Physica E 12 (2002) 319–322www.elsevier.com/locate/physe

Photo-assisted dynamical transport in multiple quantum wellsRosa L#opez ∗, David S#anchez, Gloria Platero

Instituto de Ciencia de Materiales de Madrid (CSIC), Cantoblanco, 28049 Madrid, Spain

Abstract

We study the dynamical transport in weakly coupled superlattices in the presence of intense radiation in the THz regime.We derive a general model for the time dependent tunneling current within the Keldysh nonequilibrium-Green-functionformalism. For the particular situation in which fast scattering procceses drive the system to local equilibrium within thewells, drastic changes are found in the current vs. voltage curves. ? 2002 Elsevier Science B.V. All rights reserved.

PACS: 73.40.Gk; 73.21.Cd

Keywords: Superlattices; Photo-assisted tunneling; Keldysh formalism

Interaction with external time-dependent ;elds inmesoscopic systems leads in many cases to completelynew ways of electrical transport [1]. In particular,it has been reported in semiconductor superlattices(SLs) negative pumping of electrons, dynamical local-ization, photo-induced electric ;eld domains (EFDs),and bistability between positive and negative current[2,3]. In voltage biased weakly coupled SLs, station-ary EFDs arise if the well doping is large enough.They usually consist of two homogenous electric ;eldregions separated by a domain wall (DW) of accu-mulated electrons. When the carrier density decreasesbelow a critical value, the DW is unable to ;nd astable position and starts to move over several pe-riods. Its motion and recycling gives rise to spon-taneous self-oscillations of the electric current [4].As doping is experimentally hard to manipulate, thisstatic-to-dynamic-state transition has been achievedby the application of transverse magnetic ;elds [5],laser illumination in undoped SLs [6], and careful vari-ation of temperature [7].

∗ Corresponding author. Fax: +34-913720623.E-mail address: rosalg@icmm.csic.es (Rosa L#opez).

Due to the strongly nonlinear behavior of a weaklycoupled SL (stemming from the interplay betweentunneling processes and averaged Coulomb inter-actions), perturbing the system with an AC po-tential involves the rising of even more complexaspects in the physics of transport. In particular,analyses of time-dependent current in the presenceof a low-frequency signal both experimentally andtheoretically have been reported [8]. In that case,the AC frequency is of the order of tens of MHzand the AC potential results in an adiabatic mod-ulation of the system. In this paper, we are inter-ested in a very diEerent regime. We investigate thetime-dependent current through a multiple quan-tum well driven by a high-frequency AC potential,V ac

i (t) = V aci cos(!0t), where the AC frequency

fac = !0=2� is of the order of several THz and V aci

is the AC amplitude in the ith quantum well (QW).It is well known that in this case, photo-assistedtunneling takes place, and the electronic statesdevelop side-bands which act as new tunnelingchannels.The Keldysh Green-function formalism [9] al-

lows to obtain general expressions for the tunneling

1386-9477/02/$ - see front matter ? 2002 Elsevier Science B.V. All rights reserved.PII: S 1386 -9477(01)00355 -1

320 Rosa L&opez et al. / Physica E 12 (2002) 319–322

current. It also permits to include electron–electroninteraction in the system. Both the limits of noninter-acting systems and local equilibrium can be deducedwithin this scheme. We shall restrict ourselves toanalyze the case of a weakly coupled SL where theelectron–electron interaction will be included in amean-;eld (Hartree) manner. From the time evolu-tion of the occupation number operator of the ith QW(Ni), the change in the number of electrons in the ithQW is e〈N i〉= Ii−1; i − Ii; i+1. This continuity equationrelates the change of the number of electrons in theith QW with the current density Jowing from thei−1st (ith) QW to the ith (i+1st) QW, Ii−1; i (Ii; i+1).By using the standard techniques of motion equation[9] one arrives at

Ii; i+1(t) =2e˜ Re

∑ki ki+1

Tki ki+1

∫d�

×[Grki+1

(t; �)g¡ki(�; t) + G¡

ki+1(t; �)ga

ki(�; t)] ;

(1)

Tki ki+1being the transmission coeKcient. ki represents

the set of quantum numbers which label the elec-tronic states in the ith QW. Here, ga(¡)

kiis the ad-

vanced (lesser) Green function (GF) for a decoupledQW in the presence of an AC potential and scatter-ing processes. Gr(¡)

kiis the retarded (lesser) QW GF

when scattering, AC signal and tunneling are present.In the case of a SL, the GFs which appear in Eq. (1)can be written down by making several assumptions.Firstly, one considers the scattering self-energy (dueto, e.g., impurities or LO phonons) in a phenomeno-logical way as an energy independent constant (whichis denoted here by � = Im �sc). As in weakly coupledSL’s the scattering lifetime (∼ 1 ps) is much shorterthan the tunneling time (∼ 1 ns), we can assume localequilibrium within each QW and neglect the tunnelingself-energy in the expressions of the GFs. The electrontransport is thus sequential. Once these approxima-tions are made, the expression for the retarded (lesser)AC-driven QW GF in the presence of scattering isGr(¡)

ki(t; t′) = e(ieV ac

i =˜!0)(sin !0t − sin !0t′) NGr(¡)ki

(t −t′), where NG

r(¡)ki

(t − t′) is the static retarded (lesser)QW GF’s in the presence of scattering. Similar re-lations hold for ga(¡)

ki(t; t′). By using these expres-

sions, the current in the case of local equilibrium

reads [10]

Ii; i+1(t)

=2e˜

∑ki ki+1

Tki ki+1

m=∞∑m=−∞

Jm(�i; i+1)

×{cos(�i; i+1 sin !0t − m!0t)

×∫

d� [Aki+1(� + m˜!0)Aki

(�)

×(fi(�)− fi+1(� + m˜!0))]

+ sin(�i; i+1 sin(!0t − m!0t)∫

d�

×[Aki+1(� + m˜!0)Re Nga

ki ki(�)fi+1(� + m˜!0)

+Re NGrki+1 ki+1

(� + m˜!0)Aki(�)fi(�)]

};

(2)

where Akidenotes the spectral function for the

ith isolated QW in the presence of scattering.Jp is the pth Bessel function whose argumentis given by � = e(V ac

i − V aci+1)=˜!0 and fi(�)

is the Fermi–Dirac distribution function for theith QW.Notice that the current may be written as I(t)= I0+∑l¿0 I cosl cos(l!0 t) + I sinl sin(l!0 t), where I0 is the

time-averaged current [2]. I cosl and I sinl contain higherharmonics for l ¿ 0. Now, since we are interested inthe photo-assisted current, !0 is much larger than thetunneling rate. In addition, the scattering lifetime rep-resents the lowest temporal cutoE above which our as-sumption of local equilibrium within each QW holds.In other words, to ensure the vality of Eq. (2) oneis restricted to study dynamical processes whose timevariation is longer than ˜=�. As intense THz ;elds typ-ically have ˜!0 ¿ � we must carry out a time averageof I(t). This implies that the explicit time variation ofI(t) vanishes and we are left with the implicit changeof I0 with respect to time. This variation (in scaleslarger than ˜=�) results from the evaluation of the con-tinuity equation for i=1; : : : ; N , where N is the numberof wells, supplemented with Poisson equations, con-stitutive relations, and appropiate boundary conditions(see Ref. [10] for details). Of course, by incorporating

Rosa L&opez et al. / Physica E 12 (2002) 319–322 321

0 1 2Voltage (V)

0

10

20

30

40

50

Cur

rent

den

sity

(A

/cm

2 )

β=0β=0.25β=0.5β=0.75β=1

0.00 0.10 0.20Time (µs)

16

20

24

Cur

rent

den

sity

(A

/cm

2 )

β=0β=0.25β=0.5β=0.75β=1

0.8 1

21.9

22.2

β=0β=0.25

(b)(a)3

Fig. 1. (a) Time-averaged current vs. applied voltage (for param-eters see text). (b) Time dependence of the current for V =1:1 Vfor the values of � shown in (a).

an accurate microscopic model for the scattering dy-namics and calculating the nonequilibrium distributionfunction for each QW in the presence of the AC sig-nal, one could investigate the entire parameter range.This is out of the scope of the present work.The total current traversing the sample con-

sists of the tunneling plus displacement currents,I(t)=Ii; i+1+(�=d)(dVi=dt), where (� is the static per-mittivity, d the barrier width, and Vi the voltage dropin the ith barrier. Given reliable initial conditions,I(t) is self-consistently calculated for a N = 40 SLwith 13.3-nm GaAs wells and 2.7-nm AlAs barriers.Well doping is 2× 1010 cm−2 and we take �=8 meVand fac = 3 THz. In Fig. 1(a) the average of I(t)is plotted as a function of the applied DC bias, V .Without AC forcing the I–V curve shows branchesafter the peak corresponding to resonant tunnelingbetween the lowest subbands (see inset). This featureis distinctive of static EFD formation [2–4]. In thepresence of an AC signal, the branches coalesce anda plateau is formed—this is the key signature of cur-rent self-oscillations. I(t) for V = 1:1 V is depictedin Fig. 1(b). For � = 0, the current achieves a con-stant value after a transient time. As � is increased,damped oscillations are observed until a stable cur-rent self-oscillations arise when � = 1. This is anindication that the AC potential induces a transitionfrom a stationary con;guration towards a dynamicstate possibly via a supercritical Hopf bifurcation.The oscillation frequency is of the order of 150 MHz,

0 0.5 1 1.5 2 2.5 3Voltage (V)

0

10

20

30

40

50

Cur

rent

den

sity

(A

/cm

2 )

β=0β=0.25β=0.5β=0.75β=1

0.00 0.05 0.10Time (µs)

12

16

20

24

Cur

rent

den

sity

(A

/cm

2 )

β=0β=0.25β=0.5β=0.75β=1

(b)(a)

Fig. 2. (a) Time-averaged current vs. applied voltage as � is varied.(b) I(t) for V = 1:1 V. Curves for � �=0 are shifted for clarity.

much smaller than !0. Then we conclude that theexistence of sidebands and its inJuence over the non-linear behavior of the system drives the SL towardsoscillations.Increasing � results in dramatic consequences. This

may be eEectively achieved by either applying a trans-verse magnetic ;eld [5] or raising the temperature[7]. In the absence of AC potentials for � = 11 meVthere is a voltage range (∼ 0:8–1:3 V) where nowself-oscillations show up [4] (see Fig. 2(a)). As � en-hances, the plateau starts to be replaced by a posi-tive diEerential resistance region. There is a similarwell-known phenomenon in weakly coupled SLs: un-der a critical value of the carrier density neither staticnor moving DWs exist and the electric ;eld drops ho-mogenously across the whole sample. In our case, thedoping density is constant and the AC ;eld is the phys-ical parameter that forces this transition. To illustratethis, we have calculatedI(t) for a ;xed bias V=1:1 V(see Fig. 2(b)). At � = 0 the current oscillates with afrequency of 170 MHz. This is a result of the motionof the accumulating layer of electrons, and its recy-cling in the highly doped contacts [4]. For �=0:75 thecurrent is already damped, and I(t) reaches rapidly auniform value for �=1. This is a striking feature—anoscilation disappearance induced by an AC potential.A qualitative explanation follows.The general transition (static EFDs)→(moving

EFDs) →(homogenous electric ;eld pro;le) dependsbasically on the particular shape of the drift velo-city between two wells [4–7]. For a suKciently high

322 Rosa L&opez et al. / Physica E 12 (2002) 319–322

peak-to-valley ratio the system evolves towards astatic EFD situation. If this ratio is very low the ap-plied voltage drops smoothly across the sample. Inbetween, self-sustained current oscillations show up.In our case the application of a THz ;eld producestunneling with absorption and emission and photons,thus strongly modifying the current between twowells (or, roughly, the drift velocity). In particular, forthe values used in the text the zero-photon tunnelingpeak (weighted by J 2

0 (�)) is depressed and the cur-rent associated with the emission of, e.g., one photonbecomes comparable when � increases from 0. Thenthe peak-to-valley ratio decreases as � increases andthe current oscillations are reached provided we startfrom a stationary current (see Fig. 1). On the otherhand, if the dynamic con;guration is stable for � = 0the AC potential will tend to drive the SL to a triv-ially homogenous electric ;eld pro;le (see Fig. 2). Amore detailed analysis will be given elsewhere [10].

Acknowledgements

We thank R. Aguado for fruitful discussions.This work is supported by the Spanish DGES grant

PB96-0875 and by the European Union TMR contactFMRX-CT98-0180.

References

[1] M. BSuttiker, J. Low Temp. Phys. 118 (2000) 519.[2] R. Aguado, G. Platero, Phys. Rev. Lett. 81 (1998) 4971.[3] B.J. Keay, et al., Phys. Rev. Lett. 75 (1995) 4102.[4] J. Kastrup, et al., Phys. Rev. B 55 (1997) 2476;

A. Wacker, M. Moscoso, M. Kindelan, L.L. Bonilla, Phys.Rev. B 55 (1997) 2466.

[5] B. Sun, et al., Phys. Rev. B 60 (1999) 8866.[6] N. Ohtani, N. Egami, H.T. Grahn, K.H. Ploog, Phys. Rev. B

61 (2000) 5097(R).[7] J.N. Wang, et al., Appl. Phys. Lett. 75 (1999) 2620;

D. S#anchez, L.L. Bonilla, G. Platero, Phys. Rev. B 64 (2001)115311.

[8] K.J. Luo, et al., Phys. Rev. Lett. 81 (1998) 1290;D. S#anchez, G. Platero, L.L. Bonilla, Phys. Rev. B 63 (2001)201,306(R).

[9] H. Haug, A.-P. Jauho, Quantum Kinetics in Transport andOptics of Semiconductors, Springer, Berlin, 1996.

[10] R. L#opez, D. S#anchez, G. Platero (2000), in preparation.