We review recent theoretical and experimental results on low- temperature tunneling and in-plane transport properties in double quantum wells (DQWs) in an in-plane magnetic field B{parallel}. These properties arise from combined effect of B{parallel}-induced relative displacement of the wave vectors in the two QWs and the interwell tunneling. In weakly coupled DQWs, the tunneling conductance has two sharp maxima as a function of B{parallel}. In strongly coupled DQWs, a partial minigap is formed due to anticrossing of the two QW dispersion curves, yielding sharp B{parallel}-dependent structures in the density of states and in- plane transport properties. Excellent agreement is obtained between theory and data from GaAs/AlGaAs DQWs

Eiles, T.M.

Harff, N.E.

Klem, J.F.

Lyo, S.K.

Simmons, J.A.

UNT Digital Library

5 i 9 P  ws-/ 7 2  3 Magne tic-Field-Induced Tunneling and Minigap Transport in Double Quantum Wells RECEIVED S. K. Lyo, J. A. Simmons, N. E. Harff, T. M. Eiles, and J. F. Klem Sandia National Laboratories, Albuquerque, NM 87185 Abstract. We review recent theoretical and experimental results on low-temperature tunneling and in-plane transport properties in double quantum wells (DQWs) in an in-plane magnetic field Bib These properties arise from the combined effect of BII-induced relative displacement of the wave vectors in the two QWs and the interwell tunneling. In weakly coupled DQWs, the tunnel- ing conductance has two sharp maxima as a function of In strongly coupled DQWs, a partial minigap is formed due to the anticrossing of the two QW dispersion curves, yielding sharp Blldependent structures in the density of states and in-plane transport properties. Excellent agreement is obtained between the theory and the data from GaAs/AlGaAs DQWs. i 1. Introduction t :, I In this paper, we discuss low-temperature tunneling and in-plane transport properties of dou- ble quantum wells (DQWs) in an in-plane magnetic field BII II x. DQWs consist of two paral- lel layers of two-dimensional (2D) degenerate electron-gases (2DEGs) separated by a barrier. Recent surge of interest in DQW structures can be attributed to the fact that DQWs display richer unique physical properties than single QWs due to an extra degree of freedom in the growth direction (I1 z). Some of the examples are the interlayer Coulomb-drag effect [l] in zero field, the tunneling Coulomb gap [2], the interlayer-tunneling excitonic effect [3] in strong perpendicular fields B,, and 2D-2D interwell tunneling in B,, [4-81. In DQWs, one can control the well-to-well separation and the interwell overlap, both sensitive to the barrier thickness. The charges in the wells can be controlled by gate voltages. We show that BII pro- vides an additional control of the effective interwell coupling by displacing the wave vectors k in the two QWs. As a result, Bll can be used to tune the tunneling current and deform the Fermi surface, introducing sharp Bll-dependent structures in the density of states @OS) and other in-plane transport properties [9-131. In contrast to the high Blcase [2, 31, Coulomb interactions are used in BII only for band bending corrections and can otherwise be neglected: our data are well explained using the combined effect of tunneling and BII within the frame work of noninteracting electrons. Three samples discussed in this paper are symmetric (except for sample 1) GaAd43Gao7As DQWs with well widths w and depths V,. The GaAs wells are separated by an &.3G& 7As barrier of thickness t. The primary effect of B,, is to induce a linear transverse displacement Aky 0~ B,, in k-space in one QW relative to the other. Here the in-plane wave vector k is a good quantum number. The linear displacement arises from the second term of the Hamiltonian: H = k + - ( k  A2 -7) 2 2  +V(Z), 2m* 21n* Y e J’e DI~TRIBUTION OF THIS DOCUMENT 1s WJL1iVIITED DISCLAIMER This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, make any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to  any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof. DISCLAIMER Portions of this document may be illegible in electronic image products. Images are produced from the best available original document. _- Page 2 where pz  = -ina/az and 4 = (fideBII)1'2. The GaAs effective mass m* = 0.067 (in units of the free electron mass) is used in both the QWs and the barriers because the confinement wave functions &(z) (n = 1, 2) penetrate negligibly into the barriers. The kinetic energy e@,) = (hkJ2/2m* is to be added to (1). The confinement potential V(z) is a superposition of the potentials Vl(z) and V~(Z) of QWl and QW2. We assume quasi-2D (Le., w < 4) thin QW's where only the ground sublevels are populated and are relevant. For the sake of physical argu- ment and for 2D-2D tunneling in Sec. 2, z in the second term of (1) can be replaced by its expectation values -1, <2>2 with respect to 4n(z) [SI. The net effect of B11 is then to shift the origin of ,$ of QW2 by A,$ = d / l2 a Blld relative to that of QWl, where d = <z>~ - The basic transport properties to be discussed in the following arise from the BII-induced shift A,$ and the tunneling between the two QWs. Spin splitting is neglected in this paper. 2. Resonant 2D-2D tunneling The tunneling conductance G, is the steady-state current flowing from QW 1 to QW2 per unit linear electric field applied between the QWs. The current flows into QWl from a source, tun- nels into QW2, and then flows out of QW2 to the drain. The source-drain resistance is related to the tunneling conductance and the Zero-BII conductances (Le., mobilities) of the QWs in terms of a transmission-line model [6,7]. The G, data is obtained from the source-drain resis- tance data using this relationship [7]. The latter depends on the geometrical structure of the source-drain current paths of the sample [6] .  The electrons are rapidly scattered inside the QWs, occasionally tunnelling into the other QW with the tunneling integral J. A weak depen- dence of J on k is ignored [6]. For tunneling from k in QWl to k' in QW2 (with k = Ikl and the sublevel energyd,), the initial and final energies are given by Only the electrons on the Fermi circles can tunnel when the energy and momentum conservations are satisfied: Elk = e k '  and k = k' [5]. For 2D densities NI > N2, the Fermi sur- faces are concentric circles of radii kl and k2 at Bll= 0 as illustrated in the inset of Fig. l(a) and the conservation conditions are not satisfied. As Bll is increased, the inner smaller circle slides relative to the outer circle until they touch each other tangentially. The conservation conditions are satisfied at this Bll, yielding a G, peak. As BII is increased further, the inner cir- cle begins to move outside the larger circle, intersecting it at two points. G, begins to decrease as the area of contact decreases. Another G, maximum is reached at a higher BII, when the smaller circle touches the larger circle from outside. G, begins to drop beyond this field as the two circles are separated as illustrated in Fig l(b). The G, maxima diverge in the absence of scattering. However, the data from sample 1 displayed in Fig. l(a) does not show the divergence due to finite scattering times as will be shown below. Sam le 1 is weakly cou- The tunneling conductance is calculated using a linear response theory. The dominant pled and has w = 150 A, t = 65 A, N1= 1.8X 10" cm-2, and N2 = 1.OX 10 P cm-2. contribution arises from the so-called bubble diagram shown in Fig. l(b) and is given by [6] wheref(0 is the first derivative of the Fermi functionfin and 1 r.. (4) Page 3 E G '. . *./- -2 -1 0 1 2 - .  2 0 0 2 4 6 8 10 0 2 4 6 8 IO Magnetic Field 0 Figure 1. Tunneling conductance per unit area from the data from sample 1 (a) and theory 0). The insets show the relative Fermi surfaces of the QWs (a) and the bubble diagram 0). 0- -2 -1 0 1 2 Wave Number ( l / l O d )  12 , I 12 , 1 <a) @) Figure 2. (a) Energy dispersion curves from sample 2 with (solid m e s )  and without (dashed curves) tunneling. (b) The DOS from the lower (dashed curves) and both branches (solid curves). Here T, = (2TJ73-l is the quantum scattering time of the electrons on the Fenni surface. The mobilities in the wells are ,XI= 6X lo4 cm2Ns and /.Q = 6X 105 cm2Ns, indicating that rr >> r2 In this case, Gzz becomes independent of r'. The theoretical GZ displayed in Fig. l(b) for several values of Fland J = 0.04 meV, show good agreement with the data. This value of J is close to J = 0.03 meV estimated from the splitting of the two ground sublevels due to tunneling. For isotropic scattering, the mobility p1 = 6X lo4 cm2Ns corresponds to r1 = 0.15 meV and T~ = 2.3X1O-l2 s. The G, maxima diverge for r1 = 0 at the fields B* = f i c & ( K  k m) / ed, where N, (NJ is the larger Oesser) of N1 and N2 3. In-plane transport properties 3.1. Energy dispersion and anticrossing DQW structures show interesting in-plane transport properties when interwell tunneling is significant. The BII-induced linear displacement At$= Bll of the origins of the transverse crys- tal momenta k, in the two QWs is shown in Fi .2(a) by dashed curves for sample 2 which has w = 150 A, t = 25 A, and Nl = A5 = 1.5X 10 cm-2. For such a thin barrier @e., large orver- lap) between the QWs, the two energy-dispersion parabolas anticross significantly and a par- tial minigap is formed as shown by the solid curves therein. For symmetric DQWs, the latter correspond to the upper and lower branches of the eigenvalues of (1) (to the lowest order in the overlap <$1(z)I&(z)>) [9,13]: lf where E($,) = (7%jJ2/2rn*, E t  = (A1!?')~/2rn*, dn = 1<2>,1, and d1= d~ Here (Az,,)~ = <4,(z)l(z - q,>)21~n(z)> is the mean square deviation of z arising from finite widths of the QWs. In (5), use is made of & I =  A q .  The minigap is independent of B I I  and is given by \ Page 4 Square-well potentials are used for the curves in Fig. 2. A similar anticrossing of dispersion curves, although of different origin occurs without BII’S in a vicinal surface such as (91 1) of a Si-inversion layer and was studied extensively many years ago [14]. 3.2. Minigap and density of states The lower minigap edges in Fig. 2(a) at 4 T and 6 T are saddle points with opposite signs of curvatures in the kx and k, directions. The DOS has a van Hove singularity diverging loga- rithmically at the saddle point as shown in Fig. 2(b) at these 2311’s. The latter is formed only at high fields (i.e., BII = 4 T) when the energy of the crossing point is large enough to overcome the energy repulsion between the two branches. The sharp step in the DOS in Fig. 2(b) is due to the contribution from the upper branch. The region between the sharp singularity and the step is the minigap and moves up in energy rigidly with increasing BII as shown in Fig. 2(b). For the samples discussed in this paper, the 2D density N2D = Nl +N2 is sufficiently high so that the chemical potential p lies in the upper branch (Le., above the step edge) at BII = 0. At low Bll’s, p, is insensitive to BII while the gap rises in energy nearly quadratically with BII [9]. Therefore the gap sweeps through p as Bll is increased. Above these BII’s, p rises in energy eventually together with the bottom of the lower branch. In Fig. 3, we compare the calculated reduced DOS at p with the capacitive DOS data [l5] from sample 3 in units of m*A/(27rh2), whereA is the cross-sectional area of the QWs. Use is made of d = 140 A, ( A Z ~ ) ~  = 621 A2 obtained by a self-consistent Hartree approximation, and Eg = 1.8 meV determined experi- mentally from the conductance data [lo]. The latter is somewhat smaller than the Hartree value Eg = 2.0 mev. Sample 3 has w = 100 A, t = 35 A, and N2D = 2.4X 10” cme2. p drops off the DOS step suddenly at -7.2 T and passes through the saddle point at - 8.9 T. Sharp edges as well as the singularity peak are rounded by damping [15]. The data are fitted at the theoret- ical asymptotic value D(p) = 2 at high BII’S, where the two QWs are uncoupled. Small oscilla- tions in the data are due to a small BI. 3.3 Cyclotron mass The cyclotron mass m, is given by m, = (h2/ 27r)aS/a+ where S is the orbit area in k-space and EF is the Fermi energy [13]. In 2D structures, m, can be rewritten as m, / m*= D@), where D(p) is the reduced DOS at the Fermi level from the orbit under consideration. In Fig. 4, we display m, / m* calculated for sample 2 using d = 200 A, ( A z ~ ) ~  = 1150 A2 obtained by a self-consistent Hartree approximation, and Eg = 1.2 meV determined experimentally from the conductance data [lo]. The latter is somewhat smaller than the Hartree value Eg = 1.4 meV. At low BII’S, ,U is above the gap and the cyclotron orbits consist of a large hour-glass orbit in the lower branch and a smaller lens orbit in the upper branch shown in the lower left corner of Fig. 4. The electrons in the hour-glass orbit have a large mass as shown by the dash-dotted curve and are scattered before completing a cycle: the oscillations from the lower-mass inner lens orbit are dominant in recent Shubnikov-de Haas measurements [12]. As BII increases, p moves toward the gap edge in Fig. 2(b). As a result, the reduced DOS (i.e., m,) from the lower and upper branch increases (dash-dotted curve) and decreases (solid curve) monotonically as a function of BII, respectively, for the hour-glass orbit and the lens orbit as shown in Fig. 4. At higher BII’s (i.e., above 6.1 T in Fig. 4), p moves into the gap, depopulating the lens orbit. Therefore m, increases abruptly to the mass of the hour-glass orbit. At 6.9 T, p lies at the saddle point. Above this Bll, the orbit splits into two separated orbits and m, drops to half. The mass thus saturates to m* of the uncoupled QWs. The calcu- lated m, yields excellent agreement with the data from sample 2 without adjustable parame- ters, as shown in Fig. 4 . Page 5 3 2.0 Q 25 .- 20 s v 3 r: 8 1.0 u" 0.5 2 0 0 4 1.5 I I I I 4 6 8 10 12 Magnetic Held 0 Figure 3. Comparison of the calculated reduced DOS at p (solid curve) with the capacitive DOS data (dashed curve) from sample 3. 0.0 0 2 4 6 8 10 Magnetic Field (T) Figure 4. Comparison of the calculated mc (solid, dash-dotted cwes)  with the data (black dots) from sample 2 with the relevant cyclotron orbits. 3.3 In-plane conductance The in-plane conductance GII in the direction u of the electric field is given by [9] where vk is the group velocity, vk = Ivkl, rk is the transport relaxation-time, and the integra- tion is along the orbit. We consider only elastic scattering. In (7), GII is proportional to vk and Tk. When pis above the gap at low BII'S, the contribution to GI, arises from both the hour-glass orbit and the lens orbit in Fig. 4. In contrast to the cyclotron mass, however, the lens orbit con- tributes little to GII because 1) the electrons in the lens orbit have slow velocities due to their small k values and 2) the number of states in the lens orbit is much smaller than in the hour-glass orbit. On the contrary, the lens orbit reduces rk and therefore GII by providing states into which the electrons in the hour-glass orbit are scattered rapidly at low BII's. As BII is increased, p falls below the upper gap edge depopulating the lens orbit. In this case, the elec- trons in the hour-glass orbit cannot be scattered into the upper branch elastically, yielding sig- nificantly larger 7k as well as GII. This behavior is shown by the solid curve (Le., ro = 0 meV) in Fig. 5, where we plot GII calculated for sample 2 by approximating r k  0: p(&J1 in (7). Here p(&) is the total DOS. The maximum in GII is due to the depopulation of the lens orbit. The GI1 minimum, on the other hand, arises when p lies on the saddle point, where GII vanishes because 7k p(&J1 = 0 due to the divergence of the DOS. Eg can be determined from the BII's of the G1 maximum and minimum [lo]. The effect of band bending is to increase the effective d from 175 8, to 200 8, by pushing the confinement wave functions away from each other. Since BII enters the Hamiltonian approximately as d/ t2  0: BI& the net effect of band bending is to rescale the BII-axis by 175xBll /ZOO as shown in the upper axis of Fig. 5. The effect of damping has been treated by a self-consistent linear response theory [16] and is shown in Fig. 5. The calculated results yield good agreement with the data [ 101 displayed in Fig. 6. 4. Summary We have discussed low-temperature tunneling and in-plane transport properties in double Page 6 0 2 4 6 8 1 2 I . 1 . I . I  0 2 4 6 8 10 MagneticField (T) Figure 5. Calculated G for sample 2 for sev- eral values of damping f= 2r,. The upper and lower scales are with and without band bending. 1.2 , 1 , - I  , .---. 0.8 I I 0 2 4 6 8 MagneticField 0 Figure 6. G data from sample 2. Char es are approximady balanced at the gate bias % = - 0.1 v. quantum wells in BII'S. These properties arise from the combined effect of BII-induced relative displacement of the wave vectors in the two QWs and the interwell tunneling. In weakly cou- pled DQW.s, the interwell tunneling conductance has two sharp maxima as a function of Bll. In strongly coupled DQWs, a partial minigap is formed due to the anticrossing of the two QW dispersion curves, yielding sharp Bli-dependent structures in the density of states, cyclotron mass, and the in-plane conductance. Excellent agreement is obtained between the theory and the data from GaAdAlGaAs DQWs. This work was supported by the Office of Basic Energy Sciences, Division of Materials Sciences, U. S .  DOE under Contract No. DE-AC04-94AL8500. References [I] Gramila T J, Eisenstein J P, MacDonaId A H, Pfeiffer L N, and West K W1990 Phys. Rev. Lett. 66 1793-6 [2] Eisenstein J P, Pfeiffer L N, and West K W 1992 Phys. Rev. Lett 69 3804-7 [3] Eisenstein J P, Pfeiffer L N, and West K W 1995 Phys. Rev. Lett 74 1419-22 141 Smoliiar J, Demmerle W, Berthold G, Gomik E, Weimann G, and Schlapp W 1989 Phys. Rev. Lett 63 [SI Eisenstein J P, GramilaT J, Pfeiffer L N, and West K W 1991 Phys. Rev. B 44 6511 [q Lyo S K and Simmons J A 1993 J. Phys.: Condens. Matter 5 L299-L306 [7] Simmons J A, Lyo S K, Klem J F, Sherwin M E, and Wendt J R 1993 Phys. Rev. B 47 157414 [SI Zheng L and MacDonaId A H 1993 Phys. Rev. B 47 10619-24 [9] Lyo S K 1994 Phys. Rev. B 50 4965-8 [lo] Simmons J A, Lyo S K, Harff N E, and Klem J F 1994 Phys. Rev. Lett 73 2256-9 [ll] Kurobe A, Castleton I M, Linfield E H, Grimshaw M P, Brown K M, Ritchie D A, Pepper M, and Jones G A C 1994 Phys. Rev. B 50 4889-92. [12] Simmons J A, Harff N E, and Klem J F 1995 Phys. Rev. B 51 11156-9 [13] Lyo S K 1995 Phys. Rev. B 51 11160-3 [14] Ando T, FowlerA B, and Stem F 1982 Rev. Mod. Phys.54 437-672 [l5] Eiles T M, Simmons J A, Lyo S K, Harff N E, and Klem J F, unpublished. [16] Lyo S K, Simmons J A, and Harff N E, 1995 i'le Physics of Semiconductors: Proceedings of the 22nd International Conference, Vancouver; 1994, VoL 1,843-6, edited by D. J. Lockwood (World Scientific, Singapore) 21 16-9 

Magnetic-field-induced tunneling and minigap transport in double quantum wells

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