Theories describing electrical transport in semiconductor superlattices can
essentially be divided in three disjoint categories: i) transport in a
miniband; ii) hopping between Wannier-Stark ladders; and iii) sequential
tunneling. We present a quantum transport model, based on nonequilibrium Green
functions, which, in the appropriate limits, reproduces the three conventional
theories, and describes the transport in the previously unaccessible region of
the parameter space.Comment: 4 Page

Jauho, Antti-Pekka

Wacker, Andreas

English

arXiv

Andreas Wacker

Antti-Pekka Jauho

Crossref

Physical Review Letters

Quantum Transport: The Link between Standard Approaches in Superlattices

Theories describing electrical transport in semiconductor superlattices can essentially be divided in three disjoint categories: (i) transport in a miniband; (ii) hopping between Wannier-Stark ladders; and (iii) sequential tunneling. We present a quantum transport model, based on nonequilibrium Green functions, which, in the appropriate limits, reproduces the three conventional theories and describes the transport in the previously inaccessible region of the parameter space

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Downloaded from orbit.dtu.dk on: Dec 17, 2017Quantum Transport: The Link between Standard Approaches in SuperlatticesWacker, Andreas; Jauho, Antti-PekkaPublished in:Physical Review LettersLink to article, DOI:10.1103/PhysRevLett.80.369Publication date:1998Document VersionPublisher's PDF, also known as Version of recordLink back to DTU OrbitCitation (APA):Wacker, A., & Jauho, A-P. (1998). Quantum Transport: The Link between Standard Approaches inSuperlattices. Physical Review Letters, 80(2), 369-372. DOI: 10.1103/PhysRevLett.80.369VOLUME 80, NUMBER 2 P HY S I CA L REV I EW LE T T ER S 12 JANUARY 1998Quantum Transport: The Link between Standard Approaches in SuperlatticesAndreas Wacker* and Antti-Pekka JauhoMikroelektronik Centret, Bldg 345 east, Danmarks Tekniske Universitet, 2800 Lyngby, Denmark(Received 25 September 1997)Theories describing electrical transport in semiconductor superlattices can essentially be divided inthree disjoint categories: (i) transport in a miniband; (ii) hopping between Wannier-Stark ladders; and(iii) sequential tunneling. We present a quantum transport model, based on nonequilibrium Greenfunctions, which, in the appropriate limits, reproduces the three conventional theories and describes thetransport in the previously inaccessible region of the parameter space. [S0031-9007(97)04960-0]PACS numbers: 73.61.– r, 72.10.–d, 72.20.HtEver since the pioneering work of Esaki and Tsu [1],which drew attention to the rich physics and potentialdevice applications of semiconductor superlattices, theseman-made structures have remained a topic of intense re-search. Semiconductor superlattices have proven to be afruitful platform for studying a wide range of transportphenomena, such as their intrinsic negative differentialconductivity [2], the formation of electric field domains[3], Bloch oscillations [4], as well as dynamical localiza-tion [5] and absolute negative conductance [6] under ex-ternal irradiation, just to mention a few.These phenomena depend crucially on the relations ofthe energy scales involved, namely, the zero-field mini-band width (which is 4 times the interwell coupling T1),the scattering rate Gyh¯, and the potential drop per pe-riod (; eFd, where F is the applied static field and d isthe superlattice period). Three distinct approaches havebeen used to describe transport in the parameter spacespanned by sT1, eFd, Gd: miniband conduction (MBC)[1,7], Wannier-Stark hopping (WSH) [8], and sequentialtunneling (ST) [9,10]. While the ranges of validity of thedifferent approaches have been addressed qualitatively be-fore [11–14], no explicit calculations have been presentedwhere the different ranges can be identified and the tran-sition between them can be studied. In the present Letterwe present such a calculation, based on nonequilibriumGreen functions. The calculated current-field relations areshown to reflect the results from the simple approaches(MBC, WSH, and ST, which will be reviewed below) intheir respective ranges of validity sketched in the “phasediagram” presented in Fig. 1. While similar diagramshave been obtained in Refs. [13,14] from more phe-nomenological arguments, we will derive the borderlinesfrom our Green function analysis here.Now, we introduce the model assumptions which willbe used in each of the following approaches. We restrictourselves to the lowest miniband of the superlattice. Ourbasis sets are orthonormal wave functions Cnszdeisk?rdyAwhere the z direction denotes the growth direction. TheCnszd ­ Csz 2 ndd are localized in well n (for example,one may use the Wannier functions). Here r and k denotetwo-dimensional vectors within the sx, yd plane (with areaA) which is assumed to be separable from the z direction.For parabolic dispersion Ek ­ h¯2k2y2m (with the effec-tive mass m of the conduction band) we thus have a con-stant density of states r0 ­ myp h¯2 per area and period.The single-particle part of the Hamiltonian within nearestneighbor coupling is then given byHˆSLn,m ­ sdn,m21 1 dn,m11dT1 1 dn,msEk 2 neFdd .(1)Furthermore we consider a phenomenological scatteringprocess at d potentials with density Nd and matrix elementdn,mVyA, leading to a scattering rate 1yt0 ­ G0yh¯ ­NdpV 2r0yh¯ between the k states within a given well.Finally, we assume that the inscattering term is deter-mined by a Fermi distribution nFsEd ­ h1 1 expfsE 2mdykBTegj21 with electron temperature Te and chemicalpotential m. This assumption establishes internal energyrelaxation without specifying the corresponding processesin detail. It has been implicitly used in the standard ap-proaches [7,8,10] as well.Miniband conduction.—For zero electric field Eq. (1)is diagonalized by a set of Bloch functions wqszd ­Pn einqdCnszd and the dispersion relation is given by theFIG. 1. Regimes where the different transport models arevalid for low electron densities and low temperatures. (Forillustrative purpose we have translated the condition a À b inthe text to a . 2b, where a and b denote the respective energyscales involved.)0031-9007y98y80(2)y369(4)$15.00 © 1998 The American Physical Society 369VOLUME 80, NUMBER 2 P HY S I CA L REV I EW LE T T ER S 12 JANUARY 1998miniband Esqd ­ 2T1 cossqdd. The stationary Boltzmannequation for the distribution function fsq, kd is theneFh¯›fsq, kd›q­nFfEsqd 1 Ekg 2 fsq, kdtfEsqd 1 Ekg, (2)where the relaxation-time approximation corresponds toour assumption on scattering mentioned above. For ourscattering model, we obtain the relaxation time tsEd ­ t0for E $ 2jT1j and tsEd ­ pt0y arccoss2Ey2jT1jd for22jT1j # E , 2jT1j. Equation (2) is solved numericallyand the current is calculated fromJsFd ­e4p3h¯Zd2kZ pyd2pyddq fsq, kddEsqddq. (3)The electron density per period is given byN2D ­d4p3Zd2kZ pyd2pyddq fsq, kd (4)and is used to determine the chemical potential [which isfield dependent due to the energy dependence of tsEd] fora given electron density. This approach can be extendedbeyond the relaxation-time approximation [15,16], but thegeneric features remain unchanged.Wannier-Stark hopping.—In the presence of an electricfield, the eigenstates of the Hamiltonian become the local-ized Wannier-Stark states,fnszd ­XnJn2nµ2T1eFd¶Cnszd , (5)with energy En ­ 2neFd where Jnszd is the Besselfunction of the first kind. Scattering causes hoppingbetween the different states. Within Fermi’s golden rule,the current is given byJsFd ­Xl.0let0Xn•Jnµ2T1eFd¶Jn2lµ2T1eFd¶‚2 12p23Zd2k fnFsEkd 2 nFsEk 1 leFddg . (6)Here the termPfJnJn2lg2 arises due to the spatial overlapof the Wannier-Stark functions, and the Fermi functionsreflect our assumption regarding inscattering. The electrondensity per period is given byN2D ­ r0kBTe ln•1 1 expµmkBTe¶‚, (7)which relates m to N2D . Again, it is possible to generalizethis approach to more realistic scattering mechanisms[17,18].Sequential tunneling.—In this approximation the phaseinformation is lost after each tunneling event between ad-jacent wells. The scattering within a well is treated self-consistently by solving for the spectral functions AsE , kd;in this work we use the self-consistent Born approximation[19] for the self-energy. The transitions to neighboringwells are calculated in lowest order of the coupling yield-ing [14,19,20],JsFd ­e2p2Zd2kZ dE2p h¯T21 AsE , kdAsE 1 eFd, kd3 fnFsE d 2 nFsE 1 eFddg . (8)The carrier density is given byN2D ­12p2Zd2kZ dE2pnFsE dAsE , kd . (9)This approach gives quantitative agreement with experi-ments in weakly coupled structures when realistic modelsfor impurity and interface scattering are employed [10,14].The important issue to recognize is that these threeapproaches treat scattering, external field, and couplingwithin different approximations. MBC does not properlyinclude field-induced localization because of its inherentassumption of extended states, WSH treats scattering inlowest order perturbation theory (in particular, there isno broadening of the states), and ST is explicitly low-est order in the interwell coupling. In contrast to theseshortcomings, a full quantum transport theory, based onnonequilibrium Green functions [21], is able to treat scat-tering, electric field, and coupling on equal footing. Suchan approach was performed in Ref. [13] using a basis ofWannier-Stark states and restricting the analysis to a highelectron temperature. Here we work within the basisCnszd and consider the general situation which allows ananalysis of transitions between the simplified approachesMBC, WSH, and ST.Nonequilibrium Green functions (NGF).—Here thecurrent and electron density are given by [19,21]JsFd ­e2p2Zd2k2h¯RehT1G,n11,nst, t, kdj , (10)N2D ­12p2Zd2k G,n,nst, t, kd , (11)where G,m,nst, t0, kd ­ ikayn st0, kdamst, kdl, and ayn st, kdand anst, kd are the creation and annihilation operatorsfor the state Cnszdeisk?rdyA in well n. We also needthe retarded Green function, Gretm,nst, t0, kd ­ 2iQst 2t0d khamst, kd, ayn st0, kdjl, where hA, Bj denotes the anti-commutator. In the stationary state the Green functionsdepend only on the time difference t ­ t 2 t0, and wedefine the Fourier transformation via [21]Gm,nsE , kd ­Zdt eifE 2eFdsn1mdy2g sty h¯d3 Gm,nst, t 2 t, kd . (12)Without scattering between the k states and at T1 ­ 0the Green functions are diagonal in the well index:Gretm,nsE , kd ­ dm,ngretn sE , kd with the free particle Green370VOLUME 80, NUMBER 2 P HY S I CA L REV I EW LE T T ER S 12 JANUARY 1998function gretn sE , kd ­ 1ysE 2 Ek 1 i01d. The fullGreen function is then determined by the Dyson equation,Gretm,nsE , kd ­ gretmˆE 1 eFd m 2 n2, k!3"dm,n 1XlSretm,lˆE 1 eFd l 2 n2, k!3 Gretl,nˆE 1 eFd l 2 m2, k!#.(13)Within the self-consistent Born approximation for thescattering the self-energy can be written asSretm,nsE , kd ­ dm,nS˜retn sE , kd 1 T1dm11,n 1 T1dm21,n ,(14)with S˜retn sE , kd ­ NdyAPk0 V2Gretn,nsE , k0d. If the scat-tering term S˜retn is neglected, the solution corresponds tothe Wannier-Stark states (5). On the other hand, neglect-ing the coupling T1 gives the spectral functions used in thesequential tunneling model. Equations (13) and (14) aresolved self-consistently for Gret. Then G, is calculatedvia the Keldysh equation [21],G,m,nsE , kd ­Xm1Gretm,m1µE 1 eFd m1 2 n2, k¶3 S˜,m1•E 1 eFdµm1 2m 1 n2¶, k‚3 Gadvm1,nµE 1 eFd m1 2 m2, k¶. (15)According to our general assumption about inscattering wereplace S˜,msE , kd by its equilibrium value 22inFsE d 3ImhS˜retm sE , kdj. Finally, the current and electron densityare calculated via Eqs. (10) and (11). The extension ofthis model to more realistic scattering processes is straight-forward by using the respective self-energies in Eqs. (14)and (15) and relaxing the assumption about inscattering,although the calculations become very tedious (see, e.g.,Ref. [22] where NGF has been applied to the resonant tun-neling diode).In Fig. 2 we display the evolution of the current-fieldrelations for the different models from weakly to stronglycoupled superlattices. The curves for MBC, ST, and NGFare qualitatively similar for all couplings. For low electricfields the current increases linearly with the electric field.Then there is a peak at intermediate fields, and negativedifferential conductivity occurs at higher field. For smallT1 as well as for high fields the result from ST is inquantitative agreement with the NGF result, while theresults deviate for larger T1. In contrast, the result fromMBC is in quantitative agreement with the NGF resultfor large T1 and small eFd. The WSH result divergesfor eFd ! 0 [17], but approaches the NGF result forFIG. 2. Current-field relations calculated from nonequilibriumGreen functions (NGF) in comparison with the standardapproaches for N2D ­ 0.2G0r0 and kBTe ­ 0.2G0. For T1 ­1.5G0 the current-field relation is also shown over a wider fieldrange in the inset. Here, one can see explicitly that the NGFresult leaves the MBC curve for eFd * T1 and approaches theST and WSH curves for large fields.large fields. These results as well as further calculationsfor kBTe , G, N2D , r0G are summarized by Fig. 1,depicting the respective regions in parameter space, wherethe different approaches approximate the NGF result.Now we want to justify these ranges of validity bystudying the quantum mechanical correlation between thewells n and m given by the retarded Green function. Fora constant scattering self-energy S˜retn sE , kd ­ 2iGy2, ofEqs. (13) and (14), we have found an analytic solutionGretm,nsE , kd ­XaJm2as2T1eFd dJn2as2T1eFd dE 2 eFds m1n2 2 ad 2 Ek 1 i G2,(16)which is a superposition of broadened Wannier-Starkstates. The Wannier-Stark ladder becomes resolved ifeFd À G. This defines the region of validity for theWSH approach, as indicated by the right region in Fig. 1.By Fourier transforming we obtainGretm,nst, t 2 t, kd ­ 2 iQstdin2meihfsm1ndy2geFd2Ek j sty h¯d3 e2Gty2 h¯Jm2n•4T1eFdsinµeFd2h¯t¶‚.(17)Here the terms Gretn61,n become of the order of Gretn,n whenjJ0j ø jJ61j, i.e., j4T1yseFdd sinseFdty2dj øp2. Thiscan be used as an estimate for the boundary between lo-calization and delocalization. Because of the exponen-tial factor in Eq. (17) only t , 2h¯yG is of relevance.If eFd . G the magnitude of the sine takes the aver-age value ø1yp2. Then we find 2jT1j ø eFd. If, onthe other hand, eFd , G we may replace sinsxd ø x andhave 2jT1j ø G at the time t ­p2 h¯yG. From these es-timates we conclude that the states are essentially delo-calized if 2jT1j À G and 2jT1j À eFd. In this case the371VOLUME 80, NUMBER 2 P HY S I CA L REV I EW LE T T ER S 12 JANUARY 1998FIG. 3. Current-field relations for (a) high electron density(N2D ­ 2G0r0 and kBTe ­ 0.2G0) and (b) high electron tem-perature (kBTe ­ 3G0 and N2D ­ 0.2G0r0) for the differentapproaches. Note that the MBC result deviates from the NGFresult at eFd * N2Dyr0 in (a).miniband states form a useful basis as indicated in the up-per left part of Fig. 1. On the other hand, for 2jT1j ¿ Gor 2jT1j ¿ eFd the correlation functions Gretm,n vanish form Þ n and the states are essentially localized so that thesequential limit can be used, as indicated in the lower partof Fig. 1.For larger electron densities the agreement between thedifferent approaches becomes better as shown in Fig. 3(a).These results together with further calculations indicatethat ST is also valid if N2Dyr0 * 2jT1j, and MBC isalso valid if N2Dyr0 * G and N2Dyr0 * eFd. A similartrend is found for higher electron temperatures [Fig. 3(b)].This agrees with the analytic findings of Ref. [13] whereit is shown that NGF gives the same result as MBC in thelimit kBTe À jT1j, eFd.In conclusion, we have explicitly shown that a transportcalculation based on nonequilibrium Green functions con-tains the simple approaches MBC, WSH, and ST as limit-ing cases. For low temperature and low electron densitythe ranges of validity of the simplified approaches aredepicted in Fig. 1, while for higher electron densities ortemperatures, these ranges are enlarged. The essentialmessage of our analysis is that for wide regions in parame-ter space (but not everywhere) a simplified theory can befound, which approximates the full theory satisfactorily.This should have important consequences for practical de-vice modeling, where other complications, such as realisticscattering mechanisms, must be considered as well.A.W. acknowledges financial support by the DeutscheForschungsgemeinschaft.*Present address: Institut für Theoretische Physik, Tech-nische Universität Berlin, Hardenbergstrasse 36, 10623Berlin, Germany.[1] L. Esaki and R. Tsu, IBM J. Res. Dev. 14, 61 (1970).[2] A. Sibille, J. F. Palmier, H. Wang, and F. Mollot, Phys.Rev. Lett. 64, 52 (1990).[3] H. T. Grahn, R. J. Haug, W. Müller, and K. Ploog, Phys.Rev. Lett. 67, 1618 (1991).[4] C. Waschke et al., Phys. Rev. Lett. 70, 3319 (1993).[5] M. Holthaus, Phys. Rev. Lett. 69, 351 (1992).[6] B. J. Keay et al., Phys. Rev. Lett. 75, 4102 (1995).[7] P. A. Lebwohl and R. Tsu, J. Appl. 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Pfeiffer, and K.W.West, Phys. Rev. B 52, 14 825 (1995).[21] H. Haug and A.-P. Jauho, Quantum Kinetics in Transportand Optics of Semiconductors (Springer, Berlin, 1996).[22] R. Lake, G. Klimeck, R. C. Bowen, and D. Jovanovic,J. Appl. Phys. 81, 7845 (1997).372

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Quantum transport: The link between standard approaches in superlattices

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