Microscopic Theory Transconductivitydownloads.hindawi.com/archive/1998/035797.pdftemperatures, the...

VLSIDESIGN1998, Vol. 6, Nos. (1-4), pp. 87-90Reprints available directly from the publisherPhotocopying permitted by license only

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Microscopic Theory of TransconductivityA. P. JAUHO, M. B. BONSAGER*, K. FLENSBERG*, B. Y.-K. HU and J. KINARET

MIC, Technical University ofDenmark, Bldg 345east. DK-2800 Lyngby, DenmarkNordita, Blegdamsvej 17, DK-21O0 Copenhagen , Denmark

Measurements of momentum transfer between two closely spaced mesoscopic electronic sys-tems, which couple via Coulomb interaction but where tunneling is inhibited, have proven tobe a fruitful method of extracting information about interactions in mesoscopic systems. Wereport a fully microscopic theory for transconductivity o12, or, equivalently, momentumtransfer rate between the system constituents. Our main formal result expresses the transcon-ductivity in terms of two fluctuation diagrams, which are topologically related, but not equiv-alent to, the Azlamazov-Larkin and Maki-Thompson diagrams known for superconductivity.In the present paper the magnetic field dependence of o12 is discussed, and we find thato2(B) is strongly enhanced over its zero field value, and it displays strong features, whichcan be understood in terms of a competition between density-of-states and screening effects.

Consider two systems containing mobile charge carri-ers so close to each other that the charges in the tworespective subsystems feel the Coulomb forces origi-nating from the other subsystem, and yet far enoughaway from each other that direct charge transferbetween the two subsystems is not possible. Experi-mental realizations of such systems are. for example.Coulomb coupled double quantum well systems, 1,2]arrangements where a 3D system is close to a 2D sys-tem, [3] or two nearby quantum wires. A scatteringevent between a carrier in one system and a carrier inthe other system leads to momentum transfer betweenthe two subsystems. Thus, if a current is driventhrough one of the systems (henceforth the drivensystem is denoted as layer 1), then an induced currentis dragged in the other subsystem (layer 2). Alterna-tively, if no current is allowed to flow in layer 2, avoltage is induced. Due to momentum conservation

the two particle number currents flow in the samedirection. Since the mechanism for the Coulomb dragis carrier-carrier scattering the drag current is propor-tional to the square of the effective interactionbetween the subsystems. The available phase spacefor electron-electron scattering tends to zero at lowtemperatures, and consequently one expects Coulombdrag to decrease with decreasing temperature. At lowtemperatures, the two Pauli factors entering the car-rier-carrier scattering rate lead to a T:Z-dependence,and this behavior is approximately seen in experi-ments. [1] Note, however, that there are small, butimportant deviations from the simple T2-1aw; thesedeviations have been the topic of much recent inter-est. 1,4,5]

The simplest possible theoretical description forCoulomb drag is based on the Boltzmann equation.

* Present address: Department of Physics, Indiana University, Bloomington, Indiana 47405-4202t Present address: Danish Institute of Fundamental Metrology, DK-2800 Lyngby, Denmark.+_ Present address: Chalmers University of Technology, G6teborg, S-41296 Sweden

87

88 A.P. JAUHO et al.

Within this formulation the following expression canbe derived for the drag rate [6]:

T,D 2rt2nlmkr dqdo3q3 [et(q

[imxo (q, t.o)]2 sinh2 (ho/2k-T). (1)

Here (q,t.0) (q)/(q,o) is e dynamicallyscreened interaction between ciers in e two lay-ers, and xo(q,o) is e be A polization func-tion. e above expression is valid only in e weak-scattering limit, d a generalization is needed if onewishes to examine, say, weak-localization coectionsor effects of external magnetic field. Only a fullymicroscopic eory can address ese questions, dwe have recently cied out such an alysis [7.8].e calculations e based on e Kubo foula,which expresses e transconductance (and hence thedrag rate) in tes of a retded cuent-cuent co,e-lation function,

ie2P(x-x;a) n (x-;a)ie

+x- 8uSO(x). (2where (throughout we use h 1)

HP’r(x-- ;t t’) --iO(t t’) ( [ff (x,t), J(,t’)] )(3)

Here (ij} indicate the subsystem, {} in e super-scripts label e Ctesian coordinates, is e exter-nal frequency, Pi(x) is e pticle density insubsystem i, d j(x,t) is e pticle cuent operator.e analysis proceeds via a systematic expsion ine interlayer interaction, and one finds in second

order [7,8]

I(,)

xA(q,;+i,-i)A(-,-;-- i,-+ i),()

where (q,) is e dynamically screened interactionand is a three-body co.elation function [7],

(q,i

(Q + ,,+

FIGURE Diagram determining transconductivity in terms of tri-angle functions

/k(q,q;iton,iOJn) fo dr’fo deeit"Xei’x’i(Txj(q 0, : 0)p(q, "l:)p(-q, xt) ). (5)

The analytic continuation from the Matsubara fre-quencies in to real frequencies is discussed in detailin Ref. [7]. Figure illustrates the diagrammaticstructure of the transconductivity. The shaded trian-gles (which correspond to three-body correlationfunctions) can be evaluated at different levels ofapproximation, and we have shown [7] that previ-ously known results, obtained either by Boltzmannequations approach or memory functions [9] can berecovered. Very recently, a similar analysis has beenpresented by Kamenev and Oreg. 10]

In the lowest order approximation the bare trianglereduces to a product of three Green functions, and itthus appears to be in disagreement with the Boltzmanresult (1) which only involves RPA-bubbles, whichconsist of two Green functions. It is, however, possi-ble to recover the connection with the following cave-ats: 1) For B 0 one must have momentum-independent transport relaxation times [7], and 2)B, 0 one must work in the high-field weak-scatteringlimit. The upshot of the analysis is that one can, if

OcX >> 1, still use an expression which formally coin-cides with the Boltzmannian result (1). but with thefollowing important modifications: 1) The polariza-tion functions o must be replaced with full X’s,including the magnetic field and impurity effects,Thus one must solve the vertex equation for in asuitable conserving approximation (e.g. including lad-der diagrams); and 2) The screened interaction must be

MICROSCOPIC THEORY OF TRANSCONDUCTIVITY 89

calculated at the same level of approximation, i.e.including magnetic field and impurity effects.

A full description of our calculations is given else-where [8], and here we will give a qualitative descrip-tion of the main physical results. First, the drag rate issignificantly enhanced: our preliminary numericalresults indicate that the enhancement factor is 60 70.11 Experimentally this is very important, since gen-

erally drag effects are small, and any enhancement ismost welcome. Secondly, the drag rate displaysstrong structure whenever the Fermi level crosses aLandau level. Finally, the temperature dependence isquite dramatic; it is qualitatively different at differentvalues of the magnetic field.

A physical understanding of these findings can beobtained with the help of Fig. 2. First, the largedegeneracy of Landau levels enhances the density-of-states at Fermi level enormously, and since the dragrate is proportional to Im2x/sinh2(ho/2kT) =-S(q,o)S(-q,t), where S is the structure factor reflect-ing the available phase-space, an enhancement of thedensity-of-states necessarily leads to an increaseddrag rate.

(b)

(c)

FIGURE 2. Transport coefficients in the Integer Quantum Hallregime (schematic). Legend: (a) Solid line: Hall conductivity inunits of e2/h, Dashed line: Longitudinal resistance (arb. units). (b)Density of states with localized states in hatched regions. (c) Pre-dicted drag resistance

It is not surprising .that the drag rate should showpeaks as the the Fermi level moves through Landaulevels as a function of magnetic field 12]. The situa-tion is quite analogous with the "normal" longitudinalresistance observed in the integer quantum Hallregime (sketched in panel (a) of Figure 2): dissipativetransport takes place only when the Fermi level lieswithin the extended states (unshaded areas in Fig.2(b)). Thus, there is drag only when states at theFermi level can undergo scatterings where momen-tum is exchanged between the layers. The structurewithin the predicted peaks in the drag rate has a moresubtle explanation. In addition to the available phase-space the drag is also strongly influenced by thescreened interaction. This, in turn, depends on thenumber of states participating in the screening proc-ess. Thus, when the Fermi level approaches the centerof a Landau level, when the number of statesapproaches a maximum, the screening becomes moreeffective, and the interlayer coupling becomesweaker. Based on this picture one expects that thedrag peaks may show internal structure, and thisbehavior is skteched in Fig. 2(c). Our numericalresults [8] fully corroborate the above discussion. Thepossibility of direct observation of carrier-carrierinteraction related effects in a transport quantity suchas the drag rate is quite unusual, and should be of sig-nificant interest to experimentalists.

References[1] T. J. Gramila, J. P. Eisenstein, A. H MacDonald, L. N.

Pfeiffer, and K. W. West, Phys. Rev. Lett. 66, 1216 1991)Phys. Rev. B 47, 12957 (1993) Physica B 197,442 (1994).

[2] U. Sivan, P. M. Solomon, and H. Shtri"kman, Phys. Rev. Lett.68, 1196 (1992).

[3] P. M. Solomon, P. J. Price, D. J. Frank, and D. C. L. Tulipe,Phys. Rev. Lett. 63, 2508 (1989).

[4] H. C. Tso and P. Vasilopoulos and F. M. Peeters, Phys. Rev.Lett. 68, 2516 (1992); ibid. 70, 2146 (1993); H. C. Tso and P.Vasilopoulos, Phys. Rev. B 45, 1333 (1992).

[5] K. Flensberg and B. Y.-K. Hu, Phys. Rev. Lett. 73, 3572(1994).

[6] A.-P. Jauho and H. Smith, Phys. Rev. B 47, 4420 (1993).[7] K. Flensberg, B. Y.-K. Hu, A.-P. Jauho, and J. M. Kinaret,

Phys. Rev. B 52, 14761 (1995)[8] M.C. BCnsager, B. Y. -K. Hu, K. Flensberg, and A.-P. Jauho,

(unpublished)[9] L. Zheng and A. H. MacDonald, Phys. Rev. B 48, 8203

(1993).[10] A. Kamenev and Y. Oreg, Phys. Rev. B 52 7516 (1995)

90 A.P. JAUHO et al.

[11] The enhancement factor is consistent with recent experi-ments at Cambridge, U.K. We are grateful to Dr. N. Hill forcommunicating us his unpublished data.

[12] Here we restrict ourselves to the case when the densities ofthe two gases are same: under these conditions the drageffects are most pronounced.

BiographiesA. P. Jauho M. Sc. 1975 and Lic. Techn. 1979 (Hel-sinki University of Technology), Ph. D. 1982 (CornellUniversity); post-doc Nordita 1981-1984, ChalmersUniversity 1984-1985, faculty at Copenhagen Univer-sity 1985-1992, faculty at Nordita 1992-1993, Leaderof Theory Group at Mikroelektronik Centret, Techni-cal University of Denmark 1992 present

M, Bcnsager M. Sc. 1995 (Technical University ofDenmark), Graduate Student at Indiana University,Bloomington 1995 present

B. .-K. Hu B.A. 1984 (Swarthmore College),M.Sc. 1987 (Comell University), Ph.D. 1990 (ComellUniversity), post-doc U. of Maryland 1990-1993.research scientist MIC, DTU 1993 present

K. Flensberg M. Sc. 1987 (DTU), Ph.D. 1989(DTU). post-doc. U. of Tennessee 1989-91. Nordita1991-93, Carlsberg Research Fellow 1993-95. StaffScientist, Danish Institute for Fundamental Metrol-ogy, 1995 present

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