The Dynamical Franz-Keldysh Effect is exposed by exploring near-bandgap
absorption in the presence of intense THz electric fields. It bridges the gap
between the DC Franz- Keldysh effect and multi-photon absorption and competes
with the THz AC Stark Effect in shifting the energy of the excitonic resonance.
A theoretical model which includes the strong THz field non-perturbatively via
a non-equilibrium Green Functions technique is able to describe the Dynamical
Franz-Keldysh Effect in the presence of excitonic absorption.Comment: 4 pages in revtex with 5 figures included using epsf. Submitted to
  Physical Review Letter

A. Patanè

A.-P. Jauho

G. Ramian

H. Haug

J. Kono

J. S. Bakos

K. Johnsen

L. V. Keldysh

V. Nathan

W. Franz

Y. T. Rebane

Y. Yacoby

English

arXiv

K. B. Nordstrom

S. J. Allen

B. Birnir

T. Noda

H. Akiyama

H. Sakaki

Crossref

Physical Review Letters

Excitonic Dynamical Franz-Keldysh Effect

The dynamical Franz-Keldysh effect is exposed by exploring near-band-gap absorption in the presence of intense THz electric fields. It bridges the gap between the de Franz-Keldysh effect and multiphoton absorption and competes with the THz ac Stark effect in shifting the energy of the excitonic resonance. A theoretical model which includes the strong THz field nonperturbatively via a nonequilibrium Green functions technique is able to describe the dynamical Franz-Keldysh effect in the presence of excitonic absorption

Nordstrøm, K.B.

Johnsen, Kristinn

Allen, S.J.

Jauho, Antti-Pekka

Birnir, B.

Kono, J.

Noda, T.

Akiyama, H.

Sakaki, H.

Online Research Database In Technology

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Downloaded from orbit.dtu.dk on: Dec 17, 2017Excitonic dynamical Franz-Keldysh effect.Nordstrøm, K.B.; Johnsen, Kristinn; Allen, S.J.; Jauho, Antti-Pekka; Birnir, B.; Kono, J.; Noda, T.;Akiyama, H.; Sakaki, H.Published in:Physical Review LettersLink to article, DOI:10.1103/PhysRevLett.81.457Publication date:1998Document VersionPublisher's PDF, also known as Version of recordLink back to DTU OrbitCitation (APA):Nordstrøm, K. B., Johnsen, K., Allen, S. J., Jauho, A-P., Birnir, B., Kono, J., ... Sakaki, H. (1998). Excitonicdynamical Franz-Keldysh effect. Physical Review Letters, 81(2), 457-460. DOI: 10.1103/PhysRevLett.81.457VOLUME 81, NUMBER 2 P HY S I CA L REV I EW LE T T ER S 13 JULY 1998Excitonic Dynamical Franz-Keldysh EffectK.B. Nordstrom,1 K. Johnsen,2,3 S. J. Allen,1 A.-P. Jauho,2 B. Birnir,3 J. Kono,4 T. Noda,5H. Akiyama,5 and H. Sakaki4,51Center for Terahertz Science and Technology, Quantum Institute, University of California, Santa Barbara, California 931062Mikroelektronik Centret, Technical University of Denmark, Bldg 345east, DK-2800 Lyngby, Denmark3Mathematics Department, University of California, Santa Barbara, California 931064Quantum Transition Project, Japan Science & Technology Corporation, Tokyo 153, Japan5Institute of Industrial Science, University of Tokyo, Tokyo 106, Japan(Received 30 December 1997)The dynamical Franz-Keldysh effect is exposed by exploring near-band-gap absorption in thepresence of intense THz electric fields. It bridges the gap between the dc Franz-Keldysh effectand multiphoton absorption and competes with the THz ac Stark effect in shifting the energy of theexcitonic resonance. A theoretical model which includes the strong THz field nonperturbatively via anonequilibrium Green functions technique is able to describe the dynamical Franz-Keldysh effect in thepresence of excitonic absorption. [S0031-9007(98)06611-3]PACS numbers: 78.30.Fs, 71.35.Cc, 78.20.Bh, 78.66.FdWe report observation of the dynamical Franz-Keldysheffect (DFKE) by examination of the near-band-gap opti-cal properties of a semiconductor multiple quantum well(MQW) in a new experimental regime. By applying astrong electric field at frequencies near 1 THz, we per-form electroabsorption spectroscopy which is describedneither by the dc Franz-Keldysh effect (FKE) [1,2] norby optical-frequency effects like multiphoton absorption(MPA) [3]. The DFKE region, bridging these two ex-tremes, is a topic of long-standing theoretical interest [4–6], but has not previously been studied experimentally.The crossover between “high” and “low” frequencymay be quantified in terms of two parameters, EKE andg, defined asEKE ­e2E2THz4mpv2THz. (1)g ­EKEh¯vTHz­e2E2THz4mph¯v3THz. (2)EKE is the mean kinetic energy of a particle of mass mpand charge e in an electric field $ETHz cossvTHztd, i.e., theponderomotive energy. The dc FKE corresponds to g À1; multiphoton effects occur for g ¿ 1; and the case ofg , 1 is a new, distinct DFKE regime [4] (Fig. 1). Thisregime is difficult to access experimentally, as EKE mustbe comparable to or greater than any broadening energiesin the system. However, it is possible for applied fieldswith vTHz , 1 THz and ETHz , 1 10 kVycm [5].In this paper, we explore the optical absorption of asemiconductor MQW driven by an intense THz electricfield polarized in the plane of the MQW layers. Forthe case of noninteracting carriers, the DFKE predicts ablueshift of the main absorption edge in the spectrum byEKE, as well as increased subgap absorption [5] (Fig. 2a).This DFKE blueshift has not previously been observed.In order to properly address the DFKE, we must alsoconsider the effect of excitons, which dominate the ob-served near-band-gap absorption in undoped semiconduc-tors. Excitons are bound states of an electron and holewith hydrogenlike quantized energy levels. The energylevel spacings can be in the THz regime. The interactionof excitonic states with a THz field leads to shifts of thelevels via the ac Stark effect [7] (Fig. 2b). The THz fieldinteracts most strongly with the exciton 1s ! 2p tran-sition [8]. We present below a novel theoretical modelwhich shows that the experimentally observed changes inthe absorption spectrum, as a function of the THz field in-tensity and frequency, display the DFKE and its interplaywith the ac Stark effect.Our MBE-grown sample consists of a 20-period MQWwith 80 Å In0.2Ga0.8As wells separated by 150 Å GaAsbarriers on a GaAs substrate. The energy of the 2DMQW states is below the substrate band gap, allowing usto perform direct transmission spectroscopy without sub-strate removal. Strain lifts the valence-band degeneracyat k ­ 0, splitting the light-hole and heavy-hole bands byNot Well KnownWellKnownDFKEFKEMPAγ ~ 1γ >> 1Eωγ << 1FIG. 1. A schematic map of electro-optic phase space as afunction of amplitude and frequency of the perturbing field,showing DFKE regime “between” MPA and FKE.0031-9007y98y81(2)y457(4)$15.00 © 1998 The American Physical Society 457VOLUME 81, NUMBER 2 P HY S I CA L REV I EW LE T T ER S 13 JULY 1998b). c).THzNIRa).αΕΕ KEΕ = Ε gap|2>|2'>|1>|1'>ω12 ω12*ωTHz∆FIG. 2. (a) DFKE in an ideal 2D system. In a strong THzfield (dashed curve, solid curve is zero field), the “edge”shifts to higher frequency and subgap absorption increases.(b) The ac Stark effect: a strong field applied at frequencyv , v12 causes v12 to shift. For v , s.dv12, the transitionshifts to vp12 . s,dv12. D ­ sv12 2 vp12dy2 is the shift ofj1l with respect to distant energy levels. (c) Schematic ofexperiment.50 meV. The electron-heavy-hole exciton is the lowest-energy feature. It is inhomogeneously broadened by in-terface roughness and alloy disorder [9] with a FWHM of6 meV. The exciton 1s ! 2p transition energy (h¯v12)is ,8 meV, estimated from magnetotransmission experi-ments [8].A cw Ti-sapphire laser provides tunable near-band-gap radiation, in the near infrared (NIR), focused toan ,200 mm spot. Its transmission is detected by aconventional Si p-i-n photodiode, and peak NIR intensityis ,100 mW ycm 2. THz radiation is provided by theUCSB free-electron laser, a source of intense, coherentradiation tunable from h¯vTHz , 0.5 20 meV [10]. It isroughly collinear with the NIR beam and focused to a0.5–2.5 mm spot coincident with the NIR focus (Fig. 2c).We measure transmission spectra as a function of THzintensity and frequency. All measurements are performedin He vapor at ,7 K.We observe two distinct regimes of experimental be-havior for different ranges of THz frequency, vTHz.When vTHz is less than v12, we observe a redshift ofthe exciton for low THz intensities, i.e., an ac Stark shift.As THz intensity increases, this shift reaches a maxi-mum and reverses, eventually becoming a net blueshiftat the highest THz intensity; the DFKE blueshift domi-nates at high intensities as the ac Stark effect saturates(Fig. 3a). Conversely, if vTHz is greater than v12, weobserve a blueshift of the exciton which increases mono-tonically with increasing THz intensity; the ac Stark ef-fect and DFKE act in concert, each contributing to theblueshift (Fig. 3b). In both cases, the exciton peak isbroadened and suppressed, and the broadening increaseswith increased THz intensity (Figs. 3a and 3b).Since the apparent peak shifts are small, we quantifythem by taking a weighted average of the data near theexciton line. We find the geometric mean of the upperhalf of the excitonic peak, and plot this “center of mass”against g, which is directly proportional to THz power(Fig. 4). Note that our estimates of the maximum g ateach frequency are only accurate to within a factor of 2 or3. We also show results of our theoretical calculations (tobe discussed below). Again, we see an initial redshift,followed by a blueshift at low vTHz (Fig. 4a) and amonotonic blueshift at high vTHz (Fig. 4b).Theoretically, we describe the optical properties ofthe system by calculating the time dependent inducedpolarization. The system is probed using a weak NIRprobe field $ENIRstd with frequency v , Egapyh¯. Tolinear order in the NIR field, the macroscopic polarizationis the $k trace of$Ps $k, td ­Z ‘2‘dt0 xrs $k; t, t0d $ENIRst0d . (3)Energy(eV)Transmission (arb. units)I = 0I = 7b).1.39 1.4 1.41Energy(eV)Transmission (arb. units)a).1.39 1.4 1.41I = 12I = 0FIG. 3. Experimental transmission of MQW near e1h1exciton with (a) h¯vTHz ­ 2.5 meV at ITHz ­ 0, 1, 2, 4, 12(arbitrary units). (b) h¯vTHz ­ 14 meV at ITHz ­ 0, 1, 2, 4, 7(arbitrary units). Arrows connect calculated centers of experi-mental peaks and point in the direction of increasing ITHz.458VOLUME 81, NUMBER 2 P HY S I CA L REV I EW LE T T ER S 13 JULY 1998Exciton Shift (meV)γa).-0.400.40 0.4 0.800.40.81.20 0.08 0.16γExciton Shift (meV) b).FIG. 4. Center of mass of measured (squares) and calculated(triangles) exciton transmission peak vs g for (a) h¯vTHz ­2.5 meV , h¯v12 and (b) h¯vTHz ­ 10.5 meV . h¯v12.All the information about the THz field is contained inthe nonequilibrium generalization of the retarded two-time interband susceptibility xrs $k; t, t0d. In our previouswork [5], based on nonequilibrium Green functions,we studied the susceptibility starting from a two-bandHamiltonian, including the THz field nonperturbativelyin the Coulomb gauge via the vector potential $Astd ­2 $ETHz sinsvTHztdyvTHz. Here, $ETHz is oriented inthe plane of the MQW layers. Nonequilibrium Greenfunction techniques [11] yield an analytic expression forthe single particle susceptibility:x¯rs $k; t, t0d ­ 22d2h¯ust 2 t0d sin(Z tt0dsh¯efh¯ $k 2 e $Assdg),(4)where d is a dipole matrix element and ef $kg ­ k22m 1Egap , m being the electron-hole reduced mass and $k thein-plane momentum.We include excitonic effects by using the Bethe-Salpeterequation [12] to determine a susceptibility which includesthe electron-hole interaction in the ladder approximation,xrs $k; t, t0d ­ x¯rs $k; t, t0d1Z d2 $k0s2pd2Z ‘2‘dt00 x¯rs $k; t, t00d3 V sj $k 2 $k0jdxrs $k0; t00, t0d , (5)where x¯rs $k; t, t0d is the single-particle, noninteractingsusceptibility of Eq. (4). Expressing the susceptibility asxrs $k; t, t0d ­PnR dv2p xrns $k, vdeivst2t0d1in2vTHzst1t0d, theintegral equation (5) becomes a matrix equation [13],xrns $k, vd ­ x¯rn s $k, vd 1Xn0x¯rn2n0s $k, v 1 2n0vTHzdZ d2 $k0s2pd2V sj $k 2 $k0jdxrn0f $k0, v 1 2sn0 2 ndvTHzg . (6)For physically achievable THz intensities, only small val-ues of n are needed.Numerically, we solve the matrix equation includingboth s-wave and p-wave scattering; both are important,since the THz field will couple the s and p states ofthe exciton and influence the observed resonance via theac Stark effect. The solutions to the resulting equation arefound by discretizing the integrals to yield a set of linearequations which we solve by standard methods [14].Finally, we relate the results to physically measurablequantities by expressing the macroscopic polarization as$Pstd ­ $ENIRXnx˜nsvdeis2nvTHz2vdt, (7)where x˜nsvd ­P$k xrn s $k, vd. The linear absorption isthus proportional to Im x˜0svd. The terms with n Þ 0describe the nonlinear mixing of the NIR and the THzfield, resulting in optical sideband generation. Thesesidebands show a rich structure, but an analysis is beyondthe scope of this work, and will be reported separately.Using the above method, we have calculated absorptionspectra using the experimental parameters of our system.We find good agreement with experiment, using no fit-ted parameters. For vTHz , v12, theory predicts a red-shift of the exciton absorption peak at low THz intensities.With increasing THz intensity, the shift saturates and thenreverses, eventually becoming a net blueshift (Fig. 5a).Theory also predicts a blueshift for vTHz . v12, whichmonotonically increases with THz intensity for experi-mentally accessible intensities (Fig. 5b). In both cases,the exciton peak is suppressed and inhomogeneouslybroadened with increasing intensity, i.e., with increasingg (Figs. 5a and 5b). Using the same method as for themeasured spectra, we compute the center of mass of thecalculated exciton lines. We plot the shift of the theoret-ical “center of mass” as a function of g in Fig. 4, alongwith the experimental results. The theoretical and experi-mental peak shifts show identical qualitative behavior.As described above, this behavior can be understood asthe DFKE acting in competition with the ac Stark effect.The ac Stark effect, at low ETHz, results in a shift of the459VOLUME 81, NUMBER 2 P HY S I CA L REV I EW LE T T ER S 13 JULY 1998FIG. 5. Calculated absorption of MQW with (a) h¯vTHz ­2.5 meV , h¯v12 from top down at g ­ 0, 0.03, 0.06, 0.1,0.5, and 1.5. (b) h¯vTHz ­ 14 meV . h¯v12 from top down atg ­ 0, 0.01, 0.02, 0.04, 0.08, and 0.15.exciton level,D ~svTHz 2 v12dE2THzsvTHz 2 v12d2 1 G2, (8)where G is the width of the 1s ! 2p transition line [7].We estimate h¯G , 4 meV in our sample. The magnitudeof D is dominated by G, and we expect little resonantenhancement of jDj for vTHz , v12. D does not increaseindefinitely, and will saturate as ETHz increases, goingfrom quadratic to linear dependence on ETHz [7]. TheDFKE, however, always provides a blueshift proportionalto E2THz, given by EKE [5].The net effect is illustrated in Fig. 4, which shows theshift of the center of mass of the exciton transmissionpeak as a function of g for both theory and experiment.For vTHz , v12, we see a redshift which saturates andreverses with increasing intensity, showing a roughlylinear dependence on THz intensity at high fields(Fig. 4a). At low fields, D dominates, resulting in anet redshift. As D saturates with increasing field, EKEbegins to dominate, eventually overwhelming the redshiftentirely and resulting in a net blueshift. For vTHz . v12,we observe only an increasing blueshift with increasingETHz (Fig. 4b). Here, D and EKE cooperate to create ablueshift.In conclusion, we have observed the experimentalsignature of DFKE through its interplay with the acStark effect. Our theory and observations are in remark-able agreement, and show the resulting frequency- andintensity-dependent shift of the excitonic resonance in aMQW under intense THz irradiation.This work is supported by ONR Grant No. N00014-92-J-1452, the Quantum Transition Project of JSTC, QUEST(an NSF Science & Technology Center), LACOR GrantNo. 4157U0015-3A from Los Alamos National Labora-tory, National Science Foundation Grant No. CDA96-01954, and Silicon Graphics, Inc. The authors also thankD. P. Enyeart, D. T. White, J. R. Allen, and G. Ramian atthe Center for Terahertz Science & Technology for theirinvaluable technical support.[1] W. Franz, Z. Naturforsch. 13, 484 (1958).[2] L. V. Keldysh, Sov. Phys. JETP 34, 788 (1958).[3] See, e.g., V. Nathan, A.H. Guenther, and S. S. Mitra,J. Opt. Soc. Am. B 2, 294 (1985), and references therein.[4] Y. Yacoby, Phys. Rev. 169, 610 (1968).[5] A.-P. Jauho and K. Johnsen, Phys. Rev. Lett. 76, 4576(1996); K. Johnsen and A.-P. Jauho, Phys. Rev. B 57,8860 (1998).[6] Y. T. Rebane, Sov. Phys. Solid State 27, 824 (1985).[7] See, e.g., J. S. Bakos, Phys. Rep. 31C, 209 (1977), andreferences therein.[8] J. Kono, M.Y. Su, T. Inoshita, T. Noda, M. S. Sherwin,S. J. Allen, Jr., and H. Sakaki, Phys. Rev. Lett. 79, 1758(1997).[9] A. Patanè, A. Polimeni, M. Capizzi, and F. Martelli, Phys.Rev. B 52, 2784 (1995).[10] G. Ramian, Nucl. Instrum. Methods Phys. Res., Sect. A318, 225 (1992).[11] H. Haug and A.-P. Jauho, Quantum Kinetics in Transportand Optics of Semiconductors, Springer Series in Solid-State Sciences Vol. 123 (Springer, Berlin, 1996).[12] H. Haug and S. Schmitt-Rink, Prog. Quantum Electron. 9,1 (1984).[13] K. Johnsen and A.-P. Jauho (unpublished).[14] H. Haug and S.W. Koch, Quantum Theory of the Opticaland Electronic Properties of Semiconductors (WorldScientific, Singapore, 1993).460

Excitonic dynamical Franz-Keldysh effect.

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Excitonic Dynamical Franz-Keldysh Effect

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