Quantum Transport in One-Dimensional Systems

Transcription

1 Lecture 2 Quantum Transport in One-Dimensional Systems K J Thomas Department of Physics Central University of Kerala Kasaragod, Kerala Harish Chandra Research Institute, Allahabad February 23, 2016

2 ONE-DIMENSIONAL PHYSICS Ballistic, quasi-ballistic and diffusive transport The transport in low-dimensional electron systems can be classified into three regimes: diffusive, ballistic and quasi-ballistic, as illustrated below. In the diffusive regime both the length L and the width W of the conductor are much larger than the electron mean free path, W,L > l e. The transport properties are dominated by elastic scattering and can be expressed by classical Drudetheory in terms of local quantities such as mobility or conductivity, which are well defined in this regime. In the diffusive regime both the width W and length L of the constriction are larger than the mean-free path, W,L > l e.

3 For diffusive conductor, on average, twice as many scattering events will occur if the sample size is doubled in length. This means that there is a direct relationship between the conductivity σ and conductance: If the width of the conductor is smaller than the l e, but the length larger, W < l e < L, the transport is said to be quasi-ballistic. This is an intermediate regime in which both impurity and boundary scattering become important. FIGURE: Quasi-ballistic regime W < l e < L.

4 Ballistic transport occurs when the mean free path is larger than both the width and length of the conductor, l e > W,L. In this case electrons do not scatter from impurities but only from the boundaries of the conductor. The current will be limited because electrons backscatter at the entrance of the constriction and resistance will have a non-zero value. The limiting contact resistance depends on the geometry of the sample, and not its length; therefore the transport properties cannot be expressed in terms of local quantities, such as conductivity, as is the case in diffusive transport. Alternative approach known as the Landauer-B uttikerformalism is used -it describes the transport properties of the ballistic conductor in terms of the transmission probabilities of the quantum channels at the Fermi level. FIG. 13 (c) In the ballistic regime W,L < l e. Backscattering at the entrance of the constriction results in a non-zero resistance in the ballistic regime.

5 The two-dimensional electron gas (2DEG) can be laterally confined using various lithographic techniques. If a width of constriction is small enough it forms a one dimensional conductor, known as a quantum point contact, which connects the regions of 2DEG. A common technique for making QPC is by deposition of metallic gate on the surface of the heterostructure, usually by electron-beam lithography. By applying a voltage to a suitably-shaped gate electrodes, the electron gas can be locally depleted and the 2DEG can be shaped into a wire or quantum dot. The simplest 1D device is fabricated using a split-gate, as shown below. FIGURE Split-gate device of length L and width W created by metal gates deposited on the surface of the device; d is the depth of the 2DEG below the surface. With the application of a negative voltage Vg to the split-gate the electrons below the gates are depleted (regions shown in black) to leave a 1D channel connecting the two halves of the 2DEG. Typical dimensions are L = W = 0.5 μm, and d = 1000 A.

6 Split-gate devices A split-gate is a gate across the Hall bar which is split in two by a gap of width W and length L. Diffusive transport occurs if L,W > l, and ballistic transport occurs if L,W < l. 1. Take a clean 2DEG (high mobility). 2. Use electron beam lithography to define a split-gate device with dimensions W 1μm and L = 1 μm. 3. Cool device to low temperatures, for example T = 50 mk, to reduce electronphonon scattering. 4. Measure the conductance G as a function of the gate voltage. As Vg is made more negative the constriction is squeezed and the conductance characteristics G(V g ) fall monotonically. At pinch-off the channel is completely cut off, where G 0 and R = G 1.

7 Two Dimensional Electron Gas µ e = cm 2 V -1 s -1 Energy 2DEG n e = cm -2 l t = 46 µm l φ = 24 μm λ F = 45 nm

8 Quantisation and the Landauer formalism Measurements of the conductance G = di/dvreveal that it does not fall in a smooth fashion, but instead exhibits plateaus and risers, as shown in Figure. On the plateaus the conductance is quantised(to within a few%) in units of 2e 2 /h. FIG.Top left: Plan view of splitgate device. Top right: Cross-sectional view in the xzplane. Main: The conductance characteristics G(V g ) measured at 60 mk, showing plateaus quantisedat integer multiples 2e 2 /h; in a strong magnetic field the spin degeneracy is lifted. The inset shows the parabolic confinement in the y- direction that is encountered close to pinch-off.

9 The 1D device is attached to two external reservoirs: a left andright chemical potentialμ L = E F +ev sd /2 and μ R = E F ev sd /2, see Fig. below. If a small voltage V sd is applied between the two reservoirs, the resulting difference in the chemical potentials ev sd = μ L μ R will drive a current through the device. Transport will take place through the channels in the energy range from μ s -(a few k B T) to μ d + (a few k B T). For a given mode n, the current from the source drain within deis given by: while the current from drain source is given by FIG. Two-terminal electrical measurements of a device, where the chemical potentials of the source and drain (which are far away from the device) are μ L = μ s = E F +ev sd /2 and μ R = μ d = E F ev sd /2, respectively. These open up a window ev sd about the Fermi energy E F for electrical transport.

10 where g(e) is the density of states in the leads. g(e)f(e, μ) is the number of electrons at energy, E and ev(e) is the current carried by each electron, where v(e) is the group velocity. The group velocity is given by v(e) = 1/ћ de/dk, and the 1D DOS is given by g 1D (E) = 2/π dk/de, therefore the product is 2/(ћπ). This is an important result for noninteracting 1D electrons: the density of states and the Fermi velocity multiply to give only a constant. The net current in a single mode is the sum of two currents Assuming that at T = 0 K the Fermi-Dirac functions become step functions and The conductance of a single mode is

11 Equation 2.6 seems contradictory -a finite conductance (non-zero resistance) is obtained for a ballistic conductor in which there is no scattering. (Imrypointed out that the finite resistance of the ballistic conductor arises due to the contact resistance at the entrance of the conductor). A more general view of conduction through a 1D system can be understood within the Landauer formalism. Inherent disorder (impurities) present in real conductors cause scattering of incident electrons resulting in a fraction of transmitted incident electrons. In the Landauerapproach, the current through a finite system connected to several (usually semi-infinite) leads is expressed in terms of the transmission and reflection probabilities. The flow of electrons from one reservoir to the other can be described using the notion of incident, reflected and transmitted fluxes. Due to the lateral confinement, the conductor has only few transmitting modes electrons with the non-matching k-vector backscatter at the entrance giving rise to a finite resistance.

12 The physical picture of the Landauer formula can be understood as a result of standing waves set in the transverse direction due to lateral confinement. The transmission amplitude of a single channel is t. FIG. 17 Two-terminal device coupled via ideal leads to reservoirs with chemical potentials μ L and μ R. Within the con- ductorthe electrons are scattered elastically by S, whereas all inelastic processes occur in the reservoirs, which are modelled as degenerate gases with functions, f L and f R. I, T and R are the incident, transmitted, and reflected electron fluxes. where t 2 = T is the transmission probability, and which is something that can be calculated quantum mechanically. For many channels need to sum over them. Including a factor for spin degeneracy (s = 1/2) where in this picture there is no mode mixing.

13 Saddle-point model A realistic model for the electrostatic potential V (x, y) in the QPC is given by saddle-point potential: where x and y are the coordinates in the plane of the 2DEG, V 0 is the potential at the centre of the saddle, ω x is the parameter that describes the curvature of the potential barrier, and ω y is the parameter that characterises the lateral confinement (ћω y value gives the spacing between the 1D subbands). (b) (a) 2D 1D y 2D X

14 The associated Hamiltonian is separable into a transverse wavefunction, associated with the energies E n = V 0 + ћω y (n + 1/2 ), n = 0, 1, 2,..., and a wavefunction for the motion in x-direction in an effective potential V 0 + ћω y (n + 1/2 ) 1/2mω 2 xx 2. All the channels with the energy E n < E F will conduct, and those above will not. The transmission probability is given by: where ε n = 2 (E E n ) /ћω x is a dimensionless energy. δ mn is the Kronecker delta where δ nm = 1 if n = m, and δ nm = 0 otherwise.

15 Non-linear conductance When a DC source-drain voltage V sd is applied across an adiabatic constriction (one whose width varies slowly) it defines an energy strip about the Fermi energy. It can be shown that the electron states within this strip contribute to the net current through the constriction. If we assume that the applied source-drain V sd is dropped equally on both sides of the bottle-neck of the constriction, then the source (s) and drain (d) chemical potentials are given by: The current I in a single subbandis given by where T (E) is transmission probability and E n (V sd ) is the energy of the subband.

16 Let us consider a few simple cases for different alignments of the source and drain potentials. Figure shows that +kstates are occupied up to the source electrochemical potential μ s, whereas kstates are occupied up to electrochemical potential of the drain μ d. We can calculate the conductance of three different alignments of μ s and μ d shown in Fig. below. If both chemical potentials lie within the same subband, as depicted in Fig. (a), the current in that subbandis given by: Summing over all the subbands, the conductance is FIG. (a) Chemical potentials of the source μ s = E F +k B T and drain μ d = E F k B Tlie within the same subband giving G = N(2e 2 /h).

17 FIG. (b) When the number of conducting subbandsin opposite directions differs by one, half-plateaus are observed. The +k states in the Nth subbandcontribute e 2 /h to the conductance, while all N 1 subbandsconduct in both directions, contributing (N 1)2e 2 /h to conductance. The total conductance equals G = (N 1)2e 2 /h + e 2 /h. Figure (b) depicts the situation where there are N states conducting from right to left and (N 1) states conducting from left to right. N 1 subbandsconduct in both directions and they contribute to the conductance to give There are only +k states conducting in the Nth subbandand the current is carried by electrons in the strip between μ d and the bottom of the subband,

18 contributing to the conductance. The total conductance in all N subbands For certain source-drain voltages the conductance will exhibit half-plateaus, lying in between the conventional integer ones. From the G(V sd ) traces it is possible to obtain the subbandenergy spacing. If the separation between the electrochemical potential of the source and the drain is equal to the subband spacing integer, μ s μ d = ev sd, the normal plateaus will disappear altogether, and only half-plateaus will be observed.

19 Half Plateaus in a Ballistic 1D Devcie, 0.75 μn wide and 0.4 μn long Subbandspectroscopy of 1D levels. Read the crossing point value of the two dark lines (edge of subband aligning with the chemical potential of the source and drain) V sd which is equal to the subbandenergy difference between the two adjacent subbands.

20 Enhanced g-factor in a One-Dimensional Quantum Wire

21 The 0.7 Structure Thomas et al., Phys. Rev. Lett. 77, 135 (1996) Thomas et al, PRB (1998), PRB (2000)

22 Spontaneous Spin Polarization? Interaction is more important for lower density (r s =E ee /E F ~1/n l ) Gold, Calmels(1996), Berggren (1996) Absence of polarized ground state in 1D Leib and Mattis(1960) Ideal 1D model is perhaps not relevant! Wang and Berggren. PRB54 (1996) 14257

23 Other Models... A.Spontaneous splitting of the barrier due to spin polarization Lieb-Mattistheorem not necessarily violated. B. Scattering off localised S=1/2 state in the QPC Kondo model B C.No spin polarisation required. Reflection of spin excitations in a 1D (antiferromagnetic) Wigner crystalgives rise to suppression of conductance. Thomas et al., PRL 77 (1996) 135 Gold, Calmels Phil. Mag. Lett.74 (1996) 33 Wang, BerggrenPRB 54 (1996) R14257 Rokhinsonet al., PRL 96 (2006) Cronenwettet al., PRL 88 (2002) Meir et al., PRL 89 (2002) Matveev, PRL 92, (2004)

24 Conductance of a 1D Wigner Crystal

25 Wigner Crystallisation The possibility that an electron gas might freeze out into a crystalline state was suggested by E Wigner in 1934 occurs in the low-density limit, where Coulomb energy dominate kinetic energy electrons occupy equidistant positions to minimise Coulomb repulsion To be in the Wigner-crystal régime, we need r 0 >> a B 2 ηε = 2 me

26 Spin-incoherent Regime Structure at e 2 /h at weak confinement strengths Linear electron density in the weak-confinement régime was estimated to be n 1D = /cm. This value satisfies n 1D << 1/a B required for the exponential suppression of the exchange energy, i.e., that the mean separation between electrons far exceeds their effective Bohr radius. Using this density, one obtains J/k B ~2.2 mkand E F /k B ~1.6 K following [Phys. Rev. B 70, (2004).]which is consistent with J <<k B T<< E F. Spin-incoherent Luttinger Liquid Physics! W. K. Hew, K. J. Thomas, M. Pepperet al.,phys. Rev. Lett., 101, (2008).

27 Shifting the Channel Laterally

28 Parallel Magnetic Field Dependence of e 2 /h Spin-incoherent regime Spin-coherent regime

29 Temperature Dependence of Spin-incoherent e 2 /h Plateau Weakening of the plateau at e 2 /h Opposite to the behaviour of the 0.7 Structure! Spin- incoherent transport Wigner Crystal Physics

30 Spin : A New Direction? Spintronics: Electronics based on the spin degree of freedom of the electron Scheme: Adding the spin degree of freedom to conventional charge-based electronic devicesor using the spin alone Properties sought after: Non-volatility, Increased data processing speed, Decreased electric power consumption, Increased integration densities compared with conventional semiconductor devices. Technical issues: Efficient injection, transport, control and manipulation, and detection of spin polarization

31 Acknowledgements Ph. D Students MBE Growth E-beam Collaborators Sebastian Hew Luke Smith Abi Graham Michelle Simmons Ian Farrer Dave Ritchie Dan Mace Jon Griffith Geb Jones James Nicholls (RHUL) Sanjeev Kumar Mike Pepper

32 THANK YOU!

33 Reference Books [Davies] The Physics of Low Dimensional Semiconductors: An Introduction, John H Davies, Cambridge University Press (1998). [Sze] Semiconductor Devices: Physics and Technology, S. M. Sze. [Wolf] Nanophysics and Nanotechnology, E. L. Wolf (Wiley- VCH). [Kelly] Low-Dimensional Semiconductors: Materials, Physics, Technology, Devices, M. J. Kelly. Visit (look for Introduction to Semiconductors)