Coherent Emission of Electromagnetic Pulses fromBloch Oscillations in Semiconductor Superlattices

H.G. RoskosInstitut f~r Halbleitertechnik II, Rheinisch-Westf&lische Technische Hochschule(RWTH) Aachen, Sommerfeldstr. 24, D-52056 Aachen, Germany

Sunnnary: In the last few years, terahertz-wave-emission spectroscopy hasemerged as a powerful spectroscopic tool for the time-resolved investigation ofcoherent processes. We have employed this technique to study the spatial dynam-ics of coherent charge carriers in semiconductor supedattices. The observation ofBloch oscillations, i.e. the periodic motion of wave packets in a homogeneous elec-tric field, in GaAs/AIGaAs superlattices is reported. The charge oscillations aredetected via their emission of electromagnetic transients. Fundamental aspects ofBloch oscillations are discussed. We then address the proposal of Esaki and Tsufrom 1970 to employ charge carrier oscillations in superlattices as a source forbias-tunable ultrahigh-frequency electromagnetic signals. A quasi-optical approachin contrast to an all-electrical scheme for generation and amplification of terahertzelectromagnetic transientsis presented.

1 Introduction

The work described in this article is based on two developments in time-resolvedspectroscopy that both have set in around 1989 - 1990. (i.) At that time, interest inthe investigation of optically excited wave packets in semiconductors began to growtremendously. This was certainly triggeredby advances of semiconductor growthtech-niques allowing fabrication of tailor-made quantum heterostructures with large meanfree path of the charge carriers especially in the GaAs/AI=Gal_=As material system.Furthermore, the advent of the Kerr-lens-modelocked titanium:sapphire laserprovidingstable femtosecond optical pulses over the wavelength range of interest for the studyof GaAs/AI=Gal_=As quantum structures has made such investigations much moreviable. Long-standing questions concerning coherent transport of charge carriers couldfinally be addressed. The research on wave packet dynamics has culminated in the firstobservation of Bloch oscillations of charge carriers in a GaAs/Al=Gal_=As superlat-tice by degenerate four-wave-mixing (DFWM) [1, 2, 3]. (ii.) Simultaneously to thesedevelopments, time-resolved terahertz (THz) spectroscopy has emerged as a powerfultool for the study of excitations in the energy range of 0 - 20 meV, respectively dynam-ical processes covering the frequency range from 0 to 5 THz (for a review see [4]). Abenchmark in the history of time-resolved THz spectroscopy was the implementation ofthe first versatile spectroscopic setup in 1989 [5]. Since then, especially time-resolvedTHz-emission spectroscopy has matured into an extraordinarily successful subfield. Itdeals with the detection of THz electromagnetic transients emitted from the sample afterpulsed optical excitation of coherent processes [4].

298 H.G. Roskos

We have employed this novel spectroscopic technique for the investigation o f Blochoscillations in superlattices [6, 7, 8, 9, 10, 11]. This article summarizes the main results.It is organized as follows. In Chapt. 2, the physics o f Bloch oscillations is brieflyreviewed. Chapt. 3 outlines the techniques o f time-resolved THz-emission spectroscopyimportant in this context. Chapt. 4 presents the main experimental results and theiranalysis. Finally, in Chapt. 5, implications o f the coherence o fthe emission process arediscussed. An analysis o fthe cooperative (superradiant) character o fthe emission leadsto severalproposals for an enhancement o f the emission efficiency. It is shown that alsoamplitication of THzpulses in coherent semiconductorquantum systems is feasible.

2 Bloch oscillations

Charge carrier transport on a length scale below the mean free path of the carriersis the domain of coherent transport where the particle-like picture o f the carriers hasto be replaced by a fully quantum-mechanical picture taking into account the wavenature of the carriers. A problem of fundamental significance for solid-state physics isthe coherent transport o f charge carriers in a periodic potential under the influence o fa static electric bias field. This situation was addressed as early as 1928 by F. Bloch[ 12]. He tbund that an electron wave packet composed o f a superposition o f single-band states with a narrow spread of k-vectors will propagate with a constant velocitythrough k-space when subjected to a uniform electric field F. This is expressed in theacceleration theorem dlddt=eF/h. The real-space group velocity v(k) ofthe wave packetis determined by the k-derivative of the energy: hv(k)=aE(k)/c3k Consequently, thewave packet, following the energy-band contour in k-space, will experience a signreversal of its real-space velocity when it reaches the first Brillouin-zone boundary aswas first shown by Zener in 1934 [13]. Hence, when crossing the first Brillouin-zoneboundary (respectively experiencing an umklapp process at the boundary), the wavepacket reverses its propagation direction in real space and begins to travel againstthe field direction. At each Brillouin-zone boundary (or more generally at each bandminimum and maximum), the real-space velocity of the charge carrier changes signagain. The carrier is quasi-localized (dynamically localized) performing what cameto be known as Bloch oscillations, provided the coherence of the wave functions ispreserved long enough and is not destroyed by scattering before at leastone oscillationcycle is completed. The time period of a Bloch oscillation is given by rB = eFd/fi,d being the spatial period of the potential. The real-space oscillation amplitude L/2 isdetermined approximately by L = A/eF, with A denoting the zero-field width of theenergy band [14]. L is also known as the localization length.

This semi-classical model neglects the dependence of the band structure and thewave functions on the electric field itself. Surprisingly, it turned out to be difficult toestablish a firm quantum-mechanical basis for Bloch's and Zener's calculations. Oneof the reasons is that the determination of the eigenfunctions and energy eigenvaluesleads to inconsistencies when an infinitely extended electric field is considered. Afterdecades of intense theoretical debates (reviewed e.g. in [15, 16]) it is now accepted thatthe Bloch states which are the eigenstates o fthe periodic potential in the absence o f anelectric bias field evolve as Houston functions [17] when the field is switched on. TheHouston description is based on a single-band model neglecting tunneling of the wavepacket into other bands. This interband tunneling process known as Zener tunneling[13, 18] has been shown to be completely insignificant for the fields and time scales of

THz-Pulse Emission from Bloch Oscillations 299

interest here [15, 16]. In the Houston picture, the wave packet oscillations modify thedipole matrixelements o fthe optical transitions from the valence to the conductionband.The matrix elements contain time-dependent phase factors oscillating with the Blochperiod rB [19, 20]. The Fourier transforms o fthe matrixelements correspondingly havesidebands with spacing 27r/rs. Therefore, the optical interband absorption exhibits aseries ofevenly spaced resonances, the Wannier-Stark ladder, with an energy separationA E = e F dbetween adjacent resonances.

The Wannier-Stark ladder consists o f resonances, but strictly speaking these are notenergy eigenstates o fthe periodic potential [ 15, 16, 21]. Still, a resonance is characteristicfor a long-lived and spatially localized state. In the case o fstrong localization (L --, d),each state can be described in a tight-binding approximation taking into account onlycoupling between neighboring wells of the periodic potential [22, 23]. The resultingWannier-Stark functions are given by ~,~(z)=~m J~_,~(L/d) ¢(z - rnd). J~ is theBessel function of order i, C0(z) denotes the single-well wave function. The summationis performed over all periods.

Since both the Wannier-Stark and the Houston wave functions form complete setsof functions, any charge carrier wave packet can be described in either base. Whateverthe choice, Bloch oscillations in the fully quantum-mechanical picture are a quantuminterference phenomenon involving Wannier-Stark states. There is a close analogy be-tween Bloch/Zener's semi-classical and the fully quantum-mechanical picture as far asthe dynamics of wave packets is concerned. In the quantum-mechanical picture, a wavepacket undergoes oscillations in space and time with a frequency vB=eFd/h, the samedependence as predicted by Bloch/Zener's model [14, 15]. The spatial amplitude of thesemi-classical Bloch oscillator is given by L/2. A very similar value is found for theamplitude o f the quantum-mechanical wave packet oscillation [14]. Furthermore, theprobability to find the electron at a given position is similar in the semi-classical and thequantum-mechanical picture [14]. There remains, however, the fundamental differenceof the Wannier-Stark absorption resonances that are not predicted in the semi-classicalpicture. Based on the continuous-band concept of the semi-classical picture, it is oftenconsidered counter-intuitive that Bloch oscillations and the Wannier-Stark absorptionresonances are two manifestations of the same phenomenon, one in the time domain,the other in the frequency domain. It is easy to show, that Bloch oscillations can onlybe observed under the condition that the absorptionexhibits Wannier-Stark resonances.When the electric field across the periodic potential is increased,the continuous absorp-tion spectra break up into discrete Wannier-Stark resonances at a bias field Fc wherethe energy separation A E between adjacent Wannier-Stark resonances becomes largerthan their linewidth. Neglecting inhomogeneousbroadening, the linewidth is given byF = 2~/T2, with the dephasing time constant Tg. determined by scattering and Zenertunneling. The condition A E > F for the observability o f Wannier-Stark resonances isidentical to wB T9_/2 > 1, the condition forobservation of at least a single Bloch oscil-lation cycle (T9_/2 is the decay time constant of coherent signals in measurements thatmonitor intensities, e.g. DFWM measurements. In THz-emission experiments, measur-ing field amplitudes, the signal decays with 7"2 [24]). Hence, it is a general conclusionthat the coherence o f a wave packet is destroyed before a single Bloch oscillation cycleis completed if the absorptionspectra are continuous and do not exhibit Wannier-Starkresonances at the chosen bias field.

Bloch oscillations up to now have neverbeen observed experimentally in bulk solids.There exist, however, reports that their frequency-domain counterpart, the Wannier-Stark ladder, may have been observed in bulk GaAs for very high bias fields (above100 kV/cm) [25, 26, 27, 28]. These observations remain inconclusive basically because

300 H.G. Roskos

the observed effects were too small. A real breakthrough could only be reached afterEsaki and Tsu proposed in 1970 to search for Bloch oscillations in semiconductorsuperlattice structures consisting o f alternating layers of semiconductors with differentbandgaps [29]. The larger period d o f such structures as compared to the lattice constantof bulk materials leads to THz oscillation frequencies already at fields of several kV/cm.The dephasing time constant at low temperatures,on the other hand, remains rather high(on the order o f a few ps) up to fields of several ten kV/cm, because exeitonic effectsare not quenched by the applied field as in bulk solids. After progress in molecular-beam epitaxy allowed growth o f high-quality samples, Wannier-Stark ladders couldunambiguously be identified in 1988 in cwphotocurrent and photoluminescence spectraof GaAs/AIGaAs superlattice structures [30, 31]. In 1990, the negative differentialvelocity o felectrons, predicted by Esaki and Tsu as a consequence o f Bragg reflections,was found [32]. Although this observation provided strong experimental support forthe concept o f Bloch oscillations, the experiment did not yet prove their existence as anegative differential velocity does not require electrons to actually reverse their directionof propagation. It is sufficient that electrons reach aregion in k-space where their velocitydecreases with increasing k before they are scattered. Later, the simultaneous observationof Wannier-Stark ladder formation and negative differential velocity proved the closerelationship between Wannier-Stark field localization o f states and Bloch oscillations[33, 3,,1.

Figure 1Scheme for optical excitationofBloch oscillations in asuperlattice biased into theWannier-Stark regime. Onlyheavy-hole-exciton transitionsare indicated

The first direct detection of Bloch oscillations succeeded by femtosecond DFWMmeasurements on GaAs/AIGaAs superlattices [1, 2, 3]. Bloch oscillations were gener-ated optically by simultaneous excitation of at least two Wannier-Stark ladder stateswith an ultrashort (broadband) light pulse (see Fig. 1). The superlattice was biased intothe Wannier-Stark regime for the first electron miniband. Under these bias conditions,the heavy-hole wave functions are fully localized in single wells whereas the electronand the light-hole wave functions due to their lower effective mass still extend overseveral wells. In the experiments o f Refs. [1, 2, 3], only heavy-hole-exciton transitionswere excited. Bloch oscillations develop as quantum interference of the excited electronWannier-Stark states. In the two-beam DFWM measurements, they were detected as anoscillatory feature on the diffracted coherent signal measured as a function o f the timedelay between the two excitation pulses. Except for quantum beats, oscillations in theDFWM signal can also result from interference of the polarizations set up by indepen-dent optical transitions, i.e. transitions that do not lead to wave function interference.A clear dinstinction is possible by either temporally [35] or spectrally [36] resolv-ing the diffracted signal. Measurements with spectral analysis o f the DFWM signal

THz-Pulse Emission from Bitch Oscillations 301

clearly proved the quantum interference (Bitch oscillation) character o f the oscillatorydiffracted signal [9, 37].

In their pioneering paper of 1970 [29], Esaki and Tsu proposed to utilize Bitch os-cillations as a source for tunable ultrahigh-frequency electromagnetic signals. Attemptsto realize an Esaki-Tsu oscillator failed so far. Only recently, it was even refuted thatBitch oscillations are associated with a net dipole moment that may couple to a resonantelectromagnetic signal with frequency ~ = AE/h [38]. It was argued that absorptionbetween Wannier-Stark ladder states would exactly cancel stimulated emission. Beyondthe fact that this statement doesn't hold when relaxation is included in the calculations[39], this approach is only applicable for the description of incoherent excitations butnot for a situation where the charge carriers are prepared in a coherent manner as in theDFWM experiments. It is established now that Bitch oscillations are associated witha quasi-static dipole moment coupling to electromagnetic waves (quasi-static meaningwith low frequency with respect to optical frequencies) [40, 41]. The induced quasi-static macroscopic polarization density P(t) is determined by the envelopes o fthe wavefunctions [42]. In the single-particle picture, i.e. neglecting excitonic effects [23, 41],it can be calculated for a situation as that depicted in Fig. 1 (interband transitions fromheavy-hole valence band states 10) into Wannier-Stark ladder states of the first electronminiband) by

V Z [pi~(t)(zoo - z/i)] - 2 ~ [zij Re(pq(t))] . (2.1)k i > / > 0

z is the spatial coordinate in the growth (and electric bias field) direction. #ij is theintraband dipole matrix element: #ij = -ezq = -e( i l z l j ), with i, j > 0 denoting theWannier-Stark ladder states with envelope functions ffi,j (z). lntraband means betweenWannier-Stark states o f the same miniband. #oo = -ezoo is the heavy-hole dipolemoment, pq (t) denotes the intrabanddensity matrixelements, P00 (t) is the heavy-holepopulation density. The summation is performed over the in-plane k-vectors.W denotesthe number of periods of the superlattice. It is emphasized that P(t) can still containall orders of the optical electric field, not only the second-order term P(~). Assuming&function-like excitation of the levels, we can express the density matrix elements upto the second order in the optical field E0 by [43]

e2 (/lEo rl0) (0lEo r[j) e_Z(~,_iT,j)~ O(t) i # j, (2.2)p q ( t ) = r~2

e 2p~i(t) = [(i1-2 0 ( t ) . (2.3)

E,~ r[0)[2

Here, the longitudinal relaxation times are assumed to be infinitely long. "Tq is thedephasing rate between levels i and j, O(t ) is the Heaviside step function.

From Maxwell's equations it follows that any time-dependent polarization will leadto emission o f an electromagnetic transient with an electric field E(t) determined byE(t) o~ 0 2 P ( t ) / a t2. According to Eqns. 2.1-2.3, P(t) is composed of terms with dif-ferent temporal characteristics: (i.) Both terms in Eqn. 2.1 contain expressions with astep-function-like time dependence, hence radiation is emitted as a spike during theoptical excitation. This so-called instantaneouspolarization signal [44, 45, 46] is aconsequence o f the photocreation o f electrons and holes with a spatial offset with re-spect to each other. In general, the bias field F is responsible for this electron-hole

302 H.GL Roskos

polarization. (ii.) The second term in Eqn. 2.1 is a sum overexpressions with oscillatorytime dependence. This term describes the Bloch oscillations. Evidently, they emit anelectromagnetic transient provided the summation does not cancel the different contri-butions e.g. in the case o f a dipole-moment-free breathing-mode oscillation [41 ]. It isemphasized that for THz-wave emission by Bloch oscillations two conditions have to befulfilled. First, the interband transitions for the excitation of Wannier-Stark states mustbe allowed, i.e. (liEn rl0) in Eqn. 2.2 has to be nonvanishing. Second, the quantuminterference between Wannier-Stark states must be associated with nonvanishing intra-band transition dipole moments, i.e. the moments #ii = - e z i j , (i, j ~- 0) in Eqn. 2.1have to be non-zero. This second condition is also significant for the question whethernot only quantum interference but also polarization interference may lead to THz-waveemission. In contrast to DFWM, polarization-interference-induced contributions to thesignal are not expected because the non-existing intraband dipole moments preventemission at least up to the second order in the optical field E'0.

3 Time-resolved T H z - e m i s s i o n s p e c t r o s c o p y

Figure 2Subpicosecond photoconductivedipole antenna for thetime-resolved detection of THztransients (not drawn to scale)

The electromagnetic radiation emitted from optically excited coherent charge os-cillations can be detected with a photoconductive dipole antenna. Figure 2 displaysschematically an antenna as utilized in our measurements [47]. It consists o f a 50/~mlong and 5-10/~m wide Hertzian gold or aluminum dipole monolithically fabricated ona silicon-on-sapphire substrate.The two arms o fthe dipole are separated by a 5 tzm widegap that together with the silicon underneath forms a photoconductive (Auston) switch.The dipole arms are contacted by striplines that are several mm long and end in bondpads. The striplines are fairly long to facilitate handling o f the antenna. Subpicosecondspeed of the Auston switch is achieved by ion-implantation of the silicon layer afterphotolithographic definition o f the antenna structure. After implantation with 350 keVSi+-ions at a dose ofSx 10~4 cm-2, the charge carrier lifetime in the silicon is reducedto 600 fs, the minimum achievable value. At this dose, the carrier mobility has notdeclined strongly yet, hence the detectivity of the antenna remains high. To reduce thedark conductivity o f the antenna, the silicon is removed by etching except for a mesaregion at the Auston switch.

For detection o f electromagnetic fields, the antennais connected to a sensitive currentpreamplifier followed by a lock-in amplifier. The THz beam impinges onto the antennafrom the substrate side through a hyperhemisphericalsilicon lens ofseveral mm diameter

THz-Pulse Emission from Bloch Oscillations 303

Figure 3Upper part: Experimental setupfor THz-wave emissionmeasurements.Lowerpart: Illustration of theemission geometry. THztransients are emitted fromthecoherent ensemble in conespointing into the directions of thereflected and the transmittedoptical beams.

glued onto the substrate forbetter coupling of the radiation to the microantenna.A standard THz-measurement setup is displayed in the upper part o f Fig. 3. The

optical excitation pulse hits the sample, mounted on the cold finger o f a cryostat, underan angle o f 45°. Electromagnetic radiation is emitted from the ensemble o fcoherentlyexcited dipoles into the directions o f the reflected and transmitted optical beams (seelower part of Fig. 3). The polarization associated with the charge oscillations is directedperpendicular to the surface, hence optimum emission of each dipole occurs parallel tothe surface. The direction o f emission o f the dipole ensemble, however, is determinedby interference of the wavelets (generalized Fresnel's law). The THz beam propagatinginto the sample is absorbed in the highly doped substrate. The THz beam travelling inreflection direction leaves the cryostat through a high-resistivity silicon window. Thebeam is focussed with two off-axis paraboloidal mirrors onto the receiver antenna thatis gated by an optical probe pulse. The THz electromagnetic field induces a current inthe antenna whose magnitude is proportional to the electric field and whose directiondepends on the direction o f the field vector. Thus, both the amplitude and the phase ofthe electric field are recorded (time-averaged over the gating interval). By variation ofthe time delay o f the optical gating pulse, the electric field of the transient is mappedout as a function of time. The frequency response of the measurement system extendsfrom several ten GHz to about 4 THz peaking around 1 THz [4]. The spectral responseis plotted in Fig. 7. To avoid absorption of the THz-radiation by water vapor in the air,the whole setup is flushed with dry nitrogen gas during the measurements.

304 H.G. Roskos

4 Detection o f THz-radiation from Bloch oscillations

Wehave observed THz-wave emission from Bloch oscillations in aseries ofsuperlatticesconsisting of 35 periods of Al0.3Gao.rAs barriers and GaAs wells [6, 7]. The barrierwidth o f 17 ,~ is held constant. The well width has values of 61 .~, 67 ,~, 80 A. 97 ,~,, and111 A. In the following, we will concentrate on the sample with 97 ,~ well width thathas been studied most thoroughly as it exhibits superior quality [37]. Kronig-Penneycalculations yield miniband widths of 18 (1.2, 19) meV for the lowest electron (heavyhole, light hole) miniband.

The superlattice structure is grown by molecular-beam epitaxy on a n+-doped GaAssubstrate and is sandwiched between a 2500 .~, thick undoped Al0.3Gao.TAs buffer onthe substrate and another 3500 ~, thick buffer on top. A reverse-bias field can be appliedbetween the substrate and a semitransparent Cr Schottky contact on top o f the sample.

Figure 4Photocurrent spectra of asuperlattice with 18 meV widthof the first electron miniband.The dashed lines indicatedifferent excitonic Wannier-Starkladder transitions (for thelabeling see text). On the bottomof the figure, the spectrum oftheoptical excitation pulses isshown.

The formation o f Wannier-Stark ladders is verified by low-excitation photocurrent(PC) measurements at 4 K. Figure 4 displays PC spectra for different bias voltages[7, 37]. At a reverse bias o f about +0.4 V, the minibands begin to split into Wannier-Stark resonances. We can distinguish three ladders with the same bias dependence. Theheavy-hole (hh)exciton ladder consists o f fourtransitions,labeled by n=-2, - 1,0, and +1,with the index n denoting the position o fthe well where the finalelectron state is centeredwith respect to the hh state where the transition originates (see Fig. 1 for an illustrationo f the nomenclature). The two features on the short-wavelength-sideo f the 0hh excitontransition (3.5 nm and 5.5 nm away from the 0hh line) are assigned to transitionsinvolving excited hh excitons (hh2) and the light-hole (lh) exciton, respectively [48]. Inthe hh2 and the lh ladders,only the transitionswith indices n=- 1 and 0 are visible. Withincreasing reverse bias, the lines with indices n :fi 0 become weaker as the interbanddipole moments decrease with increasing localization of the wave functions [23].

THz-Pulse Emission from Bloch Oscillations 305

Figure5 Left side: Detected electromagnetic transientsfrom asuperlattice with a widthof the firstelectron miniband of 18 meV: right side: Fourier transforms ofthe time domaindata. The data are not corrected for the spectral characteristics of the detection system(sensitivity-corrected spectra are presented in [6]).The signal at -3.6 V is weaker due to ameasurement problem.

In the THz-emission experiment, we excite the sample with 100-150 fs pulses (band-width 15-22 meV) from a Kerr-lens modelocked Ti:sapphire laser with a pulse repetitionrate of 76 MHz. The pulse spectrum (shown in Fig. 4) is peaked at 802 nm. Excitonsare excited at densities of 1-2x 109 cm -2 per well (diameter of the focal spot: 350 #m).The left side of Fig. 5 (see also [6, 7]) displays the detected coherent electromagneticradiation emitted from the superlattice as a function of time for different bias voltages.Time-delay zero is chosen arbitrarily, the excitation pulse hits the sample at adelay timeof about 1.5 ps. The cold finger temperature is 10 K (sample temperature: < 15 K). Theright side of Fig. 5 shows the amplitude of the Fourier transforms of the time domaindata (the phase is not displayed). At flatband (+0.7 V), no THz-radiation is detected. Forincreasing reverse bias, we observe a rather weak 2-ps emission consisting mainly oftwo spectral components, one peaking at 250 GHz and gradually shifting with increas-ing bias to higher frequencies, and a second component peaked around I THz. Thesefeatures will be discussed later in more detail (see Fig. 10). The low-frequency signal,while shifting to higher frequencies, gains in strength whereas the 1-THz signal beginsto disappear at a reverse bias above - 1.0 V. In the time domain, the decline of the 1-THzsignal is seen as a decrease of the second maximum at 2.5 ps that is no longer visibleabove -1.5 V. For a reverse bias larger than -1.5 V, the THz signal increases considerablyin strength. Several oscillations become visible. The oscillation period decreases con-tinuously with increasingreversebias, the frequency shifts from 500 GHz to 2 THz. Upto seven oscillations are observed. Above a reverse bias of -2.9 V, the signal amplitudedecreases as the sensitivity of the measurement system declines at high frequencies.For very high bias, only a single-cycle transient with the spectral characteristics of

306 H.G. Roskos

the detection system remains. This signal results from instantaneous polarization thatapparently is important only at high fields under these excitation conditions.

Figure 6Comparison of thephoton energy of the THzradiation (full circles)withthe energyseparation of the - Ihhand Ohh Wannier-Starktransitions (open circles)determined byelectro-reflectancemeasurements under thesame excitationconditions. Full line:Wannier-Stark splittingin the low-excitationphotocurrentmeasurements; a voltageoffset of-2.9 V accountsfor field screening.Dashed line: Biasdependence as calculatedfrom simple Blochoscillation theory.

The experiments on our superlattice structures have revealed significant quasi-staticfield screening that depends on the sample temperature and the excitation conditions[6, 7, 37]. To relatethe external bias voltage to the effective internal electric field, wehaveperformed time-resolved transmission [37] and cw electro-reflectance measurements[6]. The main conclusion from these measurements is that screening is dominated bycharge accumulation at the boundaries of the superlattice to the buffer layers. The chargebuildup occurs on a time scale of many duty cycles of the laser system (time betweenconsecutive laser pulses: 13 ns). A single laser pulse has only a very weak effect on theinternal field. The field must be rather homogeneous otherwise we would not observesharp Wannier-Stark resonances. The screening shifts the onset of the Wannier-Starkregime to higher bias voltages but - within the accuracy of the measurements - does notaffect the bias dependence of the energy separation between Wannier-Stark resonances.The slope of the energy separation as a function of bias voltage remains the same forthe carrier densities of the time-resolved experiments and the much lower densities ofthe PC measurements.

To prove that the THz-radiation in the Wannier-Stark bias regime originates fromBloch oscillations, Fig. 6 compares the THz-emission data of Fig. 5, corrected for thespectral characteristics of the detection system, with the results ofcw electro-reflectancemeasurements [6] and the PC data shown in Fig. 4: (i.) As a consequence of quasi-static field screening, the emission frequency depends weakly on the bias voltage upto a reverse bias of -2.5 V. Electro-reflectance spectra taken at excitation conditionsidentical to those of the THz-emission measurements indicate a miniband regime up to-2.5 V. (ii.) For higherbias, the emission frequency depends linearly on the voltage. Theelectro-reflectance spectra reveal the excistence of Wannier-Stark ladders. The energy

THz-Pulse Emission from Bloch Oscillations 307

Figure 7Detection of Blochoscillations from asuperlattice using aTFlz-measurementsystem with improveddetection bandwidth.Left side: time-domaindata, right side: Fouriertransforms, corrected forthe spectralcharacteristics of themeasurement system thatis displayed in the graph.Bloch oscillations up to4 THzare detected. Theadditional features above3.5 THzin all spectraresult from signal noisemagnified by the spectralcorrection ofthe data.

separation AE0,_ 1 of the 0hh and the -lhh exciton transitions is plotted as open circlesin Fig. 6. A E o _ I and the THz-emission frequency at the same bias overlap very well.This is the first proofthat the emission is causedby Bloch oscillations. (iii.) The energyseparation of the 0hh and -1hh exciton transitions in the low-excitation PC spectra isplottedas a full line in Fig. 6. A screening-related offsetof-2.9 V is added to the voltage.The line follows closely the bias dependence of AEo _1. This proves that the slope ofAE0,_I(U) is not significantly affected by screening.The same should be valid for thebias dependence o f the emission frequency. Indeed, the emission frequency w has thebias dependence w oc edU/~Ls,~rr , pt~ predicted by the Bloch oscillation theory (dashedline in Fig. 6). d is the superlattice period, Lsample the thickness o f the electricallybiased region, and U the bias voltage. This is the second proof for Bloch oscillations asthe source o f the THz-emission in the Wannier-Stark bias regime.

Since our first report on THz-wave emission from a superlattice in Ref. [6], wehave improved the spectral range o f our THz-detection system. An optimized ion-implantation sequence and extremely careful alignment o f the silicon substrate lens onthe antenna have helped to increase the spectral sensitivity of the detection system to4 THz. Figure 7 shows the results of THz-emission measurements with the improvedsetup. The spectrum o fthe exciting laserpulsepeaks at 795 nm, at a higher photon energythan in the measurements shown in Fig. 5. Furthermore, the excitation density is a factoro f two lowerresulting in a lower bias offset for the onset o f the Wannier-Stark ladders.On the left side o f Fig. 7, the detected THz-radiation field is plottedas a function of time.

308 H.G. Roskos

On the right side, the amplitudes of the corresponding Fourier transforms are displayed.The spectral data are corrected with the response function shown in the figure. Theresponse function is determined independently by THz-emission measurements on InPbulk samples [44]. The general features of Fig. 5 are reproduced. The main differenceis the larger tuning range o f the emission frequency extending up to 4 THz. In theWannier-Stark bias regime, the frequency again shifts linearly with the voltage. Abovea frequency of I THz, the peak amplitude o f the sensitivity-corrected Fourier spectragradually declines with increasing bias voltage as one would expect from the strongerlocalization o f the wave functions. A stronger decrease sets in above the cut-off o four measurements system and has been observed by us with transmittive electro-opticsampling (TEOS), a technique sensitive to internal electric fields with frequencies wellabove 5 THz [49].

Figure 8Photon energy,respectively frequency ofthe THz-radiation vs.applied bias voltage. Fullline: Bias dependence ofthe energy as calculatedfromsimple Blochoscillation theory.

Fig. 8 shows the frequency of the THz-radiation as a function o f the applied biasvoltage. The full line indicates the dependence as expected from Bloch oscillationtheory. The measured data follow the prediction for reverse bias voltages above -2.0 Vifa voltageoffset of- 1.59 V is added to account for the Schottky voltage and quasi-staticscreening.

For possible practical applications of Bloch oscillations, the operation temperature isan important aspect. We have studied Bloch emission at elevated temperatures [7, 8].Figure 9 displays measured THz transients at temperatures from nominally 10 K upto 120 K [8]. The emission frequency is held constant at 2 THz by adjustment o fthe applied voltage. With rising temperature, the amplitude o f the signal decreasesand the damping becomes more rapid. It is pointed out that Bloch oscillations areclearly observed even above the technologically important temperature o f 77 K. Wehave not performed room temperature experiments yet. One may speculate whetherBloch oscillations can be observed at 300 K. To address this question, the temperaturedependence o f the damping of the oscillations is estimated with a phenomenologicalexpression for the scattering rate of excitons. The rate "/=l/T2 can be evaluated by7 = 70 + 71 T + "72 exp (-~o~t.~honon/kT), with the temperature-independent termincluding interface-roughnessscatteringand the exponential term accounting foroptical-phonon scattering [50]. With parameters 7o--0.08 ps- 1,71 =0.02 K- 1p s - i ../2=20.6 p s -1for high-quality GaAs/AIGaAs structures, one estimates "~(T=300 K)=I 1 ps -1. Fromthe condition we xT2/2 > 1 for the detection of at least a single oscillation cycle, one

THz-Pulse Emission from Bloch Oscillations 309

Figure 9Temperature dependence of thedetected THz radiation. Theemission frequency is heldconstant at 2 THz by adjustment ofthe bias voltage. The temporal shiftof the data is caused by the thermalexpansion of the cold-finger.

estimates that Bloch oscillations should be observable at 300 K for frequencies above3.5 THz.

We now discuss the bias adjustment accounting for the temperature-dependent fieldscreening. The adjustment necessitates significant changes of the applied voltage. Be-ginning with an applied bias of- 1.9 V at 10 K, we have to increase the reverse voltagecontinuously up to -4.1 V at 80 K to establish the same internal electric field. This trendreverses for higher temperatures:at 120 K, a bias of only -3.3 V is needed.A decrease ofthe offset with increasing temperature may be explained by thermionic emission o f thecharge carriers across the buffer layers. The offset increaseat low temperature, on theotherhand, points towards a reduction of the transport efficiency within the supertatticewhen the temperature is raised [37].

As the last aspect in this chapter, THz-wave emission at 10 K in the miniband biasregime is briefly addressed. Figure 10 shows THz transients measured at bias voltagesof +0.4 V (left side) and 0.0 V (right side) for different photon energies o f the opticalexcitation pulse. The signal consists of three components, (i.) a half-cycle transientfollowing excitation of the sample, (ii.) one to several cycles of a high-frequencyoscillation, and (iii.) a long-lasting low-frequency oscillation. The half-cycle transientis too slow to be an instantaneous-polarization signal in the sense of Eqn. 2.1. TheFourier transforms (not shown) peak at frequencies as low as 250-300 GHz and do notshow the typical detection-limited bandwidth.With increasing reversebias, the spectralfeature shifts to higherfrequencies.Above a reversebias of-0.4V, it exhibits oscillations[51]. As this behavior is typical for Bloch oscillations, we conclude that the half-cyclesignal originates from overdamped Bloch oscillations that in the miniband bias regimedo not evolve into full oscillations subject to the condition tab ×T2/2 < 1. The secondsignal component, the high-frequency oscillation, exhibits two different frequencies atdifferent bias voltages and wavelengths o f the excitation pulses. At +0.4 V and forwavelenghts from 810 nm to 799 rim, the signal peaks at 1.0-1.3 THz, correspondingto the frequency of the hh/lh quantum interference in this superlattice at low bias (seealso right side of Fig. 5 and Refs. [37, 49]). For shorter wavelengths, it disappears and

310 H.G. Roskos

Figure I0 THz-wave emission in the miniband bias regime.Left side: +0.4 V externalbias; right side: at 0.0 V (further away from flatband). The data are not normalized.

is replaced by a very weak signal at 1.8 THz. At 0.0 V, the 1.0-1.3-THz signal is absentfor all photon energies, but a 1.8-THz transient with 2-3 oscillation cycles is clearlyvisible forwavelengths from 803 nm to 799 nm, then weaker down to 784 nm. 1.8 THzcorresponds to an energy of7 meV, the energyseparationbetween the fundamental (s= 1)hh exciton and excited lh excitons (the energy separation o f the hh and hh2 excitonsis on the order of 4-5 meV, the hh2 exciton transition overlaps with the lh transition).We therefore assign the 1.8-THz oscillation to a quantum interference between hh andlh2 excitons [51]. The third signal component in Fig. 10 is a long-lived oscillationwith a bias- and excitation-energy-independent frequency o f 250 GHz. Its origin is notclear at the present time. Exciton/biexciton or exciton/bound-exciton interference as thesource for the signal is unlikely because both should exhibit a band-edge resonancewhen the wavelength o f the optical pump pulse is varied. The relative insensitivity forphoton-energy tuning suggests beats o f miniband states as the signal source but alsothis explanation seems rather speculative because one would expect a bias-voltage-dependence o f the oscillation frequency. Clearly, this question requires further research.

5 The superradiant character o f THz-wave emission

One driving force behind the interest in Bloch oscillations is the idea to develop tunableTHz-signal generators based on coherent charge oscillations. As the realization ofan all-electrical version o f such oscillators seems rather remote (taking into accountjust the problems associated with circuit parasitics), we have recently proposed quasi-optical oscillator concepts based on the superradiant character o f the emission [52].Superradiance in a general sense is defined as collective emission from N oscillators

THz-Pulse Emission from Bloch Oscillations 311

into the same electromagnetic mode. For quantum systems,two different situations arediscussed in the literature: (i.) The spontaneous emission rate of N two-level oscillatorswith inverted but incoherent population of the upper level is enhanced by a factorof N by collective emission (superfluorescence). (ii.) N oscillators are excited into acoherent, non-inverted state. The coherence results in cooperative emission with thesame enhancement of the spontaneous emission rate as in situation (i.). For N >> t ,the macroscopic polarization o f the coherent system is equivalent to that of a classicalphased array o fdipoles. Situation (ii.) matches the problem discussed in this paper.

To estimate the emission efficiency, we calculate the power of the electromagneticwave emitted from a two-dimensional array o f N coherent dipoles distributed evenlyover a rectangular area a × b. Dividing the integratedradiaton power Pr~d(t = 0) by theinitially available interexcitonic energyE,~a~ yields an effective radiative transition rate7~a. 1 / 7 ~ d is the radiative lifetime, i.e the time that the coherent system would needto emit the complete stored energy if dephasing would be negligibly slow. Assumingthat fields from all parts o fthe excited spot can interfere (c/27 >> a, b, with 3' being thedephasing rate), we obtain [52]

z?2%-,~,~ = (5.4)

67rhcZeoi~r = 2n2W(A/2r)2C(a,a,b). (5.5)

ZI~ is the electron-hole-separation matrix element, w12 denotes the interexcitonic fre-quency determining E , ~ = nlWabfiwx2, with n l being the population density ofthe upper excitonic levels per quantum well and W the number of wells, n~ is thesingle-well population density of the lower excitonic levels. C(a, a, b) contains thegeometrical dependencies o f the emission process. For ~V = 1, Eqn. 5.4 expresses thespontaneous transition rate of a single dipole. The main information o f Eqn. 5.4 is thatthe emission rate is considerably enhanced by the coherence of the ensemble, howevernot all N dipoles participate generally in the cooperativeemission process. The effectivenumber_~" of participating dipoles is determined by the geometry factor C(a, a, b) thatcan be asymptotically evaluated leading to the following expressions o f ~r:

2n2Wab

6 ~ n 2 W ()~/2rr)2 sin2 a / c o s a

67rn2W (A/27r) V/-~--/27r 0.376

i f a, b << A/2rrb >> ,~/27r

i f a >> ),l(2~r c o s a )a :~ 900 c~ ~ (5.6)

i f ~ a,b >> A/2r= 90o ~t

a is the angle o f emission o f the THz beam, respectively the angle of incidence o f theoptical excitation beam. For typical experimental parameters (a, b >> A/27r; c~----~5°;W=35; n2=0.5x 109cm-2; Z12=50; w12=2~'THz), Eqn. 5.6 predicts a number of co-operatively emitting dipoles o f 5 x 108. The radiative lifetime is estimated to be 1 ns.This is considerably longer than the low-temperature dephasing time constant o fseveralps. Hence, most of the energy available for emission is lost. The time-averaged powerof the emitted THz pulses is calculated to be 2 nW in each emission cone taking intoaccount reflection losses on the semiconductor interface. The experimental parametersgiven above correspond to an absorbed optical power of 2 mW with 5 #W theoreticallyavailable for THz-emission. Hence, the power conversion efficiency is on the orderof 10 -6, whereas the theoretical limit for the efficiency is more than three orders o f

312 H.G. Roskos

magnitude higher (5 #W/2 mW). It is interesting to compare these numbers with anestimate o f the radiation power in the experiments. Following Ref. [5], we estimate aTHz-radiation power o f 0.1 nW in each cone which is a factor o f 20 lower than thepower determined from Eqn. 5.4. This discrepancy may result largely from absorptionlosses in the semiconductor itself and from absorption and reflection losses associatedwith the semitransparent Schottky contact on top of the sample.

Neglecting such limitations, the emission efficiency can be enhanced by an increase o fthe radiative transition rate. To extract all available energy with the emitted THz beam,a geometry that couples as many dipoles as possible must be found. It is a remarkableresult o f Eqn. 5.6 that an increase o f the excited spot area with constant excitationdensity does not raise the transition rate for the situation a , b, >> A/2rr typical for theexperiment. The two parameters that can be optimized are the population density r~and the emission angle a . An increase of the populationdensity in praxi will be limitedby the concomitant reduction o f the dephasing time constant T2 [53]. The optimizationo f the emission angle leads to a geometry with a = 90°, i.e. the THz-radiation isemitted parallel to the semiconductor surface. Eqn. 5.6 reveals that for this and only thisspecial choice of ~, ?~" can be raised by an increase of the length ~ of the excited spotarea. For the geometry of Fig. 3, this implies excitation parallel to the surface. A morepractical approach for 90°-emission is the travelling-wave geometry [52, 54] with tiltedwave front of the optical beam. The tilt angle must be chosen such that the excitationbeam hits the sample surface at each point just at the moment when the THz wavetravelling parallel to the surface reaches that point. In this way, phase synchronizationbetween the optical and the THz beam is achieved permitting coherent amplification o fthe propagating THz wave along its path through the sample. This concept is especiallypromising for large-area THz-wave generation with the help of amplified high-powerlaser systems.

The single-pass concept for optimized power extraction can be generalized to multi-pass concepts based on amplification o f the THz wave by phase-locked feed-back o fthe wave into the coherently prepared material with the help o f an external cavity[52]. A related approach has been termed "lasing without inversion" [55], althoughthis expression should be avoided as no coherence is created by the emission processas in laser systems with inversion. The coherent material oscillation is synchronouslypumped by an optical pulse train. Ifa continuous material oscillation is to be maintained[56], stringent synchronization requirements for the pump pulse have to be fulfilled. Asthe excited interband polarization carries the optical phase information, interferometrictime precision at optical frequencies is required to ensure constructive wave-functioninterference in the material [57]. The condition for the timing precision becomes ratherrelaxed, however, if the coherence of the material oscillation is lost between successiveopticalpulses. Then, the phase o fthe returning THz wave has to match the phase o f thequasi-static intraband polarization (i.e the pump-pulse-prepared phase o f the envelopesof the wave functions), but phase matching at optical frequencies is not required. Itshould be pointed out that principally both continuous-wave THz-radiation as wellas pulsed emission from the resonator should be possible [52]. Furthermore, it hasbeen predicted [58], that coherent pumping o f the gain medium reduces the photon-number noise and the phase noise o f the radiation considerably.Hence, a quasi-opticalBloch oscillator could combine the advantages o f wide-range tunability by dc-biasing,inversionless amplification o fthe THz waves by the superradiantnature ofthe emission,pulsed operation o fthe oscillator, and finally quantum-noisequenching by the coherentgain preparation.

THz-Pulse Emission from Bloch Oscillations 313

One may speculateabout the practicability ofthequasi-optical approaches to ultrahigh-frequency oscillators discussed here. Critical points certainly are the operation temper-ature, the pump-pulse source, and cavity losses. In the preceding chapter, we haveshown that Bloch oscillations can be sustained at high-enough frequencies up to atleast 77 K, the technologically important liquid-nitrogen temperature. In DFWM ex-periments, two cycles o f 5 THz oscillations have been detected at 200 K [37]. Hence,even though low-temperature operation is necessary for a large range of frequencies(at least for GaAs/AIGaAs superlattices), it does not mean necessarily temperaturesof 10 K or below. The question of the proper laser source is different to answer for atravelling-wave approach and the cavity-based approach. As already pointed out, thetravelling-wave geometry seems suitable mainly for high-power amplified laser sources.For the cavity-based approach, in principle smaller laser systems are sufficient becauseit is not necessary to pump a long stripe of the gain medium. A practical THz oscillatormay well employ a very compact all-solid-state laser as source o f opticalpulses. On theother hand, cavity losses are a serious concern. Provided high-reflectance mirrors areutilized, the main loss source is the gain medium itself. Reflection losses at the samplesurfaces can be minimized by Brewster-angle operation, but also antireflection coatingsfor the THz-radiation may be used (at the cost o f tunability). Absorption losses mustbe reduced by minimization o f the thickness of all electrically conducting layers butcan not be avoided altogether. The metallization can be applied outside o f the opticallypumped region. It is clear from this discussion that a number o f severe problems haveto be addressed before a practicalquasi-optical oscillatormay be realized. But given thefact that only a few years ago it would have been altogetherunthinkable to implement aTHz emitter based on the concepts of Esaki and Tsu, now there is some chance that thechallenges can be met.

6 Acknowledgements

The author gratefully acknowledges the contributions from C. Waschke who has per-formed the THz-emission measurements, and from K. Victor who has worked out thetheory outlined in the preceding chapter. This work has profited tremendously fromthe investigations o f several researchers at the Institut fa r Halbleitertechnik studyingcoherent superlattice phenomena by different spectroscopic techniques: T. Dekorsy,P. Leisching, R. Schwedler, and K. Leo (now with TU Dresden), all o f us under theguidance o f the head of the institute, Prof. H. Kurz. Contributions by the students F.Brtiggemann, Y. Dhaibi, and W. Beck are gratefully acknowledged. The measurementswould not have been possible without the high-quality MBE samples supplied by K.K~Shler. The author wishes to thank E. Binder, S. L. Chuang, J. Feldmann, H. Grahn,T. Kuhn, T. Meier, M. Nuss, M. Luo, G. von Plessen, W. Sch~ifer, J. Shah (who hasinitially led the author on the track o f coherent phenomena) and the members o f theinstitute for many discussions. Financial support has been provided by the DeutscheForschungsgemeinschaft, the Volkswagen-Stiftung and the Alfried-Krupp-Stiftung.

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