Effect of parallel magnetic field on the Zero Differential Resistance State
N. Romero, S. Mchugh, M. P. Sarachik, S. A. Vitkalov, A. A. Bykov
aa r X i v : . [ c ond - m a t . o t h e r] J u l Effect of parallel magnetic field on the Zero Differential Resistance State
N. Romero, S. McHugh, M. P. Sarachik and S. A. Vitkalov
Physics Department, City College of the City University of New York, New York, New York 10031
A. A. Bykov
Institute of Semiconductor Physics, 630090 Novosibirsk, Russia (Dated: November 9, 2018)The non-linear zero-differential resistance state (ZDRS) that occurs for highly mobile two-dimensional electron systems in response to a dc bias in the presence of a strong magnetic fieldapplied perpendicular to the electron plane is suppressed and disappears gradually as the magneticfield is tilted away from the perpendicular at fixed filling factor ν . Good agreement is found with amodel that considers the effect of the Zeeman splitting of Landau levels enhanced by the in-planecomponent of the magnetic field. The nonlinear properties of highly mobile electronsin two-dimensional AlGaAs/GaAs heterojunctions havebeen the focus of a great deal of recent attention. Strongoscillations of the longitudinal resistance induced by mi-crowave radiation have been found at magnetic fieldssatisfying the condition w = nw c , where w is the mi-crowave frequency and w c is the cyclotron frequency( n = 1 , , ... ). At high levels of microwave excitationthe minima of the oscillations can reach a value close tozero. This so-called zero resistance state (ZRS) hasstimulated extensive theoretical attention.
Interesting nonlinear phenomena have also been foundin response to a dc electric field.
Oscillations ofthe longitudinal resistance, periodic as a function of theinverse magnetic field, have been observed at relativelyhigh dc bias satisfying the condition n ¯ hw c = 2 R c E H ;here R c is the cyclotron radius of electrons at the Fermilevel and E H is the Hall electric field induced by thedc bias in the magnetic field. This effect has been at-tributed to horizontal Landau-Zener tunneling betweenLandau levels, tilted by the Hall electric field E H . An-other notable nonlinear effect is a strong reduction of thelongitudinal resistance by considerably smaller dc electricfields.
This effect has been attributed to spec-tral diffusion of electrons in a dc electric field. Electronspectral diffusion occurs in the presence of a strong mag-netic field where the density of states (DOS) oscillatesdue to Landau quantization. The oscillations in the DOSresult in an oscillatory structure of the non-equilibriumelectron distribution function. When a dc electric field E dc is applied, electrons diffuse from low energy regions(occupied levels) to high energy regions (empty levels)through elastic scattering between electrons and impuri-ties. Inelastic scattering limits this process, forcing theelectron distribution function back to thermal equilib-rium. This effect also accounts for a nonlinear electronstate with zero differential resistance (ZDRS) which hasbeen recently identified. The ZDRS exhibits strongdependences on both temperature and magnetic field through the strong dependence of the electron spectraldiffusion on these parameters. In this paper westudy the effect of an in-plane magnetic field on the ZDRSand the nonlinearity of the 2D electron system induced by a dc bias. This research was also motivated by ap-parent differences between reported measurements of theZRS induced by microwave radiation in response to anin-plane magnetic field.
The sample used in this experiment was cleaved froma wafer of a high-mobility GaAs quantum well grown bymolecular beam epitaxy on a semi-insulating (001) GaAssubstrate. The quantum well was 13 nm wide, the elec-tron density n= 9 . × m − , and the mobility µ =85m /Vs at T= 1 . . µ m-wide Hallbars with a distance of 250 µ m between potential con-tacts. The differential longitudinal resistance was mea-sured at a frequency of 77 Hz in the linear regime. Directelectric current (dc bias) was applied simultaneously withan ac excitation through the same current leads (see insetto Fig. 1(a)).Figure 1(a) shows quantum oscillations of the resis-tance at T= 1 . φ =90 o ). The arrowdenotes the Shubnikov-deHaas maximum at B ⊥ = 0 . R xx is plotted as a function ofthe perpendicular component B ⊥ for magnetic field ap-plied at different angles with respect to the plane. Whileall curves display a maximum at B ⊥ = 0 .
772 T, as ex-pected, the magnitude of the resistance peaks at 0 .
772 Tdecreases as the angle φ decreases from 90 o and the totalmagnetic field increases. For the measurements reportedbelow, we rotated the sample and simultaneously variedthe magnitude of the total magnetic field in order to fixthe perpendicular magnetic field component at 0 .
772 Twhile changing the in-plane magnetic field. The fillingfactor ν is thus fixed for all curves in Fig. 1(b), while theZeeman splitting ∆ Z = gµ B B is different for differentcurves due to its dependence on the total magnetic field B .Figure 2 shows the differential resistance r xx = dV xx /dI as a function of dc bias at T = 1 . φ and fixed perpendicular magnetic field B ⊥ = 0 .
772 T corresponding to the Shubnikov de Haas(SdH) oscillation maximum indicated by the arrow inFig. 1(a). Note that the total magnetic field (denoted R xx () B (T) (a)
B , , , , R xx () Perpendicular Magnetic Field (T) (b)
FIG. 1: (a) Quantum oscillations of the resistance at T= 1 .
7K for magnetic field applied perpendicular to the electron( φ =90 o ); the arrow denotes the field of the Shubnikov-deHaas maximum for which subsequent data were obtained (seetext); the inset is a schematic of the experimental set-up;(b) Resistance R xx plotted as a function of the perpendicularcomponent B ⊥ of the magnetic field for magnetic field appliedat various angles φ with respect to the electron plane. Thelegend lists the angle φ and the total magnetic field at themaxima (marked by the dashed line). Data were taken atT= 1 . -40 -20 0 20 40-4004080120 r xx = d V xx / d I () Idc ( A ) FIG. 2: Differential resistance versus dc bias for differentangles φ between the magnetic field and the 2D electronplane, where the perpendicular magnetic field B ⊥ = 0 . . on the right-hand side of Fig. 2) and its in-plane com-ponent both increase as the angle φ decreases. The dif-ferential resistance r xx initially decreases with increas-ing bias I dc for all angles. For a perpendicular magneticfield ( φ = 90 o ) r xx exhibits a reproducible negative spikeat I dc = 7 . µ A and then stabilizes near zero. This isthe zero differential resistance state. As the angle φ be-tween the magnetic field and the plane is decreased, thespike gradually disappears and is no longer observable at φ = 21 o . For smaller angles the differential resistance r xx is increasingly positive as the angle φ decreases, anda shallow minimum develops at large bias.It is interesting to compare our results for the zero dif-ferential resistance state (ZDRS) with those reported inRef. [21] and Ref. [22] for the effect of in-plane field onthe zero resistance (ZRS) state. Both experiments wereperformed in magnetic fields smaller than those used inour experiments. Mani tilted the sample at an angle θ with respect to the magnet axis and microwave prop-agation direction. The ZRS was observable with the os-cillatory pattern unchanged at a tilt angle of θ = 80 o ( φ = 90 o − θ = 10 o ), and vanished only at θ ≈ o .The disappearance of the ZRS at θ ≈ o was attributedto the vanishing of the photon flux through the two di-mensional electron system rather than to the in-planemagnetic field. Yang et al. employed a two axis systemto provide perpendicular and parallel field components.They report the gradual reduction of the microwave in-duced ZRS and its disappearance when a parallel mag-netic field ( B || ≈ . : we find thatthe ZDRS decreases and disappears gradually with in-creasing in-plane magnetic field component, while Mani reported quenching of the ZRS only at θ ≈ o . It is pos-sible, however, that stronger magnetic fields, comparableto those applied in our experiments, are required to re-duce the dc induced nonlinearity for θ < o .We suggest that the suppression of the nonlinear re-sponse of the system and the disappearance of the zerodifferential resistance at small angles φ are due to thechange of the bias-stimulated spectral diffusion of theelectrons caused by the increase of the in-planemagnetic field component. We consider Zeeman split-ting of the Landau levels as the main mechanism leadingto a decrease of the variations of the spectral diffusionwith energy and, thus, to the reduction of the nonlin-earity. Below we compare numerical simulations of thespectral diffusion with experiment. Good agreement isfound.To estimate quantitatively the effect of Zeeman split-ting on the spectral diffusion we begin by analyzing thechange in the electron spectrum induced by the Zee-man effect. As the angle of the applied magnetic fieldis tilted away from the perpendicular and the total mag-netic field is increased, the oscillations of the density ofstates (DOS), ν ( ǫ ), split into spin up and spin down com-ponents, as seen in Eq. 1. This leads to a reduction inthe modulation of the DOS amplitude. In order to cal-culate the DOS we use a gaussian approximation givenby ˜ ν ( ǫ ) = ν ( ǫ ) ν = √ ω c τ q (cid:18) exp (cid:18) − ( ǫ/ ¯ h + ∆ z / ¯ h − nω c ) ω c /πτ q (cid:19) + exp (cid:18) − ( ǫ/ ¯ h − ∆ z / ¯ h − nω c ) ω c /πτ q (cid:19) (cid:19) , (1)where ˜ ν ( ǫ ) is the dimensionless DOS normalized by thevalue of the DOS at zero magnetic field, n is an integer, τ q is the quantum or single particle relaxation time and ω c is the cyclotron frequency. The parameter ¯ h/τ q de-termines the width of the Landau levels and is obtainedfrom comparison with experiment (shown below). A sim-ilar value of ¯ h/τ q is also found by comparison of the ex-periment with the self consistent Born approximation ofthe density of states. The inset to Fig. 3 shows the results of numericalsimulation of the effect of Zeeman splitting on the den-sity of electron states (DOS) in our sample. It canbe seen that the modulation of the DOS is weaker forsmaller angles ( φ = 8 o ) corresponding to stronger Zee-man spin splitting. Figure 3 presents the angular de-pendence of the maximum value of the differential resis-tance r xx = dV xx /dI at B ⊥ = 0 .
772 T obtained fromthe curves presented in Fig. 1(b). At fixed filling factorthe resistance decreases with decreasing angle φ (with aconsequent increase of the total magnetic field applied).Based on the evolution of the density of states displayedin the inset, the theoretically expected values of resis-tance are denoted by the circles of Fig. 3 for comparison.The resistance was estimated using a simplified expres-sion for the longitudinal conductivity in strong magneticfields ( ω c τ p ≫ σ xx = A × Z σ ( ǫ )( − df /dǫ ) dǫ, (2)where σ ( ǫ ) = σ D ˜ ν ( ǫ ) , σ D = e ν v F / ω c τ tr is the Drudeconductivity in a perpendicular magnetic field B ⊥ . Thefree parameter A accounts for possible memory effects and other deviations from Drude behavior in the presenceof strong magnetic fields. The parameters A and τ q (the quantum scattering time), were chosen to provide agood fit between experiment and theory for the angulardependence of the resistance at B ⊥ = 0 .
772 T. Fromthe comparison above, we obtain the electron g -factor, g = − . We find good agreement betweenexperiment and theory (see Fig. 3). Thus, we are able toattribute the decrease of the SdH maxima with increasingin-plane magnetic field component to the Zeeman effect.In order to estimate how the electron spectral diffu-sion and the nonlinearity of the 2D electron system inthe presence of a magnetic field is affected by the Zee-man effect, we solve numerically the spectral diffusion -1 002
Experiment
Simulation r M AX () Angle (deg)
B=5.4 T N o r m a li ze d D O S F , meV B=0.772 T
FIG. 3: The differential resistance r MAX of the quantum os-cillation maximum at B ⊥ = 0 .
772 T plotted as a function ofthe angle φ between the total magnetic field and the electronplane. The squares are the experimental results and the cir-cles represent the numerical simulation. The inset shows thedensity of electron states (normalized to its value at zero mag-netic field) in a fixed perpendicular magnetic field B ⊥ = 0 . equation for the electron distribution function f ( ǫ ): − ∂f ( ǫ ) ∂t + E dc σ Ddc ν ˜ ν ( ǫ ) ∂ ǫ (cid:2) ˜ ν ( ǫ ) ∂ ǫ f ( ǫ ) (cid:3) = f ( ǫ ) − f T ( ǫ ) τ in , (3)where f T ( ǫ ) is the Fermi distribution and E dc is the bias-induced electric field. For the normalized DOS, ˜ ν ( ǫ ), weuse the DOS obtained above from a comparison with thelinear response (see Eq. 1, Eq. 2 and Fig. 3). Spectraldiffusion is a result of elastic scattering between electronsand impurities in the presence of a bias-induced electricfield E dc ; it is limited by inelastic processes, which forcethe distribution function back to thermal equilibrium.We use the inelastic relaxation time τ in as a fitting pa-rameter. The solution of the diffusion equation f ( ǫ ) at t ≫ τ in is then inserted into Eq. 2 in order to obtain theresistivity at different dc biases.Figure 4 shows experimental and numerical results forthe longitudinal resistance R xx = V xx /I as a function ofthe dc bias plotted for two different angles, φ = 90 o and φ = 8 o . The vertical scale is fixed by the comparison withthe linear response (by the choice of the two parameters A an τ q in Eq. 1 and Eq. 2). The horizontal scale ischosen to provide the best fit between theory and exper-iment. The best result is obtained for the inelastic time τ in = 2 . × − s at φ = 90 o and for τ in = 2 . × − sat φ = 8 o . There is good agreement between theory andexperiment at small dc bias. At higher dc bias devia-tions become evident that are larger for smaller angles φ .We suggest that these deviations are due to additionalnonlinear mechanisms that occur at higher dc bias which have not been treated in this paper. , B=5.4 T R xx () I DC ( A ) , B=0.772 T FIG. 4: The solid curves show the measured resistance R xx = V xx /I dc as a function of dc bias in fixed perpendicularmagnetic field for two different total magnetic fields (angles φ ), as labeled; T= 1 . τ q = 5 . τ in = 2 . φ = 90 o and τ in = 2 . φ = 8 o . In conclusion, the effect of a dc electric field on the lon-gitudinal resistance of a highly mobile two-dimensional(2D) electron system in GaAs quantum wells was stud-ied. We observe a zero-differential resistance state in re-sponse to a direct current when the magnetic field is per-pendicular to the electron plane. 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