Proposal for the direct electromagnetic generation ofcoherent terahertz acoustic phonons in semiconductor
superlattices at the University of California, Santa Barbarafar-infrared free-electron-laser facility
T. E. Wilson
Department of Physics and Astronomy, Connecticut College, Mohegan Avenue, New London, Connecticut 06320
Received October 24, 1988; accepted January 3, 1989
We discuss the possibility of the generation of coherent packets of monochromatic acoustic waves in the far-infraredfrequency range by electromagnetic radiation. The University of California, Santa Barbara free-electron lasercould be used to excite resonantly transverse acoustic phonons in periodic space-charge layers formed in modula-tion-doped GaAs-Gal-,A 5 As superlattice structures. The conversion efficiency for the process (acoustic power/incident electromagnetic power) is estimated to be of the order of 10-3 at 0.25 THz.
INTRODUCTION
Terahertz phonons in crystalline solids have attracted in-creasing attention in recent years. Widely different meth-ods have been developed to study the generation and decaymechanisms and the scattering of dispersive high-frequencyphonons in insulators and semiconductors.'
Here we discuss a new method for the generation of coher-ent packets of monochromatic terahertz transverse acoustic(TA) phonons, based on the theory of Quinn et al.,2 which isparticularly well suited for testing at the University of Cali-fornia, Santa Barbara free-electron-laser (UCSB-FEL) fa-cility. Such phonon pulses could be used to study a numberof interactions in solids, such as phonon-phonon, phonon-defect, and electron-phonon interactions.3
DIRECT ELECTROMAGNETIC GENERATIONOF TERAHERTZ ACOUSTIC WAVES INSEMICONDUCTOR SUPERLATTICES
Quinn has postulated that periodic space-charge layersformed in modulation-doped semiconducting superlatticestructures can be used for direct electromagnetic generationof coherent, high-frequency acoustic waves. We summarizehis results below and perform a simple model calculation toarrive at an order-of-magnitude estimate for the generatedacoustic power.
The active region of the device consists of a superlattice inwhich positive and negative space-charge layers are presentin an alternating fashion, such as positively charged donorsand free electrons in modulation-doped GaAs-AlGal-,As.This active region can be grown on a GaAs substrate bymolecular-beam epitaxy.
Along the z direction perpendicular to the layers, thecharge density can be written as
p(z) a(z) > p, exp(winz/d), (1)n
where d is the period and a (z) is a slowly varying function ofz to account for the finite number of periods.
Electromagnetic radiation normally incident upon thestructure will impart a spatially periodic force to the latticeeither directly by acting upon the ionized donors or indirect-ly through collision drag of the free carriers. In the elasticcontinuum approximation, the scalar equation of motion ofthe lattice can be written as
(2)-co2 + ir - s2 2) (Z) = F() _ p(z)E(z)'
where r is a damping parameter associated with the attenu-ation rate of TA phonons in the active region, t(z) is thedisplacement field of the lattice, F(z) is the force per unitvolume on the lattice, Pm is the mass density, and s is thesound velocity. Here we have assumed a harmonic timedependence for the fields, a single transverse sound velocityfor both the active region and the substrate, and coherencebetween the collision drag force and the electric field.
The value of the electric field E(z) is determined by solv-ing the wave equation (Gaussian units)
(02 + 2 Es E = 2 AZ)-a.Z2 C2/ c 2 ()
(3)
Here es is the background dielectric constant, j(z) = cioE(z) +p(z)t is the total current density, and o is the dc conductiv-ity.
Following Quinn, as a first approximation, we neglect theeffect of acoustic generation on the electric field so that Eq.(3) describes the normal skin effect with solution E(z) =Eoe-az, where a = [(4ici/c 2)0ro - E,(cw2/c2)]l/2 is the inverse ofthe normal skin depth. For the charge density no - 1018cm-3 appropriate to a typical GaAs-AlGal-,As superlat-tice, ad << 1, so that the electric field is essentially constanton the scale of a superlattice period.
In order to arrive at an order-of-magnitude estimate forthe generated acoustic power, we now assume that we have
an infinite superlattice with a uniform electric field through-out. The Fourier coefficients of the displacement field canbe written as
- E o J dzp(z)eiGz I."
dpmC(-2 +-S 2 G2 + iwr) ''
where the reciprocal lattice vectors are given by G = 2r p/dand p is an integer.
The displacement field is obtained as a sum over its Fouri-er coefficients:
t(z) E ( ~p exp( i27 )p odd
E02ieno expi d/ /
podd 7rPmp-c2 + 2(P27r)2 + icr]
where we have assumed a space-charge density of the form
P(z) = eno Gal-,AlxAs layer-eno GaAs layer
We now proceed to estimate the damping parameter F.The TA phonon branch is forbidden to decay anharmonical-ly because of kinematic restrictions.4 The usual convexcurvature of w(k) for the lowest branch does not permitsimultaneous conservation of momentum and energy, evenfor higher-order anharmonic decay and including Umklappprocesses.5 The only mechanism that provides further en-ergy relaxation of TA phonons is elastic scattering of thetype transverse acoustic - longitudinal acoustic, so-calledmode conversion.
In the common group IV and III-V semiconductors thedominant intrinsic source for elastic scattering and modeconversion is the isotopic disorder of the crystal lattice.6The theory for elastic phonon scattering on dilute isotopicimpurities in otherwise perfect crystals has been developedby Klemens and was recently adapted to the dispersive re-gime and diatomic lattices by Tamura.7
The isotopic scattering rate is given by
Ir (M = 6 V2 (11)
where Vo is the volume per atom and D is the phonon density(6) of states per unit volume. g2 is defined by
and each layer is of thickness d/2. This is a rough approxi-mation since no is a function of z due to band bending, but wemake it to obtain the order-of-magnitude estimate desired.Equation (5) represents a sum of standing waves with wavevectors odd multiples of 27r/d. This steady-state solutionhas been obtained for an infinite superlattice (SL), but it isexpected to have this form for a SL with a large number ofperiods.
We consider the resonant condition at the fundamentalfrequency = 2rs/d. This corresponds to a Bragg scatter-ing condition at the zone center of the first Brillouin zone ofthe SL. If the phonon stopband is small, and Q >> , a pulseof electromagnetic radiation will generate a simple harmonictraveling TA wave in the substrate of the form
-2Eoeno 2ii-z~Z, t = 2Een exp it -i 27z*(7)
The time-averaged generated acoustic power P per unit areaentering the substrate becomes
1 9 2 2E02se2 n0
2 (8P = 2Ct d = p r2r2 (8)2 OZ Pr
Because the SL region is thin (typically -1 ptm) with a largeskin depth [0(103, m)], the entire device, SL and substrate,will behave as a simple dielectric in transmitting the normal-ly incident electromagnetic power per unit area W. There-fore the field in the SL is given by
E2 =4r TW= _6 W, (9)
where T is the transmission coefficient. It is easily shownthat the conversion efficiency it is given by
= P/W= 32s [ eno 2 (10)Pm7rC [( + )r(
92 = Z -yi(l -M/)2, (12)
where -yi and mi are the relative fraction and mass of the ithisotope, respectively, and mn is the average mass of all atoms.In bulk GaAs, the lattice vibrations of As do not contributeto the scattering. One needs to consider only the two stablenuclei 69Ga and 7 Ga with relative abundances of 36% and64%, respectively, resulting in g2 = 1.97 X 10-4. For pho-nons of frequency v in the dispersionless regime in bulkGaAs, r-'(v) = 7.38 X 10-42 4 (sec'1),' in reasonable agree-ment with experiment.7
In the GaAs-Gal-,AlxAs SL, the replacement of Ga by Alhas been successfully modeled with respect to phonon phe-nomena as a mass change only (no change in effective forceconstants).8 We therefore assume that the scattering ratedue to mode conversion can be obtained by treating the onestable nucleus of Al, 27AI, as an isotope of Ga in the computa-tion of g2 . The GaAs-Gaj_,AlxAs SL that we intend toemploy will have an x of at most 0.5 so that g2 = 0-099-Therefore the mode-conversion scattering rate in the SL isestimated to be
O= _ = ( 0.099 4) 7.38 X 10-422 \1.97 X 10-
P4 = 3.75 X 1O-'9 v4 (sec'1). (13)
In the model that we have used, the velocity of sound wasassumed to be the same in both layers of the superlattice. Ingeneral this will not be the case, and gaps will develop at thezone-center resonant frequency of the folded Brillouinzone of the SL with zone boundaries of +7r/d. The change indispersion associated with these gaps could have an impor-tant effect on the generation of phonons, particularly wherethe resonances should be especially sharp. However, if theindividual layer thicknesses of the SL satisfy a certain crite-rion, the gap at Q2 can be made to vanish, as we discuss below.
A forward scattering Raman scattering study of acoustic
T. E. Wilson
1060 J. Opt. Soc. Am. B/Vol. 6, No. 5/May 1989
Table 1. Parameters of GaAs-Gai_,-Al:As SL samplesa
Es sTA (100) (cm/sec) Pm (g/cm3 ) no (cm- 3 ) f (sec'1) r (sec-1) d (cm)
12.7 3.3 X 105 5.4 -2 X 1018 1.6 X 1012 2.9 X 107 1.3 X 10-6
a From Ref. 16.
zone-center gaps in GaAs-AlAs superlattices by Jusserand 9
has demonstrated that the values of the gap opening are ingood agreement with elastic continuum model predictions,which we restate below.
The gap opening, Q, at the first folded zone-center fre-quency Q can be written, to second order in ( - Q), for a SLof period d comprising layer thicknesses di (d2), sound veloc-ities si (s2), and densities P1 (P2) as
7r |i [XS1 +(1-x)s2]| (14)
where x = d2/d and a = Ip1s1 - P2S2I/(PlSlP2S2)1/2 character-izes the SL modulation. Hence the energy gap is small ingeneral for large-period superlattices with the concomitantsmall conversion of the zero-order traveling wave into astanding wave. More importantly, there is no Bragg scatter-ing when x = 2/(S1 + s2), corresponding to a vanishingstructure factor for the SL primitive cell. Therefore, pro-vided that x = x,, the TA phonons generated in the modula-tion-doped SL by direct electromagnetic excitation shouldproceed unhindered into the GaAs substrate.
To evaluate x, we need the TA sound velocity in theGal-xAlxAs alloy, which is not well known, but it may beestimated as follows.' 0 Let indices 1 and 2 denote GaAs andGal-xAlxAs, respectively. The density P, of GaAs is givenby
P = (N/2A V1)(MGa + MA), (15)
where N is the total number of atoms of both species in a unitcell and AV is the volume of the unit cell. In Gal-xAlxAs, afraction (1 - x) of Ga remains, with the fraction x replacedby Al. Ignoring the change with x of the volume of the unitcell, we have for the mass density of the alloy
P2 = (N/2AV)[(1 - X)MGa + XMAI + MAS]- (16)
By using Eq. (15), Eq. (16) can be rewritten as
P2 = p(l + x), bx = x(MAI - MGa)I(MGa + MA). (17)
An order-of-magnitude estimate for 5x can be obtained forthe case x = 0.5 by direct substitution of the atomic masses;one finds that 60.5 = -0.15.
We now use the results of Narayanmurti et al. in theirstudy of a dielectric phonon filter" involving selective trans-mission of incoherent 0.22-THz phonons that were generat-ed by a thin tunnel junction. The ratio of the acousticimpedances of the SL layers was found to be
-- = 0l l = 0.88 (18)Z 2 P2S2
for an Al concentration x = 0.5. Equations (17) and (18)then yield the speed of sound in the alloy s2 = 1.34 sl, and theSL critical geometrical factor x, = d2/d = 0.57.
Our modulation-doped GaAs-Ga1_xAlxAs SL samples,grown on (100) substrates, have been fabricated' 2 with thisvalue of x, in order to maximize the transmission of the TA
phonons into the bulk: i.e., d, = 57 A, d2 = 75 A, d = 132 A.Table 1 lists some relevant parameters that we now use to
estimate the conversion efficiency given by Eq. (10). Wenote that d was chosen to match the resonant frequency tothe output of a 1.2-mm molecular gas laser, i.e., P = 0.25 THz.When the values listed above are used, the conversion effi-ciency is q = 1 X 10-3, i.e., -30 dB.
The generated phonon packet will travel ballisticallyacross the substrate and be detected by a small (-0.5 mm X0.1 mm) tunnel heterojunction made of two superconductorswith different energy gaps. An Al-PbBi heterojunction actsas a sensitive phonon spectrometer over the frequency range130-340 GHz (Ref. 13) and hence will allow for the mono-chromaticity of the generated phonon packet to be tested.The UCSB free-electron laser provides a frequency-tunable(10-100-cm-') linearly polarized, pulsed, coherent source ofhigh-power (<10 kW peak) far-infrared radiation.l4 Such afar-infrared source would prove quite useful for investiga-tions of the resonant phonon generation technique discussedherein. Assuming that the free-electron laser radiation isfocused to a spot size of -1 cm2 at the sample, the resonantacoustic power incident upon the detector is 5.4 mW, wellabove the 10- 6-W level necessary for a reasonable signal-to-noise ratio when the heterojunction is used.'5
REFERENCES AND NOTES
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12. Grown by courtesy of C. W. Tu, AT&T Bell Laboratories, Mur-ray Hill, New Jersey 07974.
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