cm . __ + __ i!bB ELSEVIER

6 April 1998

Physics Letters A 240 (1998) 265-270

Intersubband wave packet interferometry with semiconductor nanostructures

P.I. Tamborenea, H. Metiu Center for Quantized Electronic Structures (QUEST) and Department of Chemistry,

University of California, Santa Barbara, CA 93104, USA

Received 31 December 1997; accepted for publication 2 January 1998 Communicated by L.J. Sham

PHYSICS LETTERS A

Abstract

We study the use of wave packet interferometry in modulation-doped semiconductor nanostructures. In suitably designed quantum well structures, an electronic wave packet made of a coherent superposition of subband wave functions is created by a sub-picosecond laser pulse. After a given delay time, a second pulse, identical to the first, creates another wave packet. Under appropriate conditions these two packets interfere, a process that affects the excited electron population. The dependence of this excess population on the delay time is sensitive to the dephasing of the first wave packet, and this property can be used to study the intersubband dephasing times. @ 1998 Elsevier Science B.V.

The coherent dynamics of an electron in a meso- scopic semiconductor structure can be studied by us- ing spectroscopic methods developed in atomic and molecular physics [ 121. In this Letter we investi- gate the use of wave packet interferometry [ 3-91 with intersubband electronic wave packets in suitably de- signed quantum well structures. Related studies have been performed to control the exciton population and to generate terahertz radiation [ lo-15,2]. We envi- sion two types of applications for this technique: to study intersubband dephasing, and to probe details of the potential energy [ 4,8], such as those caused by the presence of defects in artificially grown samples.

The central idea of the intersubband wave packet interferometry that we propose is the following. Con- sider, for example, the quantum well structure shown in Fig. la. In this structure excitation by a sub- picosecond laser pulse creates an electron wave packet made up of a coherent superposition of subband wave

functions. The wave packet starts in the narrow well, travels along the wide-well region, bounces back at the end of it, and returns to the narrow well. During this time the wave packet suffers a loss of coherence. To measure this dephasing, a second pulse, identical to the first, is applied when the wave packet returns to the narrow well. This pulse, identical to the first, is applied when the wave packet returns to the nar- row well. This pulse creates another wave packet that interferes with the first. The strength of the interfer- ence is diminished by the dephasing suffered by the first packet. If the wide well is narrow, the travelling time will be short and hence the dephasing of the first packet will be small. By gradually increasing the width of the wide well, the first packet is dephased more, and this increase in dephasing diminishes the interference between the two packets. This decrease in interference can be used as a measure of dephasing.

We consider an electron in a modulation-doped

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266 PI. Tamborenea, H. Metiu/Physics Letters A 240 (1998) 265-270

semiconductor quantum well structure and work in the effective mass approximation [ 161. The Hamilto- nian is

H= -& + Vi(z) - ?$40-?$ (1)

this the inteeerence population. This experiment is a temporal analog of a two-slit experiment where in- terference takes place because we are unable to tell through which slit the particle goes in its way to the lo- cation of the detector. Here, we cannot tell from which pulse the photon is taken.

where m* is the effective electron mass (we use GaAs In the remainder of this Letter we study P,(t) at a parameters) and VO is the confining potential. We take time t after both pulses have acted on the system. In the vector potential A(t) parallel to the confinement this case all terms in Eq. (4) are time independent. direction z^ and work in the dipole approximation. The The interference population depends on the relative vector potential phase 6 and on the delay time r.

A(t) = A~e-~‘-‘0)2/AZ~~~[~~(t - to)]

+ AOe-(+f0-T)2/AZ cos[q)(t-to-T) i-61 (2)

describes two Gaussian laser pulses having the same envelope and carrier frequency wa. The second pulse arrives at the electron’s location with a delay time 7. The pulses described by this expression are not phase locked: The phase tier+6 of the second pulse depends on the delay time. Phase-locked pulses are described by setting we7 equal to zero in the second term in Eq. (2). An experimental method for phase locking has been proposed and implemented by Scherer et

al. [6].

In all systems examined here the initial electron wave function is localized. Because of this the wave packet created by an ultrashort pulse is located, at the time of its creation, in a region we call the formation region, which roughly coincides with the location of the initial wave function. To have wave packet interfer- ence the delay time must be such that the wave packet produced first has returned to the formation region of the second packet. Unless the second packet overlaps - at the time of its creation - with the first one, there will be no interference.

We are interested here in one-photon absorption. We use two pulses to give the electron an alternative: it can absorb the photon from the first or the second pulse. Because of this ambiguity the excited state created by the interaction with the two pulses is a coherent

superposition

The wave function (z IP~( t)) of the excited elec- tron is calculated by first-order time-dependent per- turbation theory, when the pulses are weak; for strong pulses we solve the time-dependent Schriidinger equa- tion with the Hamiltonian ( 1) . In the latter case the numerical calculations use an FIT method proposed by Fleck et al. [ 17-191.

Ipe(t)) = ITe,l(t>) + lpe,2(f)) (3)

of two wave packets: I!Pe,r (t)) represents the electron state created if the photon is absorbed from the first pulse, and ]!Pe,2( t)) the state created if the photon is absorbed from the second pulse. The excited-state population is

Wavepacket interferometry can be applied to vari- ous structures. Here we consider the ones shown in Fig. 1, henceforth called structures A, B, C and D. These can be grown with semiconductor alloys such as GaAs-AlGaAs. The ground state of an electron in each of these structures is localized in a narrow spatial region, and therefore the ultrashort laser pulses cre- ate narrow wave packets (superpositions of excited- subband wave functions).

P,(l) = (~e(f)lly,<t>) = 9 (t) + f’z(t) + 2Re(P,,I (f) I pe,2(f)). (4)

If the pulses are weak then Pi(t) = (We,i( t) 1 ‘Pe,i( t)) (i = 1, 2) are the populations that would be created if each pulse acted alone. If they are strong P2 < PI because the first pulse depletes the ground-state pop- ulation. The last term in Eq. (4)) denoted Pt2( f), is caused by the interference of the two packets. We call

The upper panel in Fig. 2 shows snapshots of the probability ]?Pe ( z, t) I2 of finding the excited electron at Z, in structure A, at different times after excitation with one laser pulse. The lower panel shows the prob- ability P,,(t) that the excited electron is in the narrow well, together with the probability P,(t) that the elec- tron is excited. P,,(t) shows an initial buildup, a de- cay due to tunneling into the wide well, and subsequent returns of the electron to the narrow well. The round- trip time can be controlled by changing the width of

?I. Tamborenea, H. Metiu/Physics Letters A 240 (1998) 265-270 267

-sol..~..*....n . ...? . . . U..l -100 -50 0 50 100

,““...“I”.......,........‘I”‘.‘....I

-loot...-.....~.........~.........~.........i u

-200 -100 0 100 200

= bd

Fig. 1. Conduction-band potential-energy profile of the semicon-

ductor structures considered for wave packet interferometry. The

short dashes give the energy levels (the middle figure shows the

energy levels for structures B and C).

the wider well and the carrier frequency of the pulse. A similar calculation for structure D shows that the excited wave packet propagates almost without dis- tortion, and returns to the narrow well practically un- changed. This effect is typical of a harmonic system. In this structure the recurrence time is controlled by the harmonic frequency of the parabolic parts of the well.

Fig. 3 shows the excited-state population as a func- tion of the delay time r between two pulses, after the electron has interacted with both pulses. If the phase difference between the two pulses is not locked, the evolution of the population with r has a slowly vary- ing envelope and rapid oscillations whose frequency is very close to the carrier frequency of the pulse. If the pulses are phase locked only the slowly varying envelope is observed. This can be understood from a first-order perturbation theory analysis of the excita- tion process, which gives

(5)

Here +n are the energy eigenstates of the electron, w,t are the electron transition frequencies, and A( w,i) is the Fourier transform of the vector potential for the two pulses. For phase-locked pulses this is given by

-100 -50 0 50 100

z b-4 0.6 ““,,,“““I”““““““““‘I”’

total

P 0.4 - E

B in narrow well

:: a 3.2 - 4:

:‘.. ,’ :

:. :-. _

:: .’ : . '-_. ' : : : ,: .

0 ,.JI..,,......I.........~........,I.,. ‘._ _,_..: . . . ._ _... .~

0 1 2 3

Time (ps)

Fig. 2. Top: Time evolution of the excited wave packet created

by a short laser pulse in structure A. The duration of the pulse

is 200 fs and its carrier frequency is 18.9 meV, corresponding to

one of the transitions with highest oscillator strength. The center

of the pulse arrives at t = 0, and the intensity is 1 MWcm2. The

electron is initially in the ground state. Bottom: Probability of

being in an excited state. The solid line gives the total probability

that an electron is excited, and the dotted line gives the probability

that the excited electron is located in the narrow well.

I~(wd I2 = A2A2 --$ exp[ -A2(f+i - ~a)~/21

x {1+ cos[ (o,t - w(J)7 - S]}. (6)

The Gaussian in Eq. (6) is narrow, so only frequencies w,i close to wu will contribute to the sum in Eq. (5). This means that the difference o,i - wa appearing in the cosine of Eq. (6) is small and the oscillations of the population, as r changes, are of low frequency. They generate the envelope in the calculations shown in Fig. 3. If the pulses are not phase locked, Eq. (6) does not contain the term tia in the cosine and this leads to very rapid oscillations of the population, hav- ing the frequencies w,l close to 00. This is the behav- ior observed in the numerical calculations.

Physically, the behavior of P,(t) can be interpreted with the help of Fig. 2. Consider first the case of low-intensity pulses, for structure A (Fig. 3a). If the first wave packet is located in the narrow well when the second wave packet is being created, the spatial

268 l?I. Tamborenea, H. Me&/Physics Letters A 240 (1998) X-270

ULr 1

0.04

Q

0.03

h 0.02 0.01

0

0 1 2 3

Delay (pa)

Fig. 3. Probability that the electron is excited after interacting with two pulses, as a function of the delay time. Thick lines correspond

to phase-locked pulses: the solid line corresponds to a relative phase 6 = 0, and the dashed line to 6 = G-. The dotted line is twice

the probability of being excited by the first pulse alone. The thin,

rapidly oscillating line corresponds to pulses that are not phase

locked, with 6 = 0. Pulse widths are 300 fs. (a) Structure A, low

intensity, I = 1 kW/cm*, wo = 18.9 meV; (b) structure A, high

intensity, Z = 0.1 MW/cm2, wg = 18.9 meV, (c) structure B,

Z = 1 kW/cm*, wg = 20.8 meV; (d) structure C, Z = 1 kW/cm’,

wg = 20.8 meV; (e) structure D, Z = 10 kW/cm’, wg = 17.3 meV.

overlap between them allows interference to occur; therefore, Pt2 # 0 and P, # PI + P2. The inter- ference can be either constructive (6 = 0) or destruc- tive (S = rr). For high laser intensity (Fig. 3b), the excited-state populations can be very large, but the in- terference pattern remains qualitatively unchanged. In this case, however, even when the interference term is zero, P, = Pe,l + Pe,2 f 2P,,t, because the ground-

0 1 2 3

Time (ps) Fig. 4. Emitted radiation due to excitation with 300 fs laser pulses

in structure D. Z = 10 kW/cm2, wg = 17.3 meV; 10 = 0. Solid line:

only one pulse; Dashed: two pulses, 7 = 1040 fs, 6 = 0; Dotted:

two pulses, 7 = 1040 fs, 6 = ?r.

state population is depleted by the first pulse. Figs. 3c,d show P,(t) for the structures B and C, which differ from each other by a small perturbation in one of the wide wells. The perturbation can be due to an impu- rity or a defect occurring in the growth process. The differences between the results of Figs. 3c,d show that wave packet interferometry is sensitive to small dif- ferences in the potential, and therefore it might be used to “detect” growth imperfections in the sample. Fig. 3e shows P, for the quasi-parabolic profile, which reflects the periodicity and regularity of the return to the small well mentioned above for this structure.

Any method for measuring excited-state population can be used, in principle, to determine the interfer- ence population. For example, one can measure the far-infrared radiation emitted by the excited electron. In Fig. 4 we show the time-dependent electric field ERAD cc ( d2/dt2) (!P( t) 1.z I!P( t)) radiated by struc- ture D with one and two pulses acting on it. Notice that destructive interference is complete for S = Z-, as no excited populations are left after the second pulse (dotted line). Terahertz radiations produced by elec- trons oscillating in a double quantum well have been measured recently [ 10-131. We mention in passing that the behavior of the excited population created by wave packet interferometry can also be studied by pump-probe methods. One possibility is to use a probe pulse, fired shortly after the two pulses used to cre- ate the excited-state population (there are two pump

l?I. Tamborenea, H. Metiu/Physics Letters A 240 (1998) 265-270 269

pulses in this case). One can choose the frequency of the probe so that the excited electrons absorbing the probe photons will undergo a transition to a delocal- ized state. If the sample has a slight bias voltage, the probe induces a current that is proportional to the pop- ulation of the excited electrons.

Before concluding, we comment on the possible uses of wave packet interferometry in semiconduc- tor structures. Applications to molecular systems have shown [ 81 that the functional dependence of the in- terference population on the delay time is very sen- sitive to the nature of the potential. The calculations performed here show that this is true for quantum well systems as well. Assume now that one has grown a set of samples under identical conditions. Because of the randomness of the growth process samples are often not identical. Some may have imperfect interfaces or a few impurities or misplaced atoms inside the wells. Performing wave packet interferometry on these sam- ples and detecting emission from them can be used to detect small differences between the samples. Com- parison to calculations may also help decide which of the structures is “perfect”. One could even imagine, if one is very optimistic, trying to use simulations to de- termine the nature of the imperfections in the samples.

In the calculations presented here, we have not in- troduced any dephasing mechanism. In reality, the in- teraction with phonons and the other electrons as well as inhomogeneities (impurities, defects, and concen- tration or shape variations at different places in the wells) will cause the interference population to dif- fer from the one calculated here. Our calculations are valid only for times shorter than the dephasing times of the system. An estimate of the dephasing time in inter- subband transitions can be obtained from the linewidth of intersubband absorption or inelastic light scatter- ing spectra. In high quality AlGaAs double quantum well samples, the linewidth of inelastic light scattering spectra of intersubband charge density excitations is less than 1 meV [ 201, which corresponds to a dephas- ing time longer than 4 ps’ . This typical dephasing time is long enough for the coherent effects we pre-

1 Our intersubband transition scheme should not be analyzed

using interband free carrier dephasing times, which are more than an order of magnitude shorter than those of intersubband

transitions. For studies of interband dephasing due to Coulomb

scattering, see Ref. [Z 11.

diet here to be observed. We mention that this value corresponds to intersubband transitions whose energy is below the longitudinal optical (LO) phonon energy for GaAs (36 meV) . For excitation above this energy the emission of LO phonons leads to a rapicl dephas- ing, with dephasing times of the order of 200 fs. For this reason, we have designed our structures so that the electron excitation energy is smaller than the en- ergy of the LO phonon. The experiments we propose here could be performed at different electron densi- ties to study dephasing due to Coulomb scattering, and should be attempted first at low density to mini- mize dephasing. We mention that in our structures the excited wave packet and the remaining ground state are physically separated most of the time. They over- lap only during a short time after the wave packet is created and when it comes back to the creation re- gion. This fact will affect the scattering between ex- cited electrons and those left in the ground state, and between electrons and phonons (when it is allowed), reducing the dephasing rate compared to more conven- tional single or double quantum wells, where excited and unexcited populations physically overlap.

The sensitivity of the interference population to the effects of intersubband dephasing can be used to study the latter. One possibility is to grow a set of samples of type A or B, in which the size L of the large well differs from sample to sample. If L is small, the recur- rence time for a wave packet excited by an ultrashort pulse is very short (in structure D the recurrence time can be shortened by choosing a larger curvature). One can assume in this limit that no dephasing has time to occur and that a wave packet interferometry exper- iment will proceed according to our calculations. As we repeat the experiment with samples with larger and larger L, we can see dephasing gradually setting in. We have performed a density matrix calculation including all the subbands of the structures, with a phenomeno- logical dephasing time Tz. The described scenario is borne out by our calculations. In a longer publication we will provide further details on the effect of dephas- ing, and also on the effects of energy renormalization due to mean-field Coulomb interaction among elec- trons.

In summary, in this Letter we propose the applica- tion of wave packet interferometry to semiconductor quantum well nanostructures. We study four possible configurations, where the intersubband wave packets

270 RI. Tamborenea, H. M&u/Physics Letters A 240 (1998) 265-270

are generated with two sub-picosecond far-infrared laser pulses. The effects we discuss here are purely coherent, and can be used to study dephasing due to intersubband processes. Also, due to its sensitivity to the shape of the potential energy, wave packet inter- ferometry could be used to detect defects in artificially grown semiconductor samples.

We are grateful to Nick Blake for programming advice, and to Mark Sherwin, Holger Schmidt, Gary Woods and Atac Imamoglu for useful discussions. This research was supported in part by QUEST, the NSF Science and Technology Center for Quantized Electronic Structures (Grant No. DMR 91-20007).

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