Measurement of Coulomb drag between Anderson insulators
aa r X i v : . [ c ond - m a t . m e s - h a ll ] F e b Measurement of Coulomb drag between Anderson insulators
K. Elsayad, J. P. Carini, ∗ and D. V. Baxter Department of Physics, Indiana University, Bloomington, Indiana 47405 (Dated: February 26, 2008)We report observations of the Coulomb drag effect between two effectively 2-d insulating a-Si − x Nb x films. We find that there only exist a limited range of experimental parameters overwhich we can measure a sizable linear-response transresistivity ( ρ d ). The temperature dependenceof ρ d is consistent with the layers being Efros-Shklovskii Anderson insulators provided that a 3-ddensity of states and a localization length smaller than that obtained from the DC layer-conductivityare assumed. Materials such as a-Si − x Nb x which exhibit a disorderdriven T = 0 (Quantum Critical) Metal-Insulator Tran-sition (MIT) [1, 2] have presented many challenges tocondensed matter physics: in particular understandingthe role of long ranged electron-electron interactions inthe insulating phases [3] and in the vicinity of the MIT[2]. Since the interplay between disorder and electron-electron interactions in such systems will determine thedominant transport mechanism, the development of ex-perimental techniques to separately measure these is use-ful. In this letter we show that the Coulomb drag effectallows us to directly study long ranged electron-electroninteractions in insulating a-Si − x Nb x thin films. We findthat although linear-response Coulomb drag is only ob-servable over a limited range of sample parameters, whenobtainable, it offers unequivocal distinction to be madebetween alternative models for the electronic transportin such systems.Coulomb drag [4] arises from the Coulomb scatteringof charge carriers in spatially separated layers, in the ab-sence of charge transfer between the layers. Experimen-tally, the Coulomb drag effect between two layers (layer-1 and 2) can be observed by measuring the electric-field( E ) created in one, open circuited, layer due to a parallelapplied current-density ( j ) in the other. The (longitudi-nal) transresistivity, or linear-response Coulomb drag co-efficient, is defined as ρ d = − E /j ; whilst the measuredtotal-transresistance is the ratio of the induced voltagein layer-2 to the applied current in layer-1, i.e. − V /I .Theoretical analyses agree that the linear-response tran-sresistivity between two identical 2-d layers is given by,e.g. [5]: ρ d ∼ ~ β e n Z dω Z dq (2 π ) q (cid:12)(cid:12)(cid:12)(cid:12) Im χ ( ω, q ) U ( ω, q )sinh( ~ ωβ/ (cid:12)(cid:12)(cid:12)(cid:12) (1)where β = ( k B T ) − , T is the temperature, n is the carrierdensity in a layer, χ ( ω, q ) is the density-density responsefunction of a layer, and U ( ω, q ) is the screened interactionpotential between the layers. Im χ ( ω, q ) may be obtainedfrom the finite wavevector conductivity via:Im χ ( ω, q ) = q ( e ω ) − σ ( ω, q ) . (2)The motivation for this study comes from predictions [6]that the temperature dependence of the transresistance can be used to differentiate insulating states. In partic-ular, for the case of (2-d) Mott Anderson insulator bi-layers the low- T transresistivity should vary as ρ d ∝ T ,whilst for (2-d) Efros-Shklovskii(ES) Anderson insulatorsit should diverge when T → ρ d ∝ T exp[( T /T ) / ],where T is the ES characteristic temperature given by k B T ≈ e /κξ and κ is the dielectric constant of thelayers. This opposite behavior of the transresistivity as T → T range than intra-layer trans-port measurements do—where one would obtain differentfunctional T or ω dependencies with the same trend.In our study, samples consisted of two U-shaped200˚A thick insulating a-Si − x Nb x layers, separated bya SiO barrier (see upper left inset in figure 2). All layerswere fabricated using standard RF magnetron sputteringtechniques, with the a-Si − x Nb x layers being depositedusing the co-sputtering technique with a rotating sample-holder outlined in Ref[3]. Samples were all grown onpolished glass slides in an inert (argon) environment atambient temperature and a pressure of ≈ . ≈ ≈
5% of those obtained from optical inter-ferometry measurements on samples of similar composi-tions and thicknesses. Estimates of the uncertainty inthe thickness due to fluctuations in the deposition rateand uncertainties in the exposure time are < ± < ± − x Nb x and the barrierlayers respectively. The barrier layer was deposited indiscrete stages to reduce the formation of pinholes. Highsputtering powers and brief atmospheric exposure be- FIG. 1: Plot of the low- T transresistance per square( R drag /square) for various samples, at a driving-current of I = 1nA. (The right-vertical axis shows the ratio of the totalmeasured voltage to the total driving current). Samples 1A,1B and 1C have an average Nb concentration of x = 0 . x ≈ .
076 and ≈ . tween stages were found to increase the barrier strengthand durability. We were thereby able to fabricate barrierlayers with resistances several orders of magnitude largerthan those of the layers.Samples were cooled to T ≈ .
2K using a standardliquid helium cryostat. The low- T ( ≤ ∼ ρ d remains the same on interchanging the layers (even ifthe layers have different resistivities), the condition that( R drag /square) → = ( R drag /square) → as the driving-current is decreased, was used as a test for the linear-response regime. The transresistance at driving currentsof I = 1nA for samples with average Nb concentra-tions per layer [7] of x = 0 . → .
08 and layer sepa-rations of 50 → T ∼ → FIG. 2: Resistance per square of layers 1 (triangles) and 2(circles) in sample 1B. Open symbols are plotted against the T − / axis and closed symbols against the T − / axis. Upperinset: sketch of sample geometry. Lower inset: resistancesof layers-1 and -2 (triangles and circles) compared to lowerbounds of the barrier resistance (squares). The dashed hori-zontal line represents the maximum resistance that could bemeasured using V induced ( I applied ) techniques with existing ap-paratus. At lower temperatures the transresistance would satu-rate or decrease in magnitude. This is shown in fig-ure 1 for a selection of samples. In what follows we willpresent data for sample 1B, where the transresistanceentered the linear-response regime for I ≤ σ dc ) ∝ T − / , or the MottVRH model [9], which, for effectively 2-d films, predicts:ln( σ dc ) ∝ T − / , at low temperatures. This can be seenin figure 2, where we present T − / and T − / plots ofthe layer-resistances of sample 1B. In the temperaturerange 4K < T < T = 145( ± ≤ T = 1890( ± T = 36 , ± T -dependencereasonably well. The localization length determined fromthe ES characteristic temperature is ξ ≈ ± ≤ r c ) being larger than the widthof each layer ( W ). At these temperatures the resistanceof the barrier—obtained by measuring the tunneling cur-rent (see above)—is found to be approximately two or-ders of magnitude larger than the resistance of layers-1and -2 (see lower inset in figure 2).The observed decrease of the layer-resistance and tran-sresistance observed at T ≤ − x Nb x films with large Nb concentrations ( x ≈ . − . . < T <
15K in this letter.As the driving current is increased the transresistancedecreases (see inset in figure 3), and for I ≈ I = 1nA. This is likely due tolarger driving currents both increasing the effective sam-ple temperature and producing significant non-linear re-sponses. For I < ≈
5% between1, 0 .
75 and 0 . R drag /square)/( R layer /square) as a functionof temperature. Doing so we find that our data isbest described (see figure 4) by the 2-parameter equa-tion: ( R drag /square)= aT b ( R layer /square), with a =1 . ± . × − K − b and b = 2 . ± . b = 3 for a bilayer system comprised of 2-dES Anderson insulators.The observed discrepancy can be explained if thescreening in the layers is not 2-d in the studied regime—i.e. if the response is dominated by a finite wave-vector q > W − (where W ≈ ξ − > W − > r − = ξ − ( T /T ) − / , at the temperatures of interest, this oc-curs at momentum transfers ( q > r − ) that dominate thetransresistance (see Ref[6]).Repeating the calculations for the finite- ω, q conduc-tivity in Ref[14] for 3-d systems, we find it takes theasymptotic forms: σ ( ω, q ≪ r − ω ) ∼ C ( e / ~ )( ω/ω ) r − ω (3) FIG. 3: Temperature dependence of linear-response transre-sistance observed at driving currents I ≤ I ≥ T increase of the I < T -dependence of the ratio ( R drag /square)/ ( R layer-2 /square) at I = 1nA for sample 1B (circles), the pre-dicted slope for the case of two 2-d Mott-Anderson insulatorlayers (dotted line), the predicted slope for the case of two2-d Efros-Shklovskii Anderson insulator layers (dash-dottedline), and the predicted slope for two effectively 3-d Efros-Shklovskii Anderson insulator layers (solid line). As can beseen, the last prediction describes the data best. Inset: T -scaled transresistance per square for the same sample. σ ( ω, r − ω ≪ q ≪ ξ − ) ∼ C ( e / ~ )( ω/ω ) q − r − ω (4)where r ω = ξ ln( ω /ω ), ω = k B T / ~ , and C i , i = 1 , q > r − and we can assume weak static screening, would changewith T as: ρ d ∝ T exp[( T /T ) / ]. The observed ρ d ∝ T exp[( T /T ) / ] temperature dependence can howeverarise if one or both of the following are the case: (1) The relevant localization length is much smaller thanthat obtained from the T -dependence of the DC conduc-tivity so that: ξ ( T /T ) / < d. (5)In this way the r − < q momentum transfers (which areotherwise dominant) are suppressed, and the q < r − contribution determines the low- T transresistance. Sub-stituting equation (4) into (2) and (1), the transresis-tance would now change with temperature as: ρ d ∝ T exp[( T /T ) / ]. (2) Finite- ω transport is effectively 3-d due to pair-arms( r ω ) reaching into the barrier and the second layer. Fromthe strongly localized nature of electrons in the barrierlayer, the effective ξ would be much smaller and condition(5) may be satisfied, even if the effective layer separationalso decreases significantly.We note that since several of the relevant length scalesare comparable ( r c ∼ ξ ∼ W ∼ d ), a small modifica-tion of the effective values that these parameters take(due to e.g. finite-size effects, correlated hopping or sur-face effects) could cause a change between the ξ > d and ξ < d regime in the temperature range probed, resultingin a different T -dependence than that predicted. We alsonote that we do not observe the expected transition fromthe q < r − to the q > r − regime, as the temperatureis increased. However, if the apparent change in the T -dependence of the transresistance (see figure 4) at T ≈ q > r − regime kicks in, then we predictthat ξ ≈ (4K βκ/e ) d ∼ ρ d ∝ T exp[( T /T ) / ] for T > q < r − , as we have observed.In conclusion, we have observed the Coulomb drag ef-fect between two 200˚A thick insulating a-Si − x Nb x films(with 0 . ≤ x ≤ .
08) separated by a 50 − x = 0 . ± . ± − x Nb x layers is smaller than that inferred fromthe temperature dependence of the DC layer-resistances.Our study suggests that whilst theoretically the Coulombdrag effect is a useful technique for distinguishing theinsulating states of thin films, it is experimentally chal-lenging due to the complex dielectric properties of disor-dered thin films at low energies, which result in non-linearinter- and intra- layer excitations becoming dominant atpractical temperatures and sample dimensions. Experi-mental studies of the non-linear crosstalk regime betweenthin insulating films, along with simulations of the non-linear (current-dependent) transresistance in such sys-tems, may prove to be the most productive method ofstudying the detailed nature of the observed excitations.We would like to thank Prof. E. Shimshoni for fruitfuldiscussions, and D. Sprinkle and M. Hosek for help withthe experiments. ∗ [email protected][1] S.L. Sondhi, S.M. Girvin, J.P. Carini, and D. Shahar,Rev. Mod. Phys. , 315 (1997).[2] D.J. Bishop, E.G. Spencer, and R.C. Dynes, Solid StateElectron. , 73 (1985); H.-L. Lee, J.P. Carini, D.V. Bax-ter, W. Henderson, and G. Gr¨uner, Science , 633(2000).[3] E. Helgren, G. Gr¨uner, M.R. Ciofalo, D.V. Baxter, andJ.P. Carini, Phys. Rev. Lett. , 116602 (2001).[4] P.M. Solomon, P.J. Price, D.J. Frank, andD.C. La Tulipe, Phys. Rev. Lett. , 2508 (1989).[5] L. Zheng and A.H. MacDonald, Phys. Rev. B , 8203(1993).[6] E. Shimshoni, Phys. Rev. B , 13301 (1997).[7] Average niobium concentrations were estimated by com-parison of the deposition rate and characteristic ES andMott temperatures to that of a-Si − x Nb x films for which x was determined from Electron microprobe analysis.[8] A.L. Efros and B.I. Shklovskii, J. Phys. C: Solid StatePhys. , L49 (1975).[9] N.F. Mott, Phil. Mag. , 7 (1970).[10] H. Aubin, C.A. Marrache-Kikuchi, A. Pourret,K. Behnia, L. Berge, L. Dumoulin, and J. Lesueur,Phys. Rev. B , 094521 (2006).[11] T.J. Gramila, J.P. Eisenstein, A.H. MacDonald,L.N. Pfeiffer, and K.W. West, Phys. Rev. Lett. , 1216(1991).[12] T.J. Gramila, J.P. Eisenstein, A.H. Macdonald,L.N. Pfeiffer, and K.W. West, Phys. Rev. Lett. , 12957(1993).[13] K. Flensberg and B. Y-K. Hu, Phys. Rev. Lett. , 3572(1994).[14] I.L. Aleiner and B.I. Shklovskii, Intern. Journ. of Mod.Phys. B8