PHYSICA [~I.qEVIER
Phystca D 83 (1995) 229242
Nonlinear quantum dynamics in semiconductor quantum wells Mark S Sherwln d, Kelth Craig d, Bryan Galdrlklan d, James Heyman ~ 1, Andrea Markelz d, Ken Campman b, Simon Fafard b'e, Pete F. Hopkins b'3, Art Gossard b 'Ph~sws Department, ~entet for FreeElectron Laser Studte~, and Center ~or Nonhnear 5cwn~e, University o] Cahfornta at Santo Barbara Santa Barbara, C 4 93106, USA bMatertals Department, Untversm' of Cahfornta at Santa Barbara, Santa Barbara, CA 93106, U~A
Abstract
We discuss recent measurements of the nonlinear response of electrons m wide quantum wells driven by intense electromagnetic radiation at terahertz frequencies The theme is the interplay of quantum mechanics, strong periodic driving, the electronelectron interaction and dissipation We discuss harmonic generation from an asymmetric double quantum well m which the effects of dynamic screening are important Measurements and theory are found to be in good agreement We also discuss intensitydependent absorption in a 400A square quantum well A new nonhnear quantum effect occurs, in which the frequency at which electromagnetic radiation is absorbed shifts to the red with increasing intensity The preliminary experimental results are in agreement with a theory by Zaluzny, in which the source of the nonlinearity ts the selfconsistent potential m the Hartree approximation for the electron dynamics
1. Introduction
T h e original inspiration for this work came f r o m the experiments of Peter K o c h [1] and Jim Bayfield [2] on R y d b e r g h y d r o g e n atoms in strong microwave fields In the range of p a r a m e ter values used in their experiments, the classical phase space contains both regular and chaotic
~Current address Department o1 Physics, Macalaster, St Paul, MN 551115 USA 2 Current address National Research Council of Canada, Ottowa, ON K1A036, Canada National Inshtutes ot Standards and Technology Boulder CO 80303, USA
regions The underlying classical d y n a m i c s leave a clear Imprint on the true q u a n t u m d y n a m i c s of the system The physics is welldescribed within the f r a m e w o r k of the q u a n t u m mechanics of a single p a r t i c l e  t h e influences of the e l e c t r o n electron interaction and dissipation are neghglble If one solves the Schrodlnger e q u a t i o n for a single electron in a wide (1 e , 400A) 1D square q u a n t u m well driven by a strong ac field with frequency near 1 T H z , the classical d y n a m i c s are indeed chaotic, and the q u a n t u m d y n a m i c s retain some footprints of the underlying classical dynamics [3] H o w e v e r , the singleelectron Schrodlnger e q u a t i o n is not a particularly g o o d approximation to the physics In wide q u a n t u m wells at
01672789/95/$09 50 © 1995 Elsevmr Scmnce B V All rights reserved S S D I (/1672789(94)002665
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M S Sheram et al / Ph~szca D $3 (1995) 22q 242
typical sheet denslhes (1() ~ cm 2), the electronelectron lnteracnon cannot be neglected and indeed leads to some fascinating new phenomena Furthermore, diss~panon plays an ~mportant role m scattering electrons between quantxzed subbands Such scattering ~s induced by emission of acousnc and optic phonons, electronelectron scattering, and electronimpurity scattering [4] The theme of this paper is the interplay of quantum mechamcs, electronelectron interactions, strong driving and dissipation in wide quantum wells driven by intense THz frequency electromagnetic radmt~on The current m o n v a n o n for th~s work is twofold First, quantum wells are very clean and wellcontrolled systems m which to study the quantum mechamcs of an interacting electron gas Second, the electromagnenc spectrum between 0 2 T H z and 3 0 T H z (1 5 mm and 10 #.m wavelength) ts very poor in technology At the present time, there exist no compact sohdstate sources of radianon which operate in this range of frequencies Quantum wells and other semiconductor heterostructures are good candidates for detectors and sources m this range of frequencies In fact, a "quantum cascade l a s e r " based on mtersubband transmons m narrow quantum wells  has recently been demonstrated at 4 ~m wavelength [5] The remainder of the paper is dlwded into three mare parts Section 2 gives a brief mtroductxon to the physics of electrons m quantum wells (for a more complete dtscussaon, see Ref [4]) Section 3 discusses experiments on resonant harmomc generanon from a quantum well m which the electronelectron interaction, in the form of dynamic screening, Is important Experimental and theorencal results are reported, and are m good agreement Section 4 d~scusses experiments which demonstrate a new, nonlinear quantum effect the frequency of the peak of the lntersubband absorption line decreases with mcreasing intensity of the incident radlanon The paper concludes by discussing some open questions. Including the possibdity that manybody
interactions may, under certain condmons, lead to chaotic dynamics in simple quantum systems
2. Quantum wells, intersubband transitions, and our experimental approach Quantum wells m I I I  V semiconductors are grown by molecular beam epltaxy The A1,Ga 1 ,As system which we are studying here is exceptional because AlAs and G a A s are almost perfectly latticematched, an arbitrary alloy A1,Ga t ,As is also almost perfectly latticematched to G a A s Since the bandgap vanes almost hnearly with A1 concentration, one may confine one electronic degree of freedom (m what we will call the " z "  d l r e c t l o n ) m an almost arbitrary potential, while leaving the other two degrees ot freedom unconfined The allowed states m the quantum well are usually computed within the effective mass approxlmahon it is assumed that an electron behaves hke a parhcle with charge  e and band mass m ~ O 067m , where m~ is the bare electromc mass The effectzve mass approximation is, for our purpose~, almost perfect within 20 meV of the band edge of GaAs, the range ot energies with which we are concerned The conduchon band breaks up into '~subbands "" Within each subband, the lowest energies e,, are the elgenvalues of the 1D Schrodmger equation for an electron moving m the zdirection m the potennal defined by the variation of the band gap with Aluminum concentration x (we ignore for the time being the effects of the electronelectron interaction) The energies of the states within the various subbands are given by E,, = e,, + h  k ±  / 2 m ~, where k z is the wavevector associated with the free motion of the electrons within the plane of the quantum well Our general approach, then, is to drive the electrons in a quantum well with an oscillating electric field polarized m the zdirection This oscillating electric field induces currents in the zdirection within the quantum well These oscfl
M S Sherwm et al / Physwa D 83 (1995) 229242
latang currents, which an general have Fourier c o m p o n e n t s at the driving frequency to as well as at its harmonics, absorb and emit electromagnetic radmtaon We use the absorption and emission as probes of the dynamics of the electrons in the quantum wells Studies of absorption and emission from quantum well mtersubband transmons are not new Intersubband absorption in inversion and accumulation layers in S1 was observed m the 1970s [6,7] The first observanon of antersubband absorption in quantum wells of order 100A. wade ( m t e r s u b b a n d spacings ~100meV) were made by West and Eglash m 1981 [8] In 1983, G u r m c k and D e T e m p l e [9] predicted that asymmetric quantum wells would display grant secondorder electric susceptlbllatles Th~s has been supported by m e a s u r e m e n t s of resonant secondharmonac generation [10,11] in A 1 G a A s / G a A s asymmetrac wells with intersubband spacings of the order of 100meV In addition, large thirdorder susceptabflitms in asymmetric wells have been observed through m e a s u r e m e n t s of the intensityd e p e n d e n t refractave mdex [12], and thirdharmonic generation [13] Crudely, the reason for the large enhancements of quantum well nonhnear susceptlbflmes over the corresponding bulk values m insulating material is that a given electric field can cause an electron m a quantum well (100400A wide) to move a much larger distance than an electron that ~s bound up m the atomichke orbatals of the valence band ( < 3 A ) [9,14] The work we are reporting here is concerned wath the nonlinear response of "wide" quantum wells which have mtersubband spacangs of the order of 10meV These differ from " n a r r o w " q u a n t u m wells (lntersubband spacing >50 meV) m two important respects (1) The electronelectron anteractlon cannot be neglected The average Coulomb potential energy of two electrons with a density of 101~ cm 2 is 10meV, comparable to the lntersubband spacing The plasmon energy
231
( t i ( 4 a v n e 2 / m ~ e ) 1'2, where n is the 3D denslty, e is the electronic charge, m* is the band mass rne/15 and e ~ 13 as the dielectric constant) for a 3D electron gas with a density of 2 5 × 1016cm ~ (corresponding roughly to a 400A, wide well containing a sheet density of 1011 cm 2) is 5 5 meV, again of the same order of magmtude as the mtersubband spacings we are interested in (2) Dissipation is much weaker If the spacing between subbands is greater than 36meV, then an electron in an excited subband can relax to the lowest subband in a time T~ < 1 psec by emitting an optical phonon If the spacing between subbands is less than 36 meV, a cold electron an an excited subband can relax only by slower processes such as acoustic phonon emission, scattering from an impurity, or scattering from another electron Some recent experiments of ours re&cite that, at low t e m p e r a t u r e s (e g , 10K), T 1 ~ 1 2 _ + 0 4 n s e c [15]
3. Harmonic generation and dynamic screening in a double quantum well This section describes a study of resonant harmonic generation in a modulationdoped double quantum well which approximates a twostate system (for a detailed account of this work, see [16]) The structure used m these measurements (Fig 1) consists of a single period of an asymmetric double quantum well ( A D Q W ) with 85A and 73A, G a A s wells separated by a 23A. A10 ~Ga 0 7As barrier The structure was grown by M B E on a semiinsulating G a A s substrate The barrier region on either side of the well consists of 5200,~ Al~3 ~Ga 0 7As grown as a digital alloy with a 20A period (14A. G a A s and 6A. AlAs) The digital alloy was used to improve the Interface smoothness The well is remotely d o p e d on both sides with $1 deltadoped layers 700A from the well Fig 2 shows the results of a slmulanon of the conducnon band potentml in
232
M S Sherwm et al / Phystca D 83 (1995) 229242
A1 Ga As 03 07
4500h
700A .85A 23i " 73A 
Y~
700A 4500A
O
Substrate
Fig 1 Composition and contact configuration of the modulationdoped double quantum well used in experiments on secondharmonic generation Electrons may be depleted from the well by applying a negative voltage between the "gate" (Schottky contact on surface) and the ohmic contact to the electrons in the well (spike on left side of figure) t h e well, which was u s e d to e s t i m a t e the s u b b a n d e n e r g i e s a n d e n v e l o p e w a v e f u n c t l o n s T h e results w e r e o b t a i n e d by selfconsistently solving t h e S c h r o d l n g e r a n d PoIsson e q u a t i o n s In t h e well for a c a r r i e r d e n s i t y of 2 x 10 x~ cm 2 T h e c a l c u l a t e d s u b b a n d spacings are E~2 = 11 9 meV, E t 3 = 110 m e V a n d E14 = 156 m e V F o r F I R excit a t i o n at h u ~  E t 2 t h e s y s t e m can be a p p r o x i mated by a twosubband model
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,
A n A l S c h o t t k y c o n t a c t ( " g a t e " ) was e v a p o r a t e d o n t o the surface o f the s a m p l e , a n d an In O h m i c c o n t a c t was a l l o y e d to c o n t a c t the elect r o n s in the well A n e g a t i v e v o l t a g e b e t w e e n the gate and the O h m i c c o n t a c t d e p l e t e s t h e structure of e l e c t r o n s a n d also a p p l i e s a d c e l e c t r i c field across the q u a n t u m well T h e ability to t u n e the e l e c t r o n d e n s i t y with a gate v o l t a g e is cructal for the e x p e r i m e n t a l results we r e p o r t Fig 3 shows the c a p a c i t a n c e vs v o l t a g e for the d o u b l e q u a n t u m well n e a r 1 0 K T h e c a p a c i t a n c e is i n d e p e n d e n t of gate v o l t a g e at low n e g a t i v e g a t e v o l t a g e s , a n d t h e n quickly d r o p s to n e a r l y z e r o T h e c h a r g e d e n s i t y at a given g a t e v o l t a g e in o u r well can be e s t i m a t e d m sttu by i n t e g r a t i n g the C ( V ) curve T o a g o o d a p p r o x i m a t i o n , the c h a r g e d e n s i t y v a n e s linearly with gate v o l t a g e f o r v o l t a g e s less n e g a t i v e t h a n the d e p l e t i o n v o l t a g e T h e charge d e n s i t y at z e r o gate bias in o u r well was 2 0 × 1 0 ~ c m 2 b e l o w 5 0 K , as m e a s u r e d by b o t h C ( V ) a n d H a l l effect A m o b t h t y of 1 0 x 1 0 5 c m 2 / V s e c was d e d u c e d f r o m m e a s u r e m e n t s of the H a l l effect at 4 2 K T h e a b s o r p t i o n of o u r s a m p l e at low m t e n s m e s was m e a s u r e d using a B O M E M D A 3 002 F T I R s p e c t r o m e t e r Fig 4 plots the a b s o r b a n c e vs f r e q u e n c y at a gate bias of  0 6 V T h e abs o r b a n c e p e a k has a L o r e n t z i a n h n e s h a p e with a p e a k n e a r 1 0 5 c m ' T h e p e a k p o s i t i o n varies a l m o s t l i n e a r l y with gate bins f r o m 115
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5450 Depth (~) Fig 2 Selfconsistent potentml, wavefuneUons and energy levels for the quantum well depicted in Fig 1
Fig 3 Typical capacitance vs voltage trace for the quantum well near 10 K in Fig 1 The capacitance is initially constant, (C = eA/d, where A is the area ot the gate, e is the dielectric constant and d is the distance) The capacitance quickly drops to zero near  1 6 V, when all of the electrons have been depleted from the well
233
M 5 Sherwm et al / Physwa D 83 (1995) 229242 Photolummescence from Double Quantum Well Experiment and Simulation .
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Energy (eV)
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Frequency (cm" 1)
Fig 4 Absorbance ~s lrequency tor the well shown in Fig 1 Below schematic experimental conhguratlon tor absorptton measurements
(14 6 m e V ) cm ~ at 0 gate bias to 8 0 c m ~ at the depletion voltage of 1 6 V (See Fig 7a) This is a relatively large tuning range, and may have apphcatlons for voltagetunable nonhnear optical elements The spacing between subbands can be measured in a separate expertment, photolummescence excitation, PLE) Ftg 5 shows the photoluminescence data and our assignments of the PL lines in terms of transitions between bound states xn the conduction and valence bands In the PL experiment, the sample was p u m p e d with green hght from an A r + laser ( ~ 5 0 0 n m , wellabove the AI. ~Ga. 7 A s band gap), which creates free e l e c t r o n  h o l e pairs The photoexctted electrons (holes) quickly relax nonradlatively to the lowest (highest) energy bound states in the conduction (valence) bands in the quantum well Radiative recombination from these lowlying states ts detected as photolummescence Because the double quantum well is asymmetrtc, all four transitions between the el and e2 levels in the conduction band and the h h l and hh2 levels in the valence band are allowed and have been detected The mtersubband spacing c 2  e l ~ 11 meV for zero gate bias was determined in
e2:i=1: HHI ~   '   '  
el HH2
Fig 5 (a) Photolummescence intensity (bottom trace) and computed electronhole overlap (top trace) ~s energy for the sample described in Fig 1 The largest peak corresponds to e I  h h l transitions The mtenstttes of the higher peaks are reduced because they are only populated at the 4 2 K temperature of these measurements (b) Schematic diagram for assignment of the transttlons observed in (a)
dependently from ( e 2  h h l )  ( e l  h h l ) and ( e 2 h h 2 )  ( e l  h h 2 ) Unfortunately, tt was not possible to measure PL at nonzero gate bins because of photoconductlvlty between the gate and the channel Elementary quantum mechamcs of a single particle suggests that electromagnetic radiation should be absorbed when its frequency is equal to the spacing between subbands H o w e v e r , m this sample, at zero gate bias, the peak m F I R absorption occurs at 14 6 meV, whereas the intersubband spacing determined from PL ms only 11 meV The shift between lntersubband spacing and lntersubband absorption frequencies is a wellknown manybody effect in q u a n t u m wells
M S Sherwm et al / Phvstca D 83 (1995) 229242
234
[17,18] For slmplloty, we will refer to the entire shift as the "depolarization shift" even though this term is usually reserved for that portion of the shift which arises from the Hartree term (as opposed to the ExchangeCorrelation term) in the K o h n  S h a m equation The depolarization shift can be understood physically in the following way The lntersubband spacing is the amount of energy required to remove one electron from the lowest subband and place it in the next subband If there were only a single electron in the quantum well, this would also be the energy at which light is absorbed However, when there are many electrons m the well, there are many modes of oscillation To a good approximation, the farlr radiation with which we are driving the system displaces all of the electrons simultaneously in the same d i r e c t i o n  I t couples to a collective mode of oscillation, rather than a singleparticle mode (see Fig 6) The resonant frequency of this mode is higher than the resonant frequency of the single particle mode, since it requires a modulation of the denstty of the electron gas in
,~
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c~,cle Pos~t~on Fig 6 Light couples to a collecnve mode of oscdlatmn ot the electrons m a q u a n t u m well The frequency ol the collective m o d e is higher than the frequency of the single p a m c l e mode because a spatml d l s t o m o n of the electron densLty is reqmred, and thxs d l s t o m o n is resisted by a Coulomb force (schematically depicted by the arrow in the figure)
addition to an mtersubband transition An extreme example of the depolarization shift occurs in a bulk classical metal Here, the spacing between "subbands" is z e r o  it costs no energy to move a single electron in space However, light couples to collective oscillations (plasmons) of the metal, which occur at the plasma frequency which can be several eV We are now m a position to study the nonhnear response of the collective mode in the double quantum well depicted m Fig 1 The simplest form of nonlinear response is secondharmonic generation For a single electron in a twolevel system, the secondorder nonlinear susceptlblhty for secondharmonic generation can be simply computed within timedependent perturbation theory One finds resonances when either the pump photon or the secondharmonic photon are resonant with the spacing between energy levels [19,20] In the case of many interacting electrons, many modes of excitation are present It is not obvious a priori with which of these modes the resonances in secondharmonic generation will be associated Will the resonances occur when Okpump (20)pump) = the lntersubband spacing w12, the lntersubband absorption frequency w~2,* or some altogether different frequency w ~:~ In order to find out, it is necessary to do experiments using a source of intense, monochromatic radiation at farinfrared frequencies The FreeElectron Lasers at UCSB, tunable between photon energies of ~ 0 5 m e V and 20 meV with kW power levels, are ideal sources for this experiment [21] The following experimental protocol was used the second or third harmonic power was monitored at fixed F E L frequency and intensity while the frequency of the lntersubband resonance was tuned by applying a negatwe voltage to the gate This procedure was repeated for different F E L intensities and frequencies The details of the experimental procedure have been described elsewhere [16,22] Figs 7b and c show the second and third
M S Sherwm et al / Phvstca D 83 (1995) 229242
of K o h n  S h a m density functional t h e o r y [16,23] A s s u m i n g free m o t i o n in the plane of the well, we m a y separate out the dynamics in the well's growth direction, giving the o n e  d i m e n s i o n a l Schrodlnger e q u a t i o n
120 (a)
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Z =o 100
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235
X x X x x X x X
~, 80 ~ _~
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theory
d2 + Vetf(z)]~n(z)= En(Z )
dz 2
\ where m* is the effective mass, E n is the e n e r g y at the b o t t o m of the nth s u b b a n d , and
t
Vetf(z ) = v~[n(z), z] + Vxc[n(z ), z] :5:.~z:*~'%~,,
i~ ,~ ~ ~"~N
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VP = 357 em1   the°ry ;
i
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t
o ~.
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(1)
,,
.:× 1 0 0 5 Gate Bins (V)
0
Fig 7 (a) Absorption peak (x) position t,l*=w*t,/2"rr~vs gate bias (O) Intersubband spacing determined from photoluminescence measurements at 0 V gate bias (b) Secondharmonic power versus gate bias (O) at T < 15 K Theory is a fit to our calculated line shape [Eq (4)] Crosses show absorption coefficient a times the charge density N measured at 2 times the pump frequency % (c) Thirdharmonic power versus gate bias (0) at T < 15 K Theory is a fit by product ot a Lorentzmn and the square ot the charge density (x) The product a(3%) x N IS also shown (See Heyman et al , [22] tor further descriptions ot solid lines) h a r m o n i c p o w e r vs gate bias at low intensities (where the second (third) h a r m o n i c p o w e r increases as the square (cube) of the incident p o w e r [16] ) In Fig 7b, a clear resonance is o b s e r v e d with a p e a k at Vg =  0 67 V when the p u m p f r e q u e n c y is Up = 51 5 c m I A t this bias, the f r e q u e n c y of the linear absorption p e a k is ul2" = 102 4+ 1 2 cm ~, within 1% of 2% = 1 0 3 c m ~ In Fig 7c, a resonance in the third h a r m o n i c is o b s e r v e d near Vg =  0 47 V T h e experimental results on second h a r m o n i c g e n e r a t i o n were analyzed within the f r a m e w o r k
(2)
Is the effective singleparticle potential T h e conduction b a n d without e l e c t r o n  e l e c t r o n interactions is given by vo(z ), while v~[n(z), z] IS due to the direct C o u l o m b repulsion b e t w e e n electrons, and Vxc[n(z ), z] is the e x c h a n g e  c o r r e l a tion term N o t e that both v~[n(z),z] and vxc[n(z),z ] d e p e n d on the density n(z)= Ej aj[{Z/(z)[ 2, where G is the s u b b a n d o c c u p a t i o n fraction Thus, u n h k e the o r d i n a r y Schrodlnger equation for a single electron, E q (1) is a nonhnear e q u a t i o n  this is the price we pay for treating a m a n y  e l e c t r o n p r o b l e m as a singleelectron p r o b l e m with an effective potential This nonlinearity leads to m a n y interesting consequences which will be p o i n t e d out later in this paper Eqs (1) and (2) m a y be solved selfconsistently to give the wave functions of the electrons at the s u b b a n d m i n i m a T h e calculated s u b b a n d spacings at 0 V bias are E 1 2 = l l 9 m e V , E13=ll0meV and E~4 = 156 meV, where E n m = En  E m Since E12 < < Ex3, the polarizability of a single electron in the q u a n t u m well potential u n d e r excitation at energies hv of the o r d e r or less than E~2 can be calculated using t i m e  d e p e n d e n t p e r t u r b a t i o n t h e o r y for a twostate system H o w e v e r , calculation of the macroscopic susceptiblhty at finite charge density is m o r e c o m p l e x O n e must selfconsistently a c c o u n t for the c h a n g e in the effective singleparticle potential (which includes C o u l o m b and e x c h a n g e  c o r r e l a t i o n effects) arising f r o m the t i m e  d e p e n d e n t charge density This calculation has b e e n p e r f o r m e d for the first
236
M 5 Sherwm et al / Physua D 83 (1995) 229242
order A C conductivity [17.18] To calculate the secondharmonic susceptibility, we have carried this selfconsistent treatment out to second order [231 We assume that in the system's initial state, only the ground subband is occupied When an external perturbation eEe×t(z ) cos(wt) is applied, it modifies the charge density as Anl~/(z) cos(wt) + An{21(z) cos(2wt) We do not need to include the static modification to the charge density since it does not affect the first or secondharmonic, to within second order Selfconsistency is obtained by demanding that this change in density both arise from and also generate the secondorder perturbing potential 2
V(z, t) = ~ [eEFIRZ¢3, , + Avl,"(z)
1 6 10 7
i
 .~
one electron many parttcle expellment
1210 7
g g
8 10  8
o1
~ 0
05 m / o~12"
Fig 8 Expertment, oneelectron theory and man}electron theor) for secondharmonic generation from the double quantum well depicted m Fag 1 The error bar apphes to the expertmental xalue ot X ~e~ at resonance (who*,,_ = 0 5)
iI
+ a,, 7)(z)] cos(/,,,t)
(3)
The potentials AvC,J~(z) and kv<,~/(z) are the changes in the direct of Coulomb potential and the exchangecorrelation potential, respectively, and are treated as functlonals of An<*l(z) For a twostate system, the result of this calculation is X(2)(2w,
e
3
3 w, ca) = :N, z21z22 ~y
(0912 + 11V')2[(0212 I 1/") 2 
× [(w~,_ + 1/") 2  
"
4¢0"][(0)
*
12 + l f f ' )
o) 2] 2
w ' ] '_
,
(4)
where the dipole matrix element z .... = (~.[zlsC,.,) is calculated with the origin such that Z l~ = O, and we have inserted the phenomenologlcal dephaslng time 1/F We use the convention pI2)=XI2)E2 The secondorder susceptibility has a secondorder pole at the depolarizationshifted frequency o ) = _+w~: and a firstorder r_ pole at 2o9 = o_w~2 Our calculation shows that this frequency is exactly equal to the resonance frequency for the firstharmonic response in the twostate approximation [17] The well known sInglepamcle result IS a special case of Eq (4) obtained in the hmlt w~2 = w~2. and has simple poles at the bare frequencies w = _+w~2 and 2w = ~ O)12
Fig 8 shows comparisons of the experimental data plotted in Fig 7b with theory Recall that the p u m p frequency was held constant while the linear absorption frequency was varied with gate voltage The abclssa was converted trom gate voltage to normalized frequency (o)/w~2) using the calibration provided by the linear absorption data of Fig 7a The ordinate plots the nonlinear susceptibility X ~2) at constant sheet density N = 1 2 x 1011 cm e (this was the sheet density on resonance where %ump/W(2=05) The experimental data were not taken at constant charge density, and are adjusted in this plot by the ratio N,(o)/w12 )/N,(oo/wl2 = 0 5) The solid curve represents the predictions of the manyelectron theory, computed using Eq (4) The dotted curve represents the predictions of a single electron theory (obtained by setting w12 = to~2 In Eq (4), where 6012 is the intersubband spacing computed by solving the tlmemdependent K o h n  S h a m equation (1) selfconsistently) The computations were performed as follows at each point on the abclssa, the parameters ~0~2, %2, z12 and 2'22 were computed using both H a r t r e e and exchangecorrelation potentials, and assuming experimentallymeasured values of N, and the electric field E across the well (E =
M S Sherwm et al
/ Phvstca D 83 (1995) 229242
gate voltage/2* distance from surface to well) The width of the wells were assumed to be the nominal values The dephaslng rate F was taken from the halfwidth at halfmaximum of the absorption hneshape E x p e r t m e n t and theory agree within experimental error for both the magnitude and the position of the peak near o)/o) ~2 = 0 5 Measurements near o)/o)12 1 are clearly destrable However, since the excitation is resonant, and energy relaxatton times are of the order 1 nsec at low temperatures, population is efficiently transferred to the excited state and the saturation of X C2) is expected to occur at such low power levels that the second harmonic will be difficult to detect in the regime of quadratic response which is necessary to compare with theory This problem may be solved by working at higher temperatures we have recently determined that the lntersubband relaxation time decreases from 1 nsec to 10 s of psec as t e m p e r a t u r e is raised from 10 K to 60 K
237
o 0)
P,
c
v
L~
Shm~ted absorphon in phase
o~
=
4. Saturated absorption and intensitydependent depolarization shift: a nonlinear quantum effect This section reports prehmlnary measurements of the intensitydependent absorption hneshape m a quantum well with lntersubband transition energy below the L O  p h o n o n energy [24] In an ordinary ensemble of twolevel systems, absorptton of electromagnettc radiation tends to saturate at intenslttes that are sufficient to slgmficantly populate the excited state However, the resonant trequency does not change We observe for the first time a nonlinear quantum effect predicted by Zaluzny [25] as the second subband becomes stgnlficantly populated through resonant excitatton, not only the absorption but also the ability of the electrons to dynamtcally screening the exciting radiation IS reduced (Figs 9 and 10) The depolarization shift thus decreases, the peak of the absorption is redshifted towards the bare lntersubband absorptton frequency and the absorption line becomes
>"6 Time
Fig 9 (a) Force on and (b) velocity of a single electron for sUmulated absorption (sohd hne) and stimulated e m l s s m n (dashed hne)
asymmetric Fits to the theory by Zaluzny yield energy relaxaUon times which depend strongly on intenstty and are of order nsec The 400Awide G a A s square well used in our measurements was grown by molecular b e a m epltaxy The substrate t e m p e r a t u r e was 586°C Growth was interrupted for 20 sec at each raterface of the quantum well The potentml barrters on each side of the well are 6750A of A103Ga07As The well is symmetrically
• Large absorption • Good Screening • Large dep. shift
• Small absorpbon • Poor Screening • Small dep shift
Fig 10 Saturated absorption and intensitydependent depolarization shift (a) Wellbelow the saturation intensity, most electrons are m the lower subband, and there ts m a x i m u m screemng, absorption, and depolarization shift (b) Wellabove the saturatton intensity, the populatmns of the two subbands are nearly equal Thus, at high intensity, the current assocmted with t r a n s m o n s trom the ground to the excited subband ~s nearly canceled by the current assocmted with transitions from excited to ground s u b b a n d s Only a small fraction of the electrons are avadable for screening and absorption, and the d e p o l a n z a t m n shift xs expected to tend towards zero
238
M S Sherwm et al / Phvstta D 83 (1995) 229242
modulationdoped by sihcon layers ot sheet density 1 3 x ]012 cm 2 which are placed 1250A, from each side of the well The doping m each o~ the SI layers was sufficient to pin the Fermi level on both sides of the well and achieve a nearly flat band at the well The relaUvely large d~stance between the donors and the well ensured that the charge transferred into the well was insufficient to begin occupying the second subband at low temperatures The mobility at 4 2 K was 360,000cm2/Vsec, as measured by magnetotransport All experiments reported here were performed on a sample 1 0 cm long, 0 7 cm wide and 500 ~m thick An 800A. thick A1 gate was evaporated onto the front ot the sample, with another layer of A1 on the back of the substrate Ohmic contacts on the sample corners provide electrical connecuon to the electrons m the well The charge density m the well was measured m situ by capacitancevoltage profiling The well could be completely depleted of electrons by a  1 4 V gate bias Farinfrared radiation from the UCSB FreeElectron Laser (FEL) was used to excite our sample The pulses were 3 p,sec long, with peak powers of =1 kW The sample was mounted in a vacuum on the cold finger of a continuousflow cryostat All measurements were performed with the sample near 10 K A 3 mmthick plexlglass block prevented the FIR power from leaking around the sample F I R radmtlon was focused onto the edge of the sample, w~th electric field parallel to the growth d~rectlon of the sample The thick aluminum layers on the front and back served to confine the FIR field within the sample The metalhc boundary condition allows the field to couple strongly to the quantum well The transmxtted F I R was measured using a 4 2 K bolometer Intensitydependent. measurements were performed while taking care that the total power at the bolometer was in its regime of linear response, and the entire experiment was performed in a dry nitrogen atmosphere The intensity inside the sample was estimated by measuring the energy, duration, and spatial
profile of the F I R pulse, and then measuring and modehng the transmission of the sample Due to the lack of a direct measurement tool to probe the mtenslty reside the sample, and experimental uncertainties and fluctuations, we estimate that the intensity values (and therefore relaxation times) cited m this paper may differ by a lictor of four from an absolute intensity cahbratlon The large dots in Fig 11 are experimentally measured attenuation coefficient values for our sample, at a gate bias of  0 85 V, corresponding to a sheet density of 4 × 10 "~cm 2 The attenuation coefficient is defined as a =  l n ( T ( ~ ) / T (  2 0 V ) ) / L where L is the length of the sample and T ( V ) is the intensity transmitted at a gate voltage V~ (At  2 0 V gate bias, the well was completely depleted of electrons, providing
0 3 N=4x10'0e/cm2]
?''
VG =085V Jill "
'
,II',
I=10 4 W/cm 2, "
0 ~I1=03
6 8
i 0 3
I
I
W/cm 2
zl=5ns
"= I=1
• "~ .... ~1 = 3 ns "
W/cm 2
, ....
t..._,. I
I
1=4 W / c m 2
,'"
~1 = 1 ns
 ~ ,  ~   i   ~___ I
77
,,
I
F r e q u e n c y (cm 1 )
92
Fig 11 AbsorpUonvs frequency and intensity for N, = 4 × 10"' cm ~ FabryPerot interference fringes in the sample's transmittance (spaced by 0 15 cm ~) combine with pulsetopulse fluctuations in the frequency of the FEL to produce the scatter m the experimental data
M S Sherwm et al / Physl~a D 83 (1995) 229242
an optical reference ) The top graph shows the absorption at low intensity, where it is independent of lntensLty The peak attenuation coefficmnt Ls 0 3 c m 1, correspondmg to an easilyobserved 26% extinction of the beam The three lower graphs show the absorption hneshape for lntensltms comparable to the saturation intensity As the mtensLty increases, the peak shifts to lower f r e q u e n o e s In addLtLon, the ILneshape of the absorption becomes asymmetric The error bars shown are from a stanstlcal analysis af the data, and do not take Into account systematic errors such as interference fnnges m the transmLssLon of the sample The large dots in Fig 12 are experimentally measured a t t e n u a n o n coefficients for our sample, at a gate bias of 0V, corresponding to a
i O" i 0 6
N 2 1 1 x l[O 11 e/icm 2
VG.t =00V I=10 4 W/cm 2 O, r
....
0
"O
[''i"
I
1=0 1 W/cm 2
•
Q)
/
I
,.
....
~1/ ....
~
" I
/
I
,
:
]
/
,
•
i
:

t/ v 72
".
"[1=3ns
,
0 61 i=lOW/em 2
n
I
z 1 =7ns
:
== .... rr I 8 [ I=1 W/cm 2
°/i
e.
i
,
r
iI
1:1 = 07 ns
OO_
a
lp._O__ "
n
i
Frequency (cm 1)
P e
larger sheet density of 1 l x 1 0 1 1 c m 2 The statistical error bars for this data are smaller than the dots As in Fig 11, the top plot shows the absorption ot the well at very low intenstty to be Lorentzian The peak attenuation coefficient is 0 6 c m ~ The three lower panels show the absorption hneshape at higher lntensltLes The shift to lower frequency and distortion of the hneshape are even more marked than for Fig 11, as would be expected from the saturation of a larger depolarization shift due to the larger charge density in the well The nonhnear absorption was modeled using a theory by Zaluzny The theory uses the Hartree approximatLon for the electronelectron mteractLon The analysLs Ls based on the densLtymatrix f o r m a h s m m the relaxationtime approximation Only two subbands are considered, and the potential is assumed to have reversion symmetry The rotating wave approximation is used to model the oscillating field The parameters of the theory are the bare Lntersubband spacing hoo2x, the dressed lntersubband spacing tiw~,, the matrix element z21={X21z]x,) where IX,,) are the selfconsistent wavefunctlons that solve the Schrodlnger equation m the Hartree approximation, the energy and m o m e n t u m relaxanon rates F l and ~ (assumed by Zaluzny to be independent of the mtensLty), and the field intensity I The c o m m o n input parameters for the analysis are shown in Table 1 The theory by Zaluzny predLcts a LorentzIan hneshape for the absorption at low Intensities The dressed lntersubband absorptLon frequency w~t and the elastic relaxation rate ~ were taken from the resonant fre
Table 1 Parameters used to fit the data using the theor} by Zaluzny from e × p e n m e n t ~, from simulations
86
Fig 12 A b s o r p n o n vs trequenc~ and ]ntenslt~ tor N, 1 1 × 10 ~ cm ' The scatter in these data is smaller than m the data ot Fig 11, p r o b a b l y because the larger absorption at the h~gher charge density here d a m p s the Fabry Perot oscfllanons m the transmittance
239
Vg= o~2,/2"rr ~ w2,/2"rr z,, F~
085V
853cm ~ 76 cm 1 73A 1 63cm ~
V =0(IV 812cm 60 cm ' 48 8A 2 77cm
i
240
M S
S h e r w m et al
/ Physlca D 83 (1995) _~_ 9  _~4 _v
quency and halfwidth at halfmaximum of the Lorentzlap~ which best fit the lowintensity data We have assumed a homogeneouslybroadened line The bare lntersubband spacings were derived by solving selfconsistently the Schrodmger and Polsson equations following the procedure developed by A n d o [17] The values in Table 1 correspond to solutions for which the well width and D C electric field at the well. which were not measured experimentally, were varied slightly to bring the calculated w~x into exact agreement with experiment The data at 0 0 V gate bias were modeled in the local density approximation using the exchangecorrelation potential proposed by Hedln and Lundqvlst (see [26]) with a well width of 404 8A (within 1 monolayer of the nominal width), electronic sheet density Ns = 1 1 x 10 ~ cm 2, assuming no DC electric field (flat band) at the well The data at  0 85 V gate bins were modeled in the Hartree approximation, as we have found that the inclusion of the exchange correlation gives unphyslcally large corrections to the lntersubband transition frequencies at these relatively low charge densities The wellwidth was also 404 8.~, N ~ = 4 x 10 TM cm 2, and the D C electric field at the well was 4 5 kV/cm (of the order one might expect from the  0 8 5 V gate bias) Because of uncertainties in the actual wellwidth, charge density, and D C electric field at the well, estimates of the bare lntersubband spacing are accurate within roughly + 4 c m ~, within the approximations we have used The matrix element z2~ can be obtained directly from experiment and from our selfconsistent simulations The values in Table 1 were derived from the experimentallymeasured adsorptions at w = w 2 1 using the relation ~=4rrwe: z~l/ ce 1 ~atiF~. where e is the electronic charge, c is the speed of light, e = 13 is the D C dielectric constant, and a is the thickness of the wafer The simulations for Vg =  0 8 5 V yield z ~ = 7 3 A , within 1% of the experimentallyobtained value H o w e v e r , the simulations for Vg = 0 0 V yield a z._~ which is 1 7 times larger than the experlmen
tallydeduced value We attribute this discrepancy to screening of the F I R electric fields by the electrons in the quantum well We presently have no detailed understanding oI this screening Note also that such screening is not included in the Zaluzny theory The dotted lines in the lower three panels of Figs 11 and 12 were computed using the theory of Zaluzny with the parameters from Table 1 and the experimentallymeasured intensities as inputs For each curve, the sole fitting p a r a m e t e r was the energy relaxation rate F l =h/7~ The intensitydependent hneshape is wellreproduced by the calculated curves The energy relaxation times used for the calculated curves decrease from 5 to 1 nsec and 7 to 0 7 nsec for the low and high density data over the range of intensities studied here The geometric factors which lead to a factor of 4 uncertainty m the intensity (and hence the overall magnitude of the energy relaxation time ~~) are fixed for the data of Figs 11 and 12, and do not affect relative relaxation rates The decrease of ~~ with increasing intensity may occur because, at higher intensities, the electron gas m hotter and there will be more electrons with sufficient energy to emit L O phonons [15] Our energy relaxation times, even divided by the maximum factor of 4 we estimate from systematic errors, are longer than those recently reported by Falst et al [27] (E2~ = 19 meV, ~300psec) using excitedstate induced absorption spectroscopy, and Murdln et al [28] ( E . ~ 18 meV. 7 = 40 psec) using p u m p  a n d  p r o b e , and comparable to the time measured by LI et al [29] (Ezl = 7 meV. ~ = 1 nsec) using a tilted magnetic field to couple inplane polarized light to mtersubband transitions All these measurements were performed on different samples using different techniques, so it is difficult to pinpoint the origin of these differences The theory [25] neglects many factors which are important in this experiment, in particular the exchangecorrelation [25] potential and asymmetry in the well, as well as the variation of ~ with intensity As a result, our modeling could
241
M S Sherwtn et al / Phvseca D 83 (1995) 229242
n o t be d o n e with a c o m p l e t e l y consistent set of
J1452 ( K
assumptions
D A A L 0 3  9 2  G  0 2 8 7 (J N H
It was necessary to use e x c h a n g e 
c o r r e l a t i o n a n d / o r D C fields to fit o u r values of
Craig, J N H
and M S S ) , and
ARO
M S S ) the
wiL with calculations, a n d extract o)21 A D C electric field across the well b r e a k s i n v e r s i o n
A l f r e d P Sloan F o u n d a t i o n (M S S ), A F O S R 910214 ( K C a m p m a n , P F H , and ACG), the U C R e g e n t s u n d e r an I N C O R g r a n t in the
s y m m e t r y , a n d m a k e s possible a second n o n h n e a r q u a n t u m effect a blue shift of the inter
area of N o n l i n e a r Science (B G ) a n d the N S F C e n t e r for Q u a n t l z e d E l e c t r o n i c S t r u c t u r e s
s u b b a n d t r a n s i t i o n which arises from modifications of the bare l n t e r s u b b a n d energies as charge
D M R 9120007 (S F )
is t r a n s f e r r e d to an excited state [25]
Prelimin
ary s i m u l a t i o n s were carried out in the H a r t r e e a p p r o x i m a t i o n Including a small D C electric field which is p r e s e n t at o u r well large
difference from
s y m m e t r i c well f u r t h e r study
the
presented
T h e s e showed n o s i m u l a t i o n s of the here,
and
require
5. Conclusion We have s h o w n that the i n t e r p l a y of strong p e r i o d i c driving, manybody effects, and dissipat i o n leads to rich b e h a v i o r in wide q u a n t u m wells W h e n viewed within the f r a m e w o r k of a m e a n  f i e l d t h e o r y such as the H a r t r e e approxim a t i o n , the richness can be traced to the n o n l i n e a r effective singleparticle S c h r o d m g e r e q u a tion
In principle, such n o n l i n e a r i t i e s could lead
to truly c o m p l e x d y n a m i c s which are n o r m a l l y f o r b i d d e n in simple q u a n t u m systems, such as chaotic oscillations of the w a v e f u n c t l o n i n d u c e d by periodic driving [23] F u r t h e r e x p e r i m e n t s a n d significant i m p r o v e m e n t s to theory are req u i r e d to d e t e r m i n e if such complexity is in fact possible A solid theoretical and e x p e r i m e n t a l f o u n d a t i o n will also be i m p o r t a n t for e n g i n e e r i n g devices such as l n t e r s u b b a n d lasers [5] in the 100 Ixm w a v e l e n g t h range
Acknowledgement We gratefully a c k n o w l e d g e helpful discussions with S J A l l e n , B Blrnlr, J C e r n e , a n d C L Felix a n d K U n t e r r a l n e r This work was supp o r t e d by the following grants O N R N0001492
References [1] See, e g , B E Sauer, M R W Bellermann and P M Koch, Phys Re,~ Lett 68 (1992) 1633 [21 See, e g J E Bayfield, G Casatl, I Guarnerl and S D W, Phys Rev Lett 63 (1989) 364 [3] B Blrnir, B Galdrlklan R Grauer and M S Sherwln, Phvs Rev B 47 (1993) 6795 [4] M Sherwln, in Quantum Chaos, eds G C'asatl and B Chlrlkov (Cambridge Unlv Press, Cambridge, UK) [5] J Falst F Capasso, D L Slvco, C Slrtorl, A L Hutchinson and A Y Cho, Science 264 (1994) 553 [6] A Kamgar, P Kneschaurek, G Dorda and J F Koch, Phys Rev Lett 32 (1974)1251 [7] R G Wheeler and H S Goldberg IEEE Trans Electron Devices ED22 (1975) 1001 [8] L C West and S J Eglash Appl Phys Lett 46 (1985)
1156 [9] M K Gurnlck and T A DeTemple, IEEE J Quantum Electron QE19 (1983) 791 [10] M M Fejer, S J B Yoo, R L Byer, AHarwlt andJ S Harris, Jr , Phys Rev Lett 62 (1989) 1041 [11] P Boucaud F H Juhen, D D Yang JM Lourtloz, E Rosencher P Bols and J Nagle, Appl Phys Lett 57 (1990) 215 [12] A Sa'ar N Kuze, J Feng, I Grave and A Yarlv, in Intersubband Transitions in Quantum Wells, eds E Rosencher, B Vinter and B Levine (Plenum. New York and London, 1992) p 197 [13] C Slrtorl, F Capasso, D L Slvco and A Y Cho, Phys Re~ Len 68 (1992) 1010 [14] WW Bewley, C L Felix, J J Plombon, M S Sherwln, M Sundaram, PF Hopkins and A C Gossard, Phys Rev B 48 (1993) 2376 [15] J N Heyman K Unterralner, K Craig, B Galdrlklan, M S Sherwln, K Campman, PF Hopkins and A C Gossard, Phys Rev Lett in press [16] J N He~man, K Craig, B Galdrlklan, K Campman, P F Hopkins, S Fafard A C Gossard and M S Sherwln Phys Re~ Lett 72 (1994)2183 [17] T Ando, A B Fowler and F Stern, Rev Mod Phys 54 (1982) 436 [18] S J Allen, D C Tsul and B Vintner Solid State Commun 20 (1976) 425 [19] Y R Shen, The Principles of Nonlinear Optics (Wiley New York, 1984)
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/ Ph~stca D 83 (1995) 229242
[20] J F Ward, Rev Mod Phys 37 (1965) 1 [21] G Ramlan Nuclear Instrum Methods Phys Res A 318 (1992) 225 [22] J N Heyman, K Craig M S Sherwln, K Campman P F Hopkins, S Fafard and A C Gossard, in Quantum Well Intersubband Transitions in Physics and Devices, Whistler, Canada (Kluwer, 1993) p 467 [23] B Galdnklan Ph D Thesis, University of California at Santa Barbara (1994) [24] K Craig, C L Felix, J N Heyman A G Markelz, M S Sherwm, K L Campman, P F Hopkins a n d A C Gossard, Semlcond Scl Techn 9 (1994) 627 [25] M Zaluzny, J Appl Phys 74 (1993)4716
[26] R M Drelzler and E K U Gross, Density Functional Theory (Springer, Berlin, 1990) [27] J Falst, C Sltorl, F Capasso, L Pfeltler and K West, Appl Phys Lett 64 (1994)872 [28] B N Murdm, G M H Knlppels A F G v a n d e r M e e r , C R Pldgeon, C J G M Langerak, M Helm W Helss, K Unterralner, E Gornlk, K K Geerlnck, N J Hovenler and W T Wenckebach, Semlcond Scl Tech (1994) in press [29] W J El B D McCombe, J P Kammskl, S J Allen M J Stockman, L S Muratov, L N Pande}, T F George and W J Schaff, Semtcond Scl Techn (1994) 63O