A double quantum well (QW) subject to in-plane magnetic fields B{sub {parallel}} has the dispersion curves of its two QWs shifted in k-space. When the QWs are strongly coupled, an anticrossing and partial energy gap occur, yielding a tunable multi-component Fermi surface. We report measurements of the resultant features in the conductance, capacitive density of states, and giant deviations in cyclotron effective masses

Eiles, T.M.

Harff, N.E.

Klem, J.F.

Lyo, S.K.

Simmons, J.A.

UNT Digital Library

r Tuning a Double Quantum Well Fermi Surface with In-Plane Magnetic Fields J. A. Simmons, N. E. Harff, T. M. Eiles, S. K. Lyo, and J. F. Klem Sandia National Laboratories Albuquerque, New Mexico 87 185- 14 15 USA Abstract RECEIVED JUN 1 9  1995 O S T I  A double quantum well (QW) subject to in-plane magnetic fields BII has the dispersion curves of its two QWs shifted in k-space. When the QWs are strongly coupled, an anticrossing and partial energy gap occur, yielding a tunable multi-component Fermi surface. We report measurements of the resultant features in the conductance, the capacitive density of states, and giant deviations in the cyclotron effective masses. DISCLAIMER This report was prepared as an account of work sponsored by an agency of the United States Government. 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Images are produced from the best available original document 2 Double quantum wells (DQWs) constitute a system in which an additional degree of freedom can be controllably introduced to a two dimensional electron gas by changing the thickness and height of the grown interwell barrier. In the case of purely in-plane fields BII, single particle tunneling dynamics dominate the interactions, with the primary effect of B I ~  being a linear transverse shift in the momenta Ak of electrons in one QW relative to those in the other [1,2]. Here we describe our studies of a strongly coupled DQW subject to a high BII. The strong tunneling causes the shifted QW dispersion curves to distort into an anticrossing, opening a partial energy gap or minigap. The presence of the gap mixes the two (displaced) QW Fermi circles to produce a Fermi surface (FS) with orbits of greatly differing area. At the edges of the gap, van Hove singularities in the density of states D(E) appear, and the electron group velocities approach zero. By sweeping BII, the FS can be tuned and the gap can be made to pass through the chemical potential p, with the gap edges producing dramatic effects in the transport properties [3,4]. These include sharp features in both the conductance and the capacitance as a function of BII, and giant deviations in the cyclotron mass Q. The two samples studied were grown by MBE. Sample A (B) consisted of a modulation-doped pair of GaAs QWs of equal width w = 150 A (100 A) separated by an &.3G~.7As barrier of thickness t = 25 A (35 A). Each QW had density n = 1.5 x 1011 (1.2 x loll) cm-2 as measured by Shubnikov-de Haas oscillations (SdHO) for Blld.  Top and bottom QW mobilities were -2.7 and -2.2 x IO5 (-1.2 and -0.6 x 105) cm*Ns. Four- terminal 17 Hz lock-in measurements were performed on Hall bars with Cr/Au top gates. Measurements of m, required a simultaneous small B l  and large Bli, achieved by applying a uniform total field BT at angles 0 = O.O", 2.5", 3.0", and 3.5" from parallel, with 3 rotations performed in situ. In Fig. l(a) we show the normalized in-plane resistance Rll(B11) for sample A, at Td.5 K and 8=0. A minimum occurs at 5.9 T, followed by a maximum at 6.5 T. We now briefly discuss the origin of these two features [3,4]. Shown in Fig. 2(a) is a tight- binding calculation for sample A of the DQW dispersion &(ky), for Bli=7.5 T in the x- direction [5]. BII shifts the two QW dispersion curves relative to one another by an amount Aky=edBl@. The strong QW coupling produces an anticrossing and minigap whose width Q-j (of order 1 meV) is independent of Bii and c- ASAS, the zero-field symmetric- antisymmetric gap [6]. The minigap dramatically distorts D(&) and the electron group velocities [Fig. 2(b)]. Since the upper branch of &(k) is nearly parabolic, a step-like increase occurs in D(&) at the upper gap edge. Thus as B ~ I  is increased and the gap moves upward, the (low velocity) states at the upper gap edge pass through p and are abruptly emptied, reducing the scattering and causing a sharp minimum in R I ~ .  By contrast, &(k) at the lower gap edge is saddle-shaped, yielding a logarithmic divergence in D(&). When B11 is further increased and the lower gap edge crosses p, electrons are divergently scattered into these (zero-velocity) states, yielding the sharp maximum in RII. The shape of the FS is deterrnined by the intersection of p and E@). For BII < 5.9 T, p lies above the gap and the FS has a small lens-shaped orbit lying within a larger hourglass-shaped orbit, shown in Fig. 2(c). For 5.9 T < BII < 6.5 T, p lies in the gap and only the hourglass orbit is present. As BII is increased further, p touches the lower gap edge, causing the hourglass "waist" to pinch off. For Bii > 6.5 T, the FS separates into two Fermi circles of roughly equal size. Changes AD(&) in D(&) can be measured directly via changes AC(Bi1) in the capacitance between the surface gate and the DQW. For AD(&) small relative to the full 2D density of states, AC(Bl1) = AD(&) [7]. This condition appears to hold for our sample, based on a calculation of the geometrical capacitance. In Fig. 3 we show AC(B11) of sample B (gate area -5.4 mm2) for several different gate voltages. In contrast to the conductance data, C(Bi1) remains relatively constant at low BII, until a sudden step-like decrease appears. This corresponds to the disappearance of the lens orbit, occuring when the upper gap edge crosses p. C(B11) then increases monotonically, and ends in a slight peak before settling at a value nearly the same as just before the step-decrease. The peak is due to the lower gap edge, with a logarithmic singularity in D(E), crossing p, and is washed out due to damping. The relative constancy of C(B11) outside the anticrossing region is in stark contrast to RII [see Fig. l(a)], which convolves the changing Fermi velocities. The overall shape of C(Bii) bears a striking resemblance to the calculated D(E) of Fig. 2(b). The slight oscillations just below the step-decrease are SdHO due to a slight misalignment of the sample from 0, which brings us to our next topic. By adding a small BL and measuring the temperature (2") dependence of the resulting (SdHO), we probed mc near the minigap [SI. BII and B l  could not be swept independently, mixing the SdHO with the anticrossing features. To remove this mixing, for each Texamined we subtracted the resistance at 0=0 [e.g. Fig. l(a)] from the resistance in a tilted field, to yield AR(B11,0) = R(B11,0) - R(B11,0=0). Fig. l(b) shows AR(B11,8=3.0°) for several T. The SdHO are uniform up to -5.9 T; then both the period and amplitude change significantly, indicating a drastic change in q. To determine m, from the T-dependence of the SdHO, we first consider which FS orbits contribute. For BII < 5.9 T, the lens and hourglass orbits enclose greatly different areas in k-space. Electrons traverse these orbits at a rate mddt = -(efi)vk x B l ,  where vk is the Fermi velocity. When 0 and hence BI are sufficiently small, electrons in the hourglass are unable to complete the - . .,... .... . ... 5 orbit without scattering, contributing negligibly to the SdHO. However, because Vk is nearly the same for the two orbits (except very close to the gap edge), enough electrons complete the lens orbit to produce SdHO. This assumption agrees with the data for BII c 5.9 T, where only a single period occurs for all three 0. For 5.9 T < BII < 6.5 T, the FS consists of only the hourglass, while for BII > 6.5 T, the FS consists of two separated Fermi circles of equal size, producing SdHO of the same period and T-dependence. Thus for all three ranges, the SdHO arises from a single orbit of the FS. For a single FS orbit, the SdHO are in general described by a series of terns [9]. -Ho.wever when BL is small enough that ocz is-of order unity (where oc = eBl/m,  and z is the scattering time), the higher order terms become negligible and the SdHO become small in amplitude and sinusoidal in shape. This is in accord with our data, where the oscillations are strongly sinusoidal and never exceed 15% of the background. When higher order terms are negligible, the SdH amplitude is given by m(T) TSinh(PTo(~&WI) -- GR(T0) - T&inh@T(mJq)/BI) where To is base T, P = 2n*kgmdfie, and mo is the free electron mass. Fig. l(b) inset shows two typical fits of Eq. (1) to the 0 = 3.0" data. (Here 6R is obtained by taking the difference between AR(B11,8) evaluated at adjacent SdHO extrema, and subtracting the magnetoresistance background.) While for 8.49 T m~ = 0.070 m ~ ,  for 5.09 T m~ = 0.025 Q, a factor of three lower than m * ~ ~ h .  In general, the fits yield mc to k 10% error. For the range -5.9 I BII  -6.5 T, however, mc is changing rapidly, yielding larger errors of roughly f 20%. Fig. 4 shows ~ ( B l l )  for all three 0. At 2.7 T, mc = 0.050 Q, considerably lower than ~ * G A .  Raising Bll causes m, to decrease, reaching 0.021 near the upper gap 6 edge at 5.6 T. As BII is increased further, m, suddenly increases, reaching a value of 0.099 n~ just below 6.5 T, where p crosses the lower gap edge, or saddle point. m, then rapidly converges to m * ~ a ~ ,  near -7 T and remains roughly constant thereafter. Following Lyo [lo], an understanding of the behavior of mc can be gained by examining Fig. 2 and employing the expression mc = (h/2z)f vi'&, where the integration is along the dominant orbit. As BII is increased, the upper gap edge moves upward towards p. As a result, the size of the lens orbit decreases, while Vk decreases more slowly, resulting in a monotonic decrease of m. The lens orbit vanishes at 5.9 T, but because Vk also vanishes, % does not go to zero. Above 5.9 T, the orbit size increases abruptly as electrons move to the hourglass, causing an abrupt increase in %. As BII increases from 5.9 to 6.5 T, p lies in the minigap and moves towards the saddle point. At the same time, becomes smaller near the saddle-point, yielding now a monotonic increase of ~ ( B l i ) .  The divergence of % at the saddlepoint, due to Vk vanishing, is suppressed in the data by damping. Above 6.5 T, p falls below the saddle point, causing the hourglass to split into two roughly circular orbits with nearly parabolic dispersion. The two QWs are thus effectively uncoupled, and Q approaches m * ~ d ~ .  In Fig. 4 we plot Lyo's calculation of ~ , J B I I )  for sample A, where BII has been scaled by 0.96. Excellent agreement is seen. In summary, an anticrossing and minigap appear in strongly coupled DQWs subject to a strong Bit. These give rise to singularities in the density of states, sharp features in the in-plane resistance and the gate capacitance, and giant deviations in the cyclotron effective masses. This work was supported by the U. S. Dept. of Energy under Contract No. DE- AC04-94AL85000. 7 References [l] J.P. Eisenstein. T.J. Gramila, L.N. Pfeiffer, and K.W. West, Phys. Rev. B 44, (1991) 6511. J.A. Simmons, S.K. Lyo, J.F. Klem, M.E. Sherwin, and J.R. Wendt, Phys. Rev. B 47, (1993) 15 741; S.K. Lyo and J.A. Simmons, J. Phys.: Condens. Matter 5, (1993) L299. J.A. Simmons, S.K. Lyo, N.E. Harff, and J.F. Klem, Phys. Rev. Lett. 73, (1994) [2] [3] 2256. S.K. Lyo, Phys. Rev. B 50, (1994) 4965. Including the Hartree term increases the center-to-center distance between the wavefunctions of the two QWs, thus lowering the BII at which the minigap crosses p, and reducing &G. G.S. Boebinger, H.W. Jiang, L.N. Pfeiffer, and K.W. West, Phys. Rev. Lett. 64, (1990) 1793. M. Buttiker, J. Phys.: Condens. Matter 5, (1993) 9361, and references therein. N.E. Harff, J.A. Simmons, S.K. Lyo, J.E. Schirber, J.F. Klem, and S.M. Goodnick, in The Physics of Semiconductors, Vol.1, Ed. D.J. Lockwood, (World Scientific, Singapore, 1995) p. 83 1. 291 T. Ando, A.B. Fowler, and F. Stern, Rev. Mod. Phys. 54, (1982) 437. [lo] S.K. Lyo, Phys. Rev. B 51, (1995) 11 160. [4] [5] [6] 171 [SI " *,. , 8 Figure Captions FIG. 1. (a) Normalized Rli(B11) of sample A at 6=0. (b) AR(B11,6=3.0") for several T. Inset: SdHO amplitudes vs. T, and fits to Eq. (1) for two BII. FIG. 2. Sample A calculations at BII = 7.5 T. (a): &(Icy) with (solid lines) and without (dotted lines) coupling. (b): D(E) for lower energy branch (dotted line) and both branches (solid line). (c): Fermi surface sketch for p 9 meV. FIG. 3. Change in capacitance AC(Bll), proportional to AD(&), for several gate biases. The curves are strikingly similar to Fig. 2(b). FIG. 4. Summary of measured values vs. BII. Solid line is theoretical calculation of Ref. [lo]. n 0 0 0 I I  a CT 4 W n 0 I I  ce Q & W 65 45 25 5 1 1 h W t-"U I- U Lo \ h  Lo n 3.0 4 6 3 8 10 FIG. 1 h > v >r a, S W P lo  pa- gap \::: - step '5 van 1 Hove sing. t I I I I I I I -1 0 1 4  6 8 k (1 /I 00 kl) D(E) (1 O1 O/cm2meV) (a hourglass lens orbit FIG. 2 15 I 10 n LL a W 0 5  (3 0 -5 I I I I I . I  I I I I I 2 4 6 8 10 12 FIG. 3 0.133 0.100 0.067 0.033 0.000 17 e = x o  A e=3.0° 0 8=3.5O 2.0 1.5 1 .o I I I I I I I ' 0.0 2 4 6 8 10 FIG. 4 

Tuning a double quantum well Fermi surface with in-plane magnetic fields

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