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Multisublevel transport and magnetoquantum oscillations in quantum wires
Current Applied Physics 4 (2004) 491–496 www.elsevier.com/locate/cap
Multisublevel transport and magnetoquantum oscillations in quantum wires q S.K. Lyo b
a,*
, D.H. Huang
b
a Sandia National Laboratories, Albuquerque, NM 87185, USA Air Force Research Laboratory (AFRL/VSSS), Kirtland Air Force Base, Kirtland, NM 87117, USA
Received 20 November 2003; accepted 30 January 2004 Available online 9 April 2004
Abstract Transport in one-dimensional (1D) wires show interesting properties due to restricted 1D phase space. We calculate 1D transport properties to a controlled accuracy and examine the effect of combined elastic and inelastic scattering on the conductance G and the thermoelectric power S of a single-quantum-well wire (SQWR) and tunnel-coupled double-quantum-well wires (DQWR’s) in a perpendicular magnetic field. The field dependence of G and S are strikingly different in ballistic and diffusive regimes. Interwell tunneling in DQWR’s distorts the conductance and the TEP drastically from those of a SQWR and yields a sign anomaly for S. Ó 2004 Elsevier B.V. All rights reserved. PACS: 72.15.Eb; 72.15.Jf; 72.23.Ad; 73.63.Hs; 73.63.Nm Keywords: Ballistic; Diffusive; Conductance; Thermopower; Sign anomaly
1. Introduction Transport properties of quasi-one-dimensional (1D) doped semiconductor structures are of current interest for many novel physical phenomena and possible applications for new devices. An earliest form of 1D single-quantum-well wires (SQWR’s) was realized in the so-called quantum point contact illustrated in Fig. 1(a) where the channel length is very short, of the order of a fraction of a lm. Electrons pass through this wire ballistically at low temperatures (T ’s) between the two 2D gases under a DC bias. The conductance G is quantized and decreases in steps of 2e2 =h in a spin-degenerate system (assumed in this paper) when the channel width is reduced gradually [1]. Here, e is the absolute electronic charge and h is Planck’s constant. Similar quantized G steps were observed in SQWR’s as a function of a magnetic field B applied in the perpendicular (i.e., z) q Original version presented at QTSM&QFS 2003 (Quantum Transport Synthetic Metals & Quantum Functional Semiconductors), Seoul National University, Seoul, Korea, 20–22 November 2003. * Corresponding author. Tel.: +1-505-844-3718; fax: +1-505-8441197. E-mail address: [email protected] (S.K. Lyo).
1567-1739/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.cap.2004.02.004
direction shown in Fig. 1(a) [2]. Recently, ballistic thermoelectric power (TEP) was observed in SQWR’s [3] and was also studied theoretically in zero B [3–5]. Good quality (i.e., high-mobility) long-wire samples (e.g., >20 lm) are now within reach. In long wires, transport is expected to be diffusive. Our interest lies in high-mobility wires where the wire length is longer than the mean-free-path but is shorter than the localization length. The present discussion is limited to the Fermi liquid model. We show in this paper that the B-dependences of the diffusive G and the TEP are strikingly different from those of the ballistic case and can provide insight into the basic nature of the transport. We also show that the transport properties of a tunnel-coupled double-quantum-well wires (DQWR’s) are very different from those of SQWR’s in perpendicular magnetic field including a sign reversal for the ballistic as well as the diffusive TEP. For diffusive G, a number of recent studies have focused on the effect of elastic scattering [6–13]. Here, scattering by impurities and interface roughness plays a dominant role for the momentum relaxation. Recently, we assessed the relative contribution from phonon scattering and found that the latter can be important even at relatively low temperatures in multi-sublevel structures.
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S.K. Lyo, D.H. Huang / Current Applied Physics 4 (2004) 491–496
In these systems, inter-sublevel electron–phonon scattering is important for the momentum-relaxation processes. In this case, the standard quasi-elastic-scattering approximation does not yield a reliable result because the phonon energy can be comparable to the electron energy. Therefore, a rigorous method for treating combined elastic and phonon scattering is necessary. We describe a formalism which yields G and the TEP to a desired accuracy for a general quasi-onedimensional electronic structure and present the result obtained from this formalism [14,15].
2. Effect of the magnetic field on the energy dispersion We consider only the ground sublevel for the z-confinement in the growth direction. For the channel confinement in the x direction, we consider parabolic confinement with many low-lying sublevels spaced evenly by hxx occupied as shown in Fig. 1(a). In a perpendicular magnetic field Bkz, the energy dispersion is given for a SQWR by 2 k 2 h hXx þ ; enk ¼ n þ 12 ð1Þ 2m wherep k ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi is the wave number along the wire, n ¼ 0; 1; . . ., ffi Xx ¼ x2c þ x2x , xc ¼ eB=m c, m ¼ m =½1 ðxc =Xx Þ2 , and m is the zero-field effective mass [10,13]. The sign of the displacement Dxk / k of the center of the harmonic wave functions /n ðx Dxk Þ induced by the Lorentz force depends on the sign of k. The levels are depopulated successively as m increases with B. For DQWR’s, interesting effects occur when B is in the channel plane in the x direction as shown in Fig. 1(b). In this case, the role of B is to shift the origin of k of the energy parabola of the second wire (i.e., QW2) by dk ¼ d=‘2c relative to that of the first wire (i.e., QW1),
wherepdffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi is the center-to-center distance of the QWs and ‘c ¼ hc=m x. It also localizes (i.e., separates) the wave functions for k and k at the two opposite Fermi points (e.g., black dots in (b)) into separate wires through the Lorentz force. Tunneling couples only the parabolas belonging to the same quantum number n. The energy dispersion in Fig. 1(b) shows the anticrossing between the two parabolas shifted by dk (shown in gray background) for n ¼ 0 for symmetric DQWR’s [16]. Here, the two parabolas split into the upper and lower branches separated by an anticrossing gap DSAS between the symmetric and the antisymmetric states at k ¼ 0. The parabolas move away from each other as dk / B increases, pushing the gap through the chemical potential l. The inset in Fig. 1(b) shows the relative positions of l with respect to the gap for three B’s. This level splitting occurs for each n. The levels become depopulated as B increases. The energy dispersion and DSAS depend on the QW width LW , the barrier width LB , and the barrier height Vo .
3. Ballistic transport The electron-diffusion TEP Sd is the ratio of the heat current and the charge current of the electrons divided by T in the presence of a linear DC field. Ballistic electron-diffusion Sd is given, for symmetric electronic structures, by [17] kB X X Sd ¼ Cm;c ½bðem;c lÞf ð0Þ ðem;c Þ eF m c þ lnðebðlem;c Þ þ 1Þ ; where f is the Fermi function, b ¼ 1=kB T , and XX F ¼ Cm;c f ð0Þ ðem;c Þ: m
drain contact
(a)
(b) Z Y
µ1 µ2
wire
µ3
wire-1
Cu rr en t
Cu rr en t
µ
B
wire-2
X B
source contact
Fig. 1. (a) A schematic diagram of a SQWR and the energy dispersion. A narrow channel is formed by applying a negative bias on the top split metallic gate, not shown. (b) A schematic diagram for DQWR’s and the energy dispersion. Electrons tunnel between the wires through the Alx Ga1x AS barrier in the z direction. A magnetic field B lies in the x direction for the DQWR’s and in the z direction for the SQWR. Black dots denote the Fermi points.
ð2Þ
ð0Þ
ð3Þ
c
Here, m- and c-summations indicate summing over all the energy dispersion curves and over all the energy extremum points on each curve, respectively. The quantity em;c is the extremum energy. For a given curve m, Cm;c ¼ 1 for a local energy minimum point and Cm;c ¼ 1 for a local energy maximum point. The quantity F equals the number of the pairs of the Fermi points at T ¼ 0 and is related to G by G ¼ 2e2 =hF . The results in Eq. (2) reduce to he earlier results obtained for SQWR’s with a single minimum point for each the energy dispersion curve [1,4]. In Fig. 2(a), we show G and Sd of a SQWR as a function of B when several levels are occupied at B ¼ 0 for two T ’s. The sample parameters and the electron density n1D are given in the inset. We find spikes for the TEP just before a level is depopulated near the quantum steps of G. These spikes broaden as T is raised. Fig. 2(b) shows Sd and G for DQWR’s for two T ’s for the case of
S.K. Lyo, D.H. Huang / Current Applied Physics 4 (2004) 491–496
493
(a)
(b)
Fig. 2. Ballistic Sd and G for two T ’s (a) in SQWR with three levels occupied at B ¼ 0 and (b) in DQWRs with only n ¼ 0 occupied. The parameters are defined in the text.
large hxx where only the tunnel-split n ¼ 0 doublet is occupied at B ¼ 0 as indicated in Fig. 1(b). The conductance has a minimum and is V-shaped in striking contrast with the behavior shown for a SQWR in (a). The conductance equals the number of the pairs of the Fermi points in units of 2e2 =h, namely 2 for l ¼ l1 , 1 for l ¼ l2 , and 2 for l ¼ l3 in Fig. 1(b). The TEP shows a surprising feature: it changes sign near the G minimum just before it begins to rise again. At this point, l crosses the local energy extremum (i.e., maximum) point at k ¼ 0 of the lower branch doubling the number of the Fermi points and G. The dispersion is hole-like (i.e., inverted) here, yielding a sign reversal for Sd . When many levels are occupied for small hx x DSAS , G of DQWR’s follows a V-shaped dependence on B as shown in Fig. 3 (left axis) for two T ’s. A similar Bdependence was observed earlier in DQWR’s [19]. The inset illustrates the tunnel-split sublevels for this case at a certain field near the G minimum. The B-dependence of G can be explained in a similar way as in Fig. 2(b) by
Fig. 3. Ballistic G (left axis, upper curves) and Sd (right axis, lower curves) for two T ’s in DQWRs with many levels occupied. The parameters are defined in the text. The inset shows the tunnel-split energy dispersion at B near the G minimum.
accounting for the Fermi points from all n into account [18]. For SQWR’s, G decreases stepwise monotonically as a function of B as in Fig. 2(a) and is not shown [2]. Ballistic Sd is also displayed for the DQWR’s at T ’s (right axis). Note that Sd shows positive peaks initially when l crosses the bottom of the upper branches successively for lower n’s. It changes sign near the G minimum just before it begins to rise again. At this point, l crosses the local energy maximum point at the top (at k ¼ 0) of the lower branch increasing the number of the Fermi points and thus G. For this point, the dispersion is hole-like, yielding the sign reversal for Sd . From this point on, l keeps crossing similar local energy maximum points belonging to lower n’s, producing hole-like successive negative peaks for Sd . The Sd peak near the G minimum at 3.4 T is large because Sd is inversely proportional to G. The discussions so far are restricted to Sd . The phonon-drag contribution Spd should be small in a QPC because electrons have little time to interact with the phonons in the ballistic channel, although a recent paper concluded otherwise [5]. There has been no experimental evidence for the ballistic phonon-drag TEP.
4. Diffusive transport 4.1. Brief description of theoretical approach in 1D The theory to be described here applies to arbitrary 1D band structures including SQWR’s and DQWR’s with or without perpendicular magnetic fields studied in the two previous sections. The distribution function ð0Þ deviates from the Fermi function fk according to ð0Þ ð0Þ0 fk ¼ fk þ gk kB T ½fk eE in a linear DC electric field Eky. Here, k is the wave number in the y direction and
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the prime denotes the first energy derivative. The quantity gk satisfies the Boltzmann equation [20]: 2p X 2 vk þ hjIk0 ;k j iðgk0 gk Þdðek ek0 Þ þ Pk ¼ 0; ð4Þ h k0 where h i denotes the configurational average, vk is the Bloch velocity, Pk is the contribution from electron– phonon interaction (EPI), and Ik0 ;k is the matrix element for elastic scattering. We assume interface-roughness 2 scattering for jIk0 ;k j studied earlier [13]. The contribution from EPI Pk is given by [20] 2p X 2 ðÞ ð0Þ Pk ¼ jVsq j ½fk0 þ nsq ðgk0 gk Þ h k0 ;s;q dðek ek0 hxsq Þdk0 ;kqy Dðqx ; qz Þ: ðÞ fk0
ð5Þ
ð0Þ fk0 ,
Here, ¼ 1=2 1=2 hxsq is the phonon energy ð0Þ of mode s and wave vector q, and nsq ¼ nð0Þ ð hxsq Þ is the boson function. The conservation factor Dðqx ; qz Þ for perpendicular qx , qz and EPI Vsq are given in Ref. [15]. In Eqs. (4) and (5), curve indeces m, m0 accompany k, k 0 , respectively, and will be suppressed for simplicity. Eq. (4) can be solved for gk to a desired accuracy in 1D by subdividing the energy near the Fermi level into a sufficiently small interval de kB T over a sufficiently large range kB T . Scattering between points k, k 0 is mapped into scattering between all the discrete points lying on the intersection of the above discrete energies and the dispersions curves, yielding Ag ¼ B type equation which can be inverted for g. Here, g is the column vector which contains all the gk values on the above discrete points and A, B are known matrices involving the scattering parameters. The conductance and then heat current by electrons are then obtained from g. A detailed but somewhat complicated procedure is given in the above Ref. [15]. The electron current displaces the phonon distribution from equilibrium through EPI yielding a phonon heat current. The latter is a linear function of g. The phonon-drag contribution Spd to S is the ratio of the phonon heat current and the charge current divided by T [14].
Fig. 4. Diffusive G vs. B for a SQWR with n ¼ 0 and n ¼ 1 occupied at B ¼ 0. The dashed horizontal lines in the inset show l ¼ l1 in n ¼ 1 and l ¼ l2 in n ¼ 0. The thick horizontal (tilted) double-headed arrow illustrates intra-level (inter-level) roughness (phonon) scattering.
density of states (DOS). The rapid decrease of G at high B’s in Fig. 4 is due to this effect. It is seen that G decreases rapidly as a function of T even at low T ’s due to inter-level one-phonon scattering illustrated in the in-set. The thermal population effect with elastic scattering alone (i.e., without electron–phonon scattering) is found to yield large G as shown by the gray curve. When many levels are occupied for small hxx , G shows oscillations yielding peaks after each depopulation. Fig. 5 displays G for 1lm long symmetric DQWRs with only the n ¼ 0 doublet occupied with n1D ¼ 6:5 105 cm1 , l ¼ 3:79 meV, well-depth 270 meV, and barrier-width LB ¼ 40A. well-widths LW ¼ 80A, The abrupt rise in the range 3 < B < 8 T when l is in-
4.2. Numerical evaluation For a numerical evaluation, we employ standard parameters for the EPI including deformation-potential and piezoelectric interactions [21]. For interface-rough layer fluctuation ness scattering, we assume db ¼ 5 A with a Gaussian correlation length K ¼ 30 A. Fig. 4 shows G for a 10 lm long SQWR with two levels populated at B ¼ 0 with a n1D ¼ 106 cm1 , l ¼ 4:81 meV, well-depth 270 meV, and a well-width The conductance rises abruptly around 2 T LW ¼ 210A. due to the depopulation of the n ¼ 1 level at low temperatures (e.g., 1 K). The conductance is small when l is at the bottom of any level because of the large 1D
Fig. 5. Diffusive G vs. B for DQWRs. The inset displays the energy dispersion of two tunnel-split branches with only the n ¼ 0 level occupied. The dashed horizontal lines show l ¼ l1 above the gap, l ¼ l2 inside the gap, and l ¼ l3 below the gap. The thick horizontal (tilted) double-headed arrow illustrates roughness (phonon) scattering.
S.K. Lyo, D.H. Huang / Current Applied Physics 4 (2004) 491–496
side the anticrossing gap (e.g., l ¼ l2 ) arises from the fact that back scattering (illustrated by the doubleheaded horizontal arrow in the inset) is quenched between the initial and final state due to the B-induced localization (i.e., separation) of the initial and final wave functions [22]. At high B’s (e.g., B ¼ 8 T) and low T ’s, l falls below the gap as shown by l ¼ l3 in the inset, reducing G abruptly. At high T ’s, however, the gap region is thermally populated. Here, the separation between the initial and final states is even more severe because of large B, yielding enhanced G as shown for 8 K. However, at 16 K, phonon scattring reduces G as shown. For the TEP, we examine briefly several interesting features unique to 1D SQWR structures. Fig. 6 displays S as a function of T for B ¼ 0 for large hxx ¼ 50 meV and n1D ¼ 106 cm1 when only n ¼ 0 is occupied. The upper and right axes are linear scale for Sd which follows the well-known quasi-linear T -dependence, while Spd is given by the lower and left axes in a semi-log scale. This curve shows Spd / expðb hx2KF Þ, where x2KF is the phonon energy for the 2KF back scattering between the two Fermi points shown by the black dots in the inset. This is in contrast to the power law behavior of Spd in higher dimensions where the Fermi surface is a continuum. Note that Spd Sd except at low temperatures. Fig. 7(a) displays Sd and Spd as a function of n1D . The minimum in Spd occurs just before the occupation of the second level n ¼ 1 (marked by a vertical arrow) where the resonant 2KF phonon energy reaches a maximum. The same kind of peak occurs before the filling of the next level. Also, interesting is the sign change of Sd with a sharp variation in magnitude slightly below the occupation of the second level n ¼ 1. When l is at this
495
(a)
(b)
Fig. 7. Spd (solid curve, left axis) and Sd (dotted curve, right axis) as a function of (a) n1D and (b) B at T ¼ 2 K. Parameters not shown in the insets are given in the text. The second level begins to be populated/ depopulated at the position marked by the vertical arrow in (a)/(b). NF is the number of the Fermi points.
point, G decreases rapidly with increasing l (i.e., dG=dl < 0) because of the large DOS just inside the n ¼ 1 level. As a result, more current flows from below l, yielding the sign reversal. However, dG=dl > 0 is recovered as soon as the level n ¼ 1 is occupied, yielding a normal sign. The rapid variation of dG=dl in this range is responsible for the large variation of Sd . A very similar feature is shown in (b) where the position of l is varied by B. One difference is that Spd keeps decreasing at high fields after the depopulation of level n ¼ 1. This follows from the B-induced reduction of the overlap of the initial and final wave functions involved in 2KF phonon scattering, which weakens the phonon scattering rate for Spd .
5. Conclusions Fig. 6. Spd (solid curve, left-bottom axis) vs. 1=T in a semi-log scale and Sd (dotted curve, right-top axis) vs T in a linear scale at B ¼ 0. Only the ground level n ¼ 0 is populated.
We showed that the B-dependence of the diffusive G, the TEP, and their quantum oscillations are strikingly different from those of the ballistic transport in SQWR’s
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S.K. Lyo, D.H. Huang / Current Applied Physics 4 (2004) 491–496
and DQWR’s. We also demonstrated interesting sign anomalies for the electron-diffusion TEP in ballistic DQWR’s and diffusive SQWR’s. Diffusive DQWR’s also show the sign anomaly. Acknowledgements Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the U.S. DOE under contract no. DE-AC0494AL85000. References [1] C.V.J. Beenaker, H. van Houton, in: H. Ehrenreich, D. Turnbull (Eds.), Semiconductor Heterostructures and Nanostructures, Solid State Physics, vol. 44, Academic, New York, 1991, and references therein.
[2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
B.J. van Wees et al., Phys. Rev. B 38 (1988) 3625. H. Van Houten et al., Semicond. Sci. Technol. 7 (1992) B215. P. Streda, J. Phys.: Condens Matter 1 (1989) L1025. M. Tsaousidou, P.N. Butcher, Phys. Rev. B 56 (1997) R10004. J. Lee, H.N. Spector, J. Appl. Phys. 54 (1983) 3921. G. Fishman, Phys. Rev. B 34 (1986) 2394. H. Sasaki et al., Appl. Phys. Lett. 51 (1987) 1934. S. Das Sarma, X.C. Xie, Phys. Rev. B 35 (1987) 9875. K.F. Berggren et al., Phys. Rev. B 37 (1988) 10118. H. Akira, T. Ando, Phys. Rev. B 43 (1991) 11676. B. Tanatar, A. Gold, Phys. Rev. B 52 (1995) 1996. S.K. Lyo, D. Huang, Phys. Rev. B 64 (2001) 115320. S.K. Lyo, D. Huang, Phys. Rev. B 66 (2002) 155307. S.K. Lyo, D. Huang, Phys. Rev. B 68 (2003) 115317. S.K. Lyo, Phys. Rev. B 50 (1994) R4965. S.K. Lyo and D. Huang (in press). S.K. Lyo, Phys. Rev. B 60 (1999) 7732. J.S. Moon et al., Phys. Rev. B 60 (1999) 11530. T. Holstein, Annals of Phys. (NY) 29 (1964) 410. S.K. Lyo, Phys. Rev. B 38 (1988) R6345. S.K. Lyo, J. Phys.: Condens. Matter 8 (1996) L703; D. Huang, S.K. Lyo, J. Phys. Condens. Matter 12 (2000) 3383.
Multisublevel transport and magnetoquantum oscillations in quantum wires q S.K. Lyo b
a,*
, D.H. Huang
b
a Sandia National Laboratories, Albuquerque, NM 87185, USA Air Force Research Laboratory (AFRL/VSSS), Kirtland Air Force Base, Kirtland, NM 87117, USA
Received 20 November 2003; accepted 30 January 2004 Available online 9 April 2004
Abstract Transport in one-dimensional (1D) wires show interesting properties due to restricted 1D phase space. We calculate 1D transport properties to a controlled accuracy and examine the effect of combined elastic and inelastic scattering on the conductance G and the thermoelectric power S of a single-quantum-well wire (SQWR) and tunnel-coupled double-quantum-well wires (DQWR’s) in a perpendicular magnetic field. The field dependence of G and S are strikingly different in ballistic and diffusive regimes. Interwell tunneling in DQWR’s distorts the conductance and the TEP drastically from those of a SQWR and yields a sign anomaly for S. Ó 2004 Elsevier B.V. All rights reserved. PACS: 72.15.Eb; 72.15.Jf; 72.23.Ad; 73.63.Hs; 73.63.Nm Keywords: Ballistic; Diffusive; Conductance; Thermopower; Sign anomaly
1. Introduction Transport properties of quasi-one-dimensional (1D) doped semiconductor structures are of current interest for many novel physical phenomena and possible applications for new devices. An earliest form of 1D single-quantum-well wires (SQWR’s) was realized in the so-called quantum point contact illustrated in Fig. 1(a) where the channel length is very short, of the order of a fraction of a lm. Electrons pass through this wire ballistically at low temperatures (T ’s) between the two 2D gases under a DC bias. The conductance G is quantized and decreases in steps of 2e2 =h in a spin-degenerate system (assumed in this paper) when the channel width is reduced gradually [1]. Here, e is the absolute electronic charge and h is Planck’s constant. Similar quantized G steps were observed in SQWR’s as a function of a magnetic field B applied in the perpendicular (i.e., z) q Original version presented at QTSM&QFS 2003 (Quantum Transport Synthetic Metals & Quantum Functional Semiconductors), Seoul National University, Seoul, Korea, 20–22 November 2003. * Corresponding author. Tel.: +1-505-844-3718; fax: +1-505-8441197. E-mail address: [email protected] (S.K. Lyo).
1567-1739/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.cap.2004.02.004
direction shown in Fig. 1(a) [2]. Recently, ballistic thermoelectric power (TEP) was observed in SQWR’s [3] and was also studied theoretically in zero B [3–5]. Good quality (i.e., high-mobility) long-wire samples (e.g., >20 lm) are now within reach. In long wires, transport is expected to be diffusive. Our interest lies in high-mobility wires where the wire length is longer than the mean-free-path but is shorter than the localization length. The present discussion is limited to the Fermi liquid model. We show in this paper that the B-dependences of the diffusive G and the TEP are strikingly different from those of the ballistic case and can provide insight into the basic nature of the transport. We also show that the transport properties of a tunnel-coupled double-quantum-well wires (DQWR’s) are very different from those of SQWR’s in perpendicular magnetic field including a sign reversal for the ballistic as well as the diffusive TEP. For diffusive G, a number of recent studies have focused on the effect of elastic scattering [6–13]. Here, scattering by impurities and interface roughness plays a dominant role for the momentum relaxation. Recently, we assessed the relative contribution from phonon scattering and found that the latter can be important even at relatively low temperatures in multi-sublevel structures.
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S.K. Lyo, D.H. Huang / Current Applied Physics 4 (2004) 491–496
In these systems, inter-sublevel electron–phonon scattering is important for the momentum-relaxation processes. In this case, the standard quasi-elastic-scattering approximation does not yield a reliable result because the phonon energy can be comparable to the electron energy. Therefore, a rigorous method for treating combined elastic and phonon scattering is necessary. We describe a formalism which yields G and the TEP to a desired accuracy for a general quasi-onedimensional electronic structure and present the result obtained from this formalism [14,15].
2. Effect of the magnetic field on the energy dispersion We consider only the ground sublevel for the z-confinement in the growth direction. For the channel confinement in the x direction, we consider parabolic confinement with many low-lying sublevels spaced evenly by hxx occupied as shown in Fig. 1(a). In a perpendicular magnetic field Bkz, the energy dispersion is given for a SQWR by 2 k 2 h hXx þ ; enk ¼ n þ 12 ð1Þ 2m wherep k ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi is the wave number along the wire, n ¼ 0; 1; . . ., ffi Xx ¼ x2c þ x2x , xc ¼ eB=m c, m ¼ m =½1 ðxc =Xx Þ2 , and m is the zero-field effective mass [10,13]. The sign of the displacement Dxk / k of the center of the harmonic wave functions /n ðx Dxk Þ induced by the Lorentz force depends on the sign of k. The levels are depopulated successively as m increases with B. For DQWR’s, interesting effects occur when B is in the channel plane in the x direction as shown in Fig. 1(b). In this case, the role of B is to shift the origin of k of the energy parabola of the second wire (i.e., QW2) by dk ¼ d=‘2c relative to that of the first wire (i.e., QW1),
wherepdffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi is the center-to-center distance of the QWs and ‘c ¼ hc=m x. It also localizes (i.e., separates) the wave functions for k and k at the two opposite Fermi points (e.g., black dots in (b)) into separate wires through the Lorentz force. Tunneling couples only the parabolas belonging to the same quantum number n. The energy dispersion in Fig. 1(b) shows the anticrossing between the two parabolas shifted by dk (shown in gray background) for n ¼ 0 for symmetric DQWR’s [16]. Here, the two parabolas split into the upper and lower branches separated by an anticrossing gap DSAS between the symmetric and the antisymmetric states at k ¼ 0. The parabolas move away from each other as dk / B increases, pushing the gap through the chemical potential l. The inset in Fig. 1(b) shows the relative positions of l with respect to the gap for three B’s. This level splitting occurs for each n. The levels become depopulated as B increases. The energy dispersion and DSAS depend on the QW width LW , the barrier width LB , and the barrier height Vo .
3. Ballistic transport The electron-diffusion TEP Sd is the ratio of the heat current and the charge current of the electrons divided by T in the presence of a linear DC field. Ballistic electron-diffusion Sd is given, for symmetric electronic structures, by [17] kB X X Sd ¼ Cm;c ½bðem;c lÞf ð0Þ ðem;c Þ eF m c þ lnðebðlem;c Þ þ 1Þ ; where f is the Fermi function, b ¼ 1=kB T , and XX F ¼ Cm;c f ð0Þ ðem;c Þ: m
drain contact
(a)
(b) Z Y
µ1 µ2
wire
µ3
wire-1
Cu rr en t
Cu rr en t
µ
B
wire-2
X B
source contact
Fig. 1. (a) A schematic diagram of a SQWR and the energy dispersion. A narrow channel is formed by applying a negative bias on the top split metallic gate, not shown. (b) A schematic diagram for DQWR’s and the energy dispersion. Electrons tunnel between the wires through the Alx Ga1x AS barrier in the z direction. A magnetic field B lies in the x direction for the DQWR’s and in the z direction for the SQWR. Black dots denote the Fermi points.
ð2Þ
ð0Þ
ð3Þ
c
Here, m- and c-summations indicate summing over all the energy dispersion curves and over all the energy extremum points on each curve, respectively. The quantity em;c is the extremum energy. For a given curve m, Cm;c ¼ 1 for a local energy minimum point and Cm;c ¼ 1 for a local energy maximum point. The quantity F equals the number of the pairs of the Fermi points at T ¼ 0 and is related to G by G ¼ 2e2 =hF . The results in Eq. (2) reduce to he earlier results obtained for SQWR’s with a single minimum point for each the energy dispersion curve [1,4]. In Fig. 2(a), we show G and Sd of a SQWR as a function of B when several levels are occupied at B ¼ 0 for two T ’s. The sample parameters and the electron density n1D are given in the inset. We find spikes for the TEP just before a level is depopulated near the quantum steps of G. These spikes broaden as T is raised. Fig. 2(b) shows Sd and G for DQWR’s for two T ’s for the case of
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(a)
(b)
Fig. 2. Ballistic Sd and G for two T ’s (a) in SQWR with three levels occupied at B ¼ 0 and (b) in DQWRs with only n ¼ 0 occupied. The parameters are defined in the text.
large hxx where only the tunnel-split n ¼ 0 doublet is occupied at B ¼ 0 as indicated in Fig. 1(b). The conductance has a minimum and is V-shaped in striking contrast with the behavior shown for a SQWR in (a). The conductance equals the number of the pairs of the Fermi points in units of 2e2 =h, namely 2 for l ¼ l1 , 1 for l ¼ l2 , and 2 for l ¼ l3 in Fig. 1(b). The TEP shows a surprising feature: it changes sign near the G minimum just before it begins to rise again. At this point, l crosses the local energy extremum (i.e., maximum) point at k ¼ 0 of the lower branch doubling the number of the Fermi points and G. The dispersion is hole-like (i.e., inverted) here, yielding a sign reversal for Sd . When many levels are occupied for small hx x DSAS , G of DQWR’s follows a V-shaped dependence on B as shown in Fig. 3 (left axis) for two T ’s. A similar Bdependence was observed earlier in DQWR’s [19]. The inset illustrates the tunnel-split sublevels for this case at a certain field near the G minimum. The B-dependence of G can be explained in a similar way as in Fig. 2(b) by
Fig. 3. Ballistic G (left axis, upper curves) and Sd (right axis, lower curves) for two T ’s in DQWRs with many levels occupied. The parameters are defined in the text. The inset shows the tunnel-split energy dispersion at B near the G minimum.
accounting for the Fermi points from all n into account [18]. For SQWR’s, G decreases stepwise monotonically as a function of B as in Fig. 2(a) and is not shown [2]. Ballistic Sd is also displayed for the DQWR’s at T ’s (right axis). Note that Sd shows positive peaks initially when l crosses the bottom of the upper branches successively for lower n’s. It changes sign near the G minimum just before it begins to rise again. At this point, l crosses the local energy maximum point at the top (at k ¼ 0) of the lower branch increasing the number of the Fermi points and thus G. For this point, the dispersion is hole-like, yielding the sign reversal for Sd . From this point on, l keeps crossing similar local energy maximum points belonging to lower n’s, producing hole-like successive negative peaks for Sd . The Sd peak near the G minimum at 3.4 T is large because Sd is inversely proportional to G. The discussions so far are restricted to Sd . The phonon-drag contribution Spd should be small in a QPC because electrons have little time to interact with the phonons in the ballistic channel, although a recent paper concluded otherwise [5]. There has been no experimental evidence for the ballistic phonon-drag TEP.
4. Diffusive transport 4.1. Brief description of theoretical approach in 1D The theory to be described here applies to arbitrary 1D band structures including SQWR’s and DQWR’s with or without perpendicular magnetic fields studied in the two previous sections. The distribution function ð0Þ deviates from the Fermi function fk according to ð0Þ ð0Þ0 fk ¼ fk þ gk kB T ½fk eE in a linear DC electric field Eky. Here, k is the wave number in the y direction and
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the prime denotes the first energy derivative. The quantity gk satisfies the Boltzmann equation [20]: 2p X 2 vk þ hjIk0 ;k j iðgk0 gk Þdðek ek0 Þ þ Pk ¼ 0; ð4Þ h k0 where h i denotes the configurational average, vk is the Bloch velocity, Pk is the contribution from electron– phonon interaction (EPI), and Ik0 ;k is the matrix element for elastic scattering. We assume interface-roughness 2 scattering for jIk0 ;k j studied earlier [13]. The contribution from EPI Pk is given by [20] 2p X 2 ðÞ ð0Þ Pk ¼ jVsq j ½fk0 þ nsq ðgk0 gk Þ h k0 ;s;q dðek ek0 hxsq Þdk0 ;kqy Dðqx ; qz Þ: ðÞ fk0
ð5Þ
ð0Þ fk0 ,
Here, ¼ 1=2 1=2 hxsq is the phonon energy ð0Þ of mode s and wave vector q, and nsq ¼ nð0Þ ð hxsq Þ is the boson function. The conservation factor Dðqx ; qz Þ for perpendicular qx , qz and EPI Vsq are given in Ref. [15]. In Eqs. (4) and (5), curve indeces m, m0 accompany k, k 0 , respectively, and will be suppressed for simplicity. Eq. (4) can be solved for gk to a desired accuracy in 1D by subdividing the energy near the Fermi level into a sufficiently small interval de kB T over a sufficiently large range kB T . Scattering between points k, k 0 is mapped into scattering between all the discrete points lying on the intersection of the above discrete energies and the dispersions curves, yielding Ag ¼ B type equation which can be inverted for g. Here, g is the column vector which contains all the gk values on the above discrete points and A, B are known matrices involving the scattering parameters. The conductance and then heat current by electrons are then obtained from g. A detailed but somewhat complicated procedure is given in the above Ref. [15]. The electron current displaces the phonon distribution from equilibrium through EPI yielding a phonon heat current. The latter is a linear function of g. The phonon-drag contribution Spd to S is the ratio of the phonon heat current and the charge current divided by T [14].
Fig. 4. Diffusive G vs. B for a SQWR with n ¼ 0 and n ¼ 1 occupied at B ¼ 0. The dashed horizontal lines in the inset show l ¼ l1 in n ¼ 1 and l ¼ l2 in n ¼ 0. The thick horizontal (tilted) double-headed arrow illustrates intra-level (inter-level) roughness (phonon) scattering.
density of states (DOS). The rapid decrease of G at high B’s in Fig. 4 is due to this effect. It is seen that G decreases rapidly as a function of T even at low T ’s due to inter-level one-phonon scattering illustrated in the in-set. The thermal population effect with elastic scattering alone (i.e., without electron–phonon scattering) is found to yield large G as shown by the gray curve. When many levels are occupied for small hxx , G shows oscillations yielding peaks after each depopulation. Fig. 5 displays G for 1lm long symmetric DQWRs with only the n ¼ 0 doublet occupied with n1D ¼ 6:5 105 cm1 , l ¼ 3:79 meV, well-depth 270 meV, and barrier-width LB ¼ 40A. well-widths LW ¼ 80A, The abrupt rise in the range 3 < B < 8 T when l is in-
4.2. Numerical evaluation For a numerical evaluation, we employ standard parameters for the EPI including deformation-potential and piezoelectric interactions [21]. For interface-rough layer fluctuation ness scattering, we assume db ¼ 5 A with a Gaussian correlation length K ¼ 30 A. Fig. 4 shows G for a 10 lm long SQWR with two levels populated at B ¼ 0 with a n1D ¼ 106 cm1 , l ¼ 4:81 meV, well-depth 270 meV, and a well-width The conductance rises abruptly around 2 T LW ¼ 210A. due to the depopulation of the n ¼ 1 level at low temperatures (e.g., 1 K). The conductance is small when l is at the bottom of any level because of the large 1D
Fig. 5. Diffusive G vs. B for DQWRs. The inset displays the energy dispersion of two tunnel-split branches with only the n ¼ 0 level occupied. The dashed horizontal lines show l ¼ l1 above the gap, l ¼ l2 inside the gap, and l ¼ l3 below the gap. The thick horizontal (tilted) double-headed arrow illustrates roughness (phonon) scattering.
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side the anticrossing gap (e.g., l ¼ l2 ) arises from the fact that back scattering (illustrated by the doubleheaded horizontal arrow in the inset) is quenched between the initial and final state due to the B-induced localization (i.e., separation) of the initial and final wave functions [22]. At high B’s (e.g., B ¼ 8 T) and low T ’s, l falls below the gap as shown by l ¼ l3 in the inset, reducing G abruptly. At high T ’s, however, the gap region is thermally populated. Here, the separation between the initial and final states is even more severe because of large B, yielding enhanced G as shown for 8 K. However, at 16 K, phonon scattring reduces G as shown. For the TEP, we examine briefly several interesting features unique to 1D SQWR structures. Fig. 6 displays S as a function of T for B ¼ 0 for large hxx ¼ 50 meV and n1D ¼ 106 cm1 when only n ¼ 0 is occupied. The upper and right axes are linear scale for Sd which follows the well-known quasi-linear T -dependence, while Spd is given by the lower and left axes in a semi-log scale. This curve shows Spd / expðb hx2KF Þ, where x2KF is the phonon energy for the 2KF back scattering between the two Fermi points shown by the black dots in the inset. This is in contrast to the power law behavior of Spd in higher dimensions where the Fermi surface is a continuum. Note that Spd Sd except at low temperatures. Fig. 7(a) displays Sd and Spd as a function of n1D . The minimum in Spd occurs just before the occupation of the second level n ¼ 1 (marked by a vertical arrow) where the resonant 2KF phonon energy reaches a maximum. The same kind of peak occurs before the filling of the next level. Also, interesting is the sign change of Sd with a sharp variation in magnitude slightly below the occupation of the second level n ¼ 1. When l is at this
495
(a)
(b)
Fig. 7. Spd (solid curve, left axis) and Sd (dotted curve, right axis) as a function of (a) n1D and (b) B at T ¼ 2 K. Parameters not shown in the insets are given in the text. The second level begins to be populated/ depopulated at the position marked by the vertical arrow in (a)/(b). NF is the number of the Fermi points.
point, G decreases rapidly with increasing l (i.e., dG=dl < 0) because of the large DOS just inside the n ¼ 1 level. As a result, more current flows from below l, yielding the sign reversal. However, dG=dl > 0 is recovered as soon as the level n ¼ 1 is occupied, yielding a normal sign. The rapid variation of dG=dl in this range is responsible for the large variation of Sd . A very similar feature is shown in (b) where the position of l is varied by B. One difference is that Spd keeps decreasing at high fields after the depopulation of level n ¼ 1. This follows from the B-induced reduction of the overlap of the initial and final wave functions involved in 2KF phonon scattering, which weakens the phonon scattering rate for Spd .
5. Conclusions Fig. 6. Spd (solid curve, left-bottom axis) vs. 1=T in a semi-log scale and Sd (dotted curve, right-top axis) vs T in a linear scale at B ¼ 0. Only the ground level n ¼ 0 is populated.
We showed that the B-dependence of the diffusive G, the TEP, and their quantum oscillations are strikingly different from those of the ballistic transport in SQWR’s
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and DQWR’s. We also demonstrated interesting sign anomalies for the electron-diffusion TEP in ballistic DQWR’s and diffusive SQWR’s. Diffusive DQWR’s also show the sign anomaly. Acknowledgements Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the U.S. DOE under contract no. DE-AC0494AL85000. References [1] C.V.J. Beenaker, H. van Houton, in: H. Ehrenreich, D. Turnbull (Eds.), Semiconductor Heterostructures and Nanostructures, Solid State Physics, vol. 44, Academic, New York, 1991, and references therein.
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