprocal of electron mean free time t 21, (i.e. Vt q 1) and the wavelength is taken to be large compared with the SL period, electron mean free path, and the de Broglie wavelength. This will enable us to use the dipole approximation. Also, the plane electromagnetic wave of frequency V satis®es V=v p . 1; where v p is the plasma frequency. As for the phonons, we shall con®ne our considerations to those for which the wave vector k~ satis®es the conditions k` q 1; where ` is the electron mean free path in the superlattice. Such phonons constitute well-de®ned elementary excitations of the system. Conditions Vt q 1 and V . v p ensure that the electromagnetic wave should penetrate well into the sample and

…4†

i

0 2 1 …s† 1 ka a ~ l ˆ …ep…s~1†k;s ~ g;s 0 lt ~ 2 ep~ †kap~;s ap~ 1k1n~ 2t p~;s p~1k1n~g;s t 1 XX 1 p C~ ‰Ms 0 s 00 …kz † N q~ s 00 n 0 k

£

ka1 ~ q1…n1n 0 †~g;s 00 …bk~ p~;s ap~ 1k2~

1

b1 †l 2k~ t

…7†

1 Ms 00 s …kz †

1 £ ka1 ~ g;s …bk~ 1 b2k~ †lŠ p~1~q2n 0 g~ ;s 00 ap~1k1n~

Solving Eq. (7) with the initial condition ka1 ~ g;s 0 ltˆ21 ˆ 0 and substituting into Eq. (6) we p~;s ap~ 1k1n~ obtain i

XX 2 C2k~ Mss 0 …2kz † kb~ l ˆ vk~ kbk~ lt 2 2t k t p~ ss 0 n £

Zt 21

e2

Rt

2t 0

…ep~2k~ 2ep~ dt 00 †

dt 0 F

…8†

O.A.C. Nunes et al. / Solid State Communications 120 (2001) 47±51

where X Fˆ Ck~ G; s 00 k 0

…9†

1 G ˆ ‰Ms 0 s 00 …kz †ka1 ~ q1…n1n 0 †~g;s 00 …bk~ 1 b2k~ †lt p~;s ap~ 1k2~ 1 2 Ms 00 s …kz †ka1 ~ g;s …bk~ 1 b2k~ †lt 0 Š p~ 1~q2n 0 g~;s 00 ap~1k1n~

Taking now the electron±phonon interaction as weak, we decouple the right-hand side of Eq. (8) in the following manner [12]: ka1 p~s ap~ 0 s 0 bk~ lt ˆ dp~ ;~p 0 ds;s 0 kbk~ lt np~ s ka1 p~s ap~s lt

where np~s ˆ is the electron occupation number. We thus obtain ! XX 2 1 ivk~ kbk~ lt ˆ C2k~ Ck1n~ ~ g Mss 0 …2kz †Ms 0 s …kz † 2t p~ ss 0 n £…kz 1 ngz †‰np~s 2 np~1k;s ~ 0Š £

Zt

0

21

dt kbk1n~ ~ g lt 0 1

!

" 0 £ exp i…ep~s 2 ep~1k;s ~ 0 †…t 2 t † 2

(10)

Eliminating kb 2k2n~ ~ g lt 0 with the help of the conjugate equation and taking the Fourier component Z1 Bk~ …v† ˆ kbk~ lt eivt dt 21

we obtain …v 2 vk~ †Bk~ …v† 1 X X qˆ2 1 ss 0 n

£

C2k~ Ck1n~ ~ g Mss 0 …2kz †Ms 0 s …kz 2 ngz †

2vk1n~ ~ g

v 1 vk1n~ ~ g 2 qV

…11†

~ v† B k~ …v 2 qV†M q …k;

1 X

~ v 1 `V† …12† J` …l=V†J `2q …l=V†P ss 0 …k;

`ˆ2 1

and ~ v 1 `V† ˆ P ss 0 …k;

X p~

n p~;s 2 np~1~q;s 0 ep~1k;s ~ 0 2 ep~;s 2 v

v 2 vk~ 2 Ck~2

1 X X p~

`ˆ2 1

J`2 …l=V†

…n p~ 2 np~1k~ †

ep~1k~ 2 ep~ 2 v

ˆ0

…15†

which is the dispersion equation for the phonons in the presence of the laser ®eld. It follows that the phonon attenuation coef®cient in the ®eld of the laser wave will be given by the imaginary part of Eq. (15), that is 1 X

J`2 …l=V†g ~…0† …vk~ 1 `V†

…16†

…np~ 2 np~1k~ †d…ep~1k~ 2 ep~ 2 v†

…17†

`ˆ2 1

X p~

k

In Eq. (16) J` …x† is the Bessel function of order ` and argument x. The expression for g…0† …v† formally coincides k~ with a similar expression for the sound attenuation coef®cient in the absence of the laser ®eld the only difference is the shift in the argument of gk~ by `V which permits the possibility of absorption or emission of ` ®eld quanta by conduction electrons upon interaction with phonons. The case ` ˆ 0 is the one discussed in Refs. [11,12]. It follows from Eq. (16) that if g k~ . 0 we have hypersound attenuation, whereas if gk~ , 0 we have hypersound ampli®cation (negative attenuation) as a result of the absorption …` . 0† and emission …` , 0† of u`u photons from the intense laser ®eld. 3. Hypersound ampli®cation coef®cient

where ~ v† ˆ Mq …k;

…14†

This result can be further simpli®ed to

g k~…0† …v† ˆ pCk~2

#

ˆ

~ v† ˆ 0 v2 2 v2k~ 2 2vk~ Ck~2 M0 …k;

where

eE0 d D V2

£ {sin…pz 1 kz †d 2 sin pz d}…sin Vt 2 sin Vt 0 †

when the electron gas is nondegenerate and only the lowest miniband is ®lled, i.e. np~s ˆ np~ and also consider the situation when the characteristic width of the potential well is small compared with the SL period whose in this case, M…kz † ˆ M…kz 2 ngz † ˆ 1: We shall also limit ourselves to q ˆ 0 since q ± 0 in the summation in Eq. (11) gives terms of higher order in perturbation. Then for n ˆ 0 we get from Eq. (11)

gk~ ˆ 2Im v ˆ

kb1 ~ g lt 0 2k2n~

49

…13†

Here l ; …eE0 Dd=V†‰sin…p z 1 kz †d 2 sin pz dŠ is the intense laser ®eld parameter …" ˆ 1†: In what follows, we shall con®ne ourselves to the case

In this section we shall calculate gk~ in the intense ®eld regime of laser ®eld and show that under certain condition the hypersound attenuation may revert its signal (ampli®cation). We shall also consider the data of a real SL to con®rm the expectation put forward in this section. In the regime of intense laser ®eld, l q V; so that only the electron±phonon collisions with the absorption or emission of ` q 1 photons are signi®cant. The laser ®eld regime, l p V where only one-photon processes …` ˆ ^1† are signi®cant has already been considered elsewhere [10]. Accordingly, in the case l q V the argument of the Bessel function J ` …l=V† is large. For large values of the

50

O.A.C. Nunes et al. / Solid State Communications 120 (2001) 47±51

argument, the Bessel function J ` …l=V† is small except when the order is equal to the argument. The sum over ` in Eq. (16) may be written approximately [18,19] 1 X `ˆ2 1

J `2 …l=V†d…E 2 `V† ù

1 ‰d…E 2 l† 1 d…E 1 l†Š; 2

pCk~2 X n {…1 2 e2…vk~ 1l†=kB T †d…ep~1k~ 2 ep~ 2 vk~ 2 l† 2cs p~ p~ 1 …1 2 e

2…v~k 2l†=kB T

† 1 d…ep~1k~ 2 ep~ 2 vk~ 1 l†} …18†

where np~ ˆ C e2ep~ =kB T is the carrier distribution function (nondegenerate semiconductor), CPbeing a factor to be determined, by the condition …1=V† p~ np~ ˆ n0 …n0 ˆ N=V†: The ®rst d -function in Eq. (18) corresponds to the emission and the second to the absorption of l=V q 1 photons. The number of photons absorbed or emitted is the same order of magnitude as the ratio of the classical oscillatory energy of the electron to the photon energy, namely [18] …" ˆ 1† `ù

…2e 2 E02 †=mV 2 V

…19†

Therefore, multiphoton absorption or emission processes …` q 1† are valid for laser ®elds such that E 0 q …V=e†…mV=2† 1=2 : We now assume that the electron temperature is low. Then KB T p l and the emission term is negligible compared to the absorption term. This is justi®ed provided D q KB T as is being considered here. Eq. (18) becomes

gk~ ˆ

pCk~2 X n …1 2 e…l2v~k †=kB T †d…ep~1k~ 2 ep~ 2 vk~ 1 l† 2cs p~ p~ …20†

The continuity of the carrier energy states as a function of the momentum p~ requires the summation in Eq. (20) be substituted by an integral. According to Eq. (2) it is convenient to choose cylindrical coordinates by making the substitution X p~

!

gk~ ˆ

1=2

p1=2 Ck~2 "Vn0

D kz cs d 3 m…kB T†3=2

e2"…v0 2cs †

£ …1 2 e"kz …v0 2cs †=kB T †

where E ; e p~1k~ 2 ep~ 2 vk~ : The attenuation coef®cient then becomes

gk~ ˆ

(z-axis), one obtains for the hypersound attenuation constant

V Z p' dp' dpz …2p†2

Substituting np~ in Eq. (20) by np~ ˆ C e2ep~ =kB T with its prefactor C evaluated for a SL in the approximation pz d p 1 and performing the indicated integration using the d -function, assuming the phonons propagating along the SL axis

2

=Dd 2 kz kB T

…21†

provided …p 2 k†d p 1: In Eq. (21), v0 ˆ eE0 Dd 2 ="2 V where cs is the hypersound velocity and n0 is the carrier density. The constant " has been recovered to provide the correct dimension of g k~ : 4. Discussion and conclusions Eq. (21) tells us that the condition for the attenuation coef®cient gk~ to be negative (hypersound ampli®cation) is 1 , e"kz …v0 2cs †=kB T or v0 . cs. One may say that the effect of the intense laser ®eld is to give a drift velocity v0 ˆ eE0 Dd 2 ="2 V to the electrons. Hence when v0 exceeds the sound velocity of the high frequency acoustic phonons, hypersound ampli®cation obtain whatever the values of kz. This is in contrast with the results of Ref. [10] where the phonon ampli®cation is obtained in the presence of a weak laser ®eld for a particular band of phonon wave vectors. This photoinduced-drift mechanism is analogous to the reversed Landau damping mechanism of plasmas in which plasma electrons drift the plasma waves under the action of a static electric ®eld. Whenever the drift velocity vD of the electrons as imposed by the external ®eld is greater than the phase velocity v f of the plasma wave energy can be given up by electrons to the plasma wave and as a consequence, the plasma wave is ampli®ed. If on the other hand vD is smaller than vf the plasma wave is damped (Landau damping). Furthermore, since gk~ / d 23 D21=2 the hypersound ampli®cation constant can be enhanced over the bulk provided we decrease the SL period d and also by the choice of semiconductors which compose the sample such that D is made smaller. However, since the gain of acoustic waves (ampli®cation) is shown to be proportional to d 23 D21=2 (D is the miniband half width), one expects that short period and narrow miniband systems can be found. Nevertheless, a short period invariably leads to wide minibands. Also, if the miniband width becomes too small, the drift velocity as imposed by the intense laser ®eld is affected. Therefore, it is important from the experimental viewpoint to consider these points in the choice of the material. Condition (19) de®nes a critical laser ®eld for multiphoton absorption processes. It follows that one has ampli®cation of hypersound provided E0 $ Ec ; where Ec ˆ …m"V 3 =2e2 †1=2 : To get a numerical estimate of gk~ we take the following typical parameters for the GaAs made superlattice: mp ˆ 0:066m0 ; cs ˆ 5:14 £ 105 cm s21 ; r0 ˆ 5:3 g cm23 ; b ˆ 2:1 £ 1025 dynes; D ˆ 0:01 eV; n0 ˆ 1015 cm23 ; kz ˆ 3 £ 105 cm21 ; e0 ˆ 18:0; T ˆ 50 K; d ˆ 1026 cm; vk~ ˆ 2:2 £ 1011 s21 ; V ˆ 10212 cm3 : It follows from Eq. (21) that for a radiation wave length 10.6 mm

O.A.C. Nunes et al. / Solid State Communications 120 (2001) 47±51 4

21

(CO2 laser) and a ®eld E0 ˆ 5 £ 10 V cm we get, gk~ , 105 cm21 which is larger than that for bulk semiconductor, thus re¯ecting the enhanced physical properties of SL and microstructures as the characteristic dimension is reduced. A point we should like to comment on here is the exact amount of gain obtained for hypersound in these superlattices. To this, we need to know what are the linear losses g 0 for typical superlattices which to our knowledge has not yet been experimentally measured, in order to see if the condition gk~ . g 0 is achieved. Speci®cally, g 0 represents the phonon decay rate which may interact with the phonons and lead to a ®nite phonon lifetime. In this respect, the present paper is incomplete. Nevertheless, we think that further experimental results on the acoustic phonons losses are awaited in superlattices to test the predictions of the current work. The hypersound ampli®cation mechanism considered here can be used for indirect experimental determination of the coef®cient of electron±phonon interaction in a SL since the hypersound ampli®cation factor is proportional to Ck~2 : Acknowledgements OACN and ALAF wish to thank the CNPq (Brazilian agency) for Doctor Research grants. References [1] E.M. Epshtein, Sov. Phys. JETP Lett. 13 (1971) 364.

[2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]

51

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