Conductance of a quantum wire at low electron density

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1 Conductance of a quantum wire at low electron density Konstantin Matveev Materials Science Division Argonne National Laboratory Argonne National Laboratory Boulder School, 7/25/2005

2 1. Quantum wires and Quantum Point Contacts 2. Experiments: Outline quantization of conductance, 0.7 anomaly, 0.5 plateau 3. Theoretical ideas: spin polarization, Wigner crystal

3 One-dimensional conductors Top view: One-dimensional electron system

4 Quantum Point Contacts 2D electron gas confined by a split gate Experiment: B.J. Van Wees et al., 1988; also, D.A. Wharam et al., 1988 Conductance (1/Resistance) vs. V g Electron motion across the channel is quantized

5 Conductance of a Quantum Wire Applied bias results in uncompensated current I = e nv F Electron density: n = 1 h p F = 1 h dp F de F ev = e hv F V Current: I = e2 h V Conductance: G = e2 h

6 Shape of the conductance steps L. I. Glazman, G. B. Lesovik, D. E. Khmel'nitskii, R. I. Shekhter, 1988 The energy of the transverse motion of electrons is inversely proportional to the square of the width d(x) of the channel V (x) π2 h 2 2md 2 (x) Constriction creates a potential barrier for electrons How does the barrier affect the conductance?

7 Landauer formula Only a fraction of electrons cross the barrier! Current: I = e nv F T (E F ) T (E F ) is the transmission coefficient of the barrier at the Fermi energy Conductance: G = e2 h T (E F )

8 Transmission coefficient Parabolic-potential approximation: V (x) = 1 2 mω2 x 2 T (E F )= exp 1 ³ 2πE F hω +1 E. C. Kemble, 1935 Conductance as a function of the Fermi energy G 0 = 2 e2 h 2 accounts for electron spins

9 Shape of the conductance steps Experiment: Kristensen et al., 1998 Good agreement with theory at low temperatures; Noticeable deviations at higher temperatures.

10 0.7 Anomaly Conductance vs. gate voltage at different temperatures: Thomas et al., 1996 Cronenwett et al., 2001 As the temperature grows, the conductance develops a shoulder near 0.7 2e2 h

11 0.5 Plateau Thomas et al., 2000 Reilly et al., 2001 Several experiments show a plateau of conductance at This 0.5 plateau tends to appear in longer samples 0.5 2e2 h

12 Magnetic field dependence Conductance vs. gate voltage at different B: Thomas et al., 1996 Cronenwett et al., anomaly evolves toward the spin-polarized plateau at 0.5 2e2 h Spin-polarization effect? Wang & Berggren, 1997 Spivak & Zhou, 2000 Flambaum & Kuchiev, 2000 Hirose, Li & Wingreen,

13 One-dimensional ferromagnetism? Two-dimensional electron gas is paramagnetic. Is one-dimensional electron system ferromagnetic? No! The ground state of a system of one-dimensional spin-1/2 fermions has the lowest spin possible. E. Lieb and D. Mattis, 1962; D. Mattis, The Theory of Magnetism.

14 Interacting one-dimensional electrons: Bosonization One-dimensional electron liquid is an elastic medium H = p2 2mn mnv2 F µ du dx 2 u(x) is displacement of the medium, p(x) is momentum density, n is electron density. In a non-interacting system the waves propagate at Fermi velocity v F In the presence of interactions v F is replaced by plasmon velocity s

15 Quantized resistance Applied current: u(0,t) = u 0 cos ωt I = en u Density of elastic energy: hhi = hmn u 2 i = m e 2 n I2 Energy carried away by plasmons in unit time: Resistance: W =2hHiv F = 2mv F e 2 n I2 = I 2 R R = 2mv F e 2 n h 2e 2 µ n = 2k F π

16 Interacting electrons One-dimensional model with non-interacting leads [Maslov & Stone; Ponomarenko; Safi and Schulz, 1995] In the dc limit ω 0 plasmon wavelength λ Plasmons are emitted in the non-interacting region Resistance is still h 2e 2

17 Wigner crystal Compare kinetic and interaction energies: Coulomb energy dominates: One-dimensional electrons at low density n 0 E kin = h2 k 2 F 2m n2, E Coul = e2 r n. Electrons form a Wigner crystal and stay near their lattice sites Density excitations are elastic waves in the crystal (plasmons) H = p2 2mn nms2 µ du dx 2

18 Spin coupling Weak exchange due to tunneling through the Coulomb barrier Ã J exp π! nab To first approximation the spins do not interact Antiferromagnetic spin chain: Hσ = X l JS l S l+1, J > 0 Note: charge and spin are not coupled!

19 Spin-charge separation? Exchange constant depends on electron density. Thus J=J(l). The density at site l depends on the number of electrons that moved through the wire: J=J[l+q(t)]. Electric current (a charge excitation) affects coupling between the spins!

20 Spin contribution to the resistance The spin Hamiltonian H σ = X J[l + q(t)]s l S l+1 depends on q(t) I. Thus the applied current excites not only charge, but also spin waves: W = I 2 h 2e 2 + I2 Rσ = I 2 R These processes give an additive contribution to the resistance R = How do we find R σ? h 2e 2 + R σ

21 Approximation: XY model Hσ = X l J l ³S x l Sx l+1 + Sy l Sy l+1, J z 0 The Jordan-Wigner transformation converts the problem to that of non-interacting fermions: X Hσ = 1 2 l J l ³ a l a l+1 + a l+1 a l J is large in the leads, but small in the constriction

22 Jordan-Wigner transformation Raising and lowering operators S l ± = Sl x ± isy l can be expressed in terms of fermionic operators: S + l = a l 1 l exp X iπ a j a j, Sl z = a l a l 1 2 j=1 Then the Hamiltonian of a spin chain H = X l [J (S x l Sx l+1 + Sy l Sy l+1 )+J zs z l Sz l+1 ] transforms to H = X 1 ³ 2 J a l a l+1 + a l+1 a l l + Jz µ a l a l + 1 µ a 2 l+1 a l XY model: Jz =0 Non-interacting fermions

23 Resistance due to scattering of excitations J l = J[l + q(t)] Exact result: Rσ = h 2e 2 n F (J) 1. At T J very few fermions are scattered by the barrier: Rσ e J/T The fermions near the Fermi level pick up energy from the oscillating barrier, leading to dissipation W = I 2 Rσ R σ = h 4e 2 Long wire limit Z Ã n F E! [1 T (E)]dE 2. At T À J most excitations are backscattered, and resistance saturates: Rσ = h 4e 2

24 Activated temperature dependence Kristensen et al., 2000 The temperature dependence of 0.7 structure fits to µ G = 2e2 h G A exp Activation temperature T A T T A 1K

25 Isotropic exchange The true excitations of the spin chain are spinons, with the dispersion relation ²(q) = πj 2 sinq, 0 <q<π J = J[l + q(t)] Spinons with energies below πj/2 pass through; all others are reflected and contribute to R σ

26 Conductance of a Wigner crystal wire At T J only an exponentially small fraction of the spinons are reflected: Rσ exp µ πj, T J 2T At higher temperatures R σ grows, and saturates at Rσ = h 2e 2, The conductance of the wire G = 2e 2 1 h 2e 2 + R σ T À J e changes from at T J to 2 at T À J. h h

27 High temperature At J T all spin excitations are reflected by the barrier (c.f. XY model) Analogy: high B When spins in the wire are polarized, no excitations pass through; expect G = e2 h B Boundary conditions at x =0 φσ = πq(t) 2 Check with bosonization: In the leads ψ = e i(φ ρ±φ σ )/ 2 φρ = πq(t) Rσ = Rρ = h 2e 2 2

28 Conclusions 1. At low density of electrons in a quantum wire, they form a Wigner crystal 2. The spins of electrons in the Wigner crystal are weakly coupled, and the propagation of spin excitations through the wire is impeded. 3. As a result, the conductance of the wire decreases by a factor of No spin polarization is required!

Conductance of a quantum wire at low electron density

PHYSICAL REVIEW B 70, 245319 (2004) Conductance of a quantum wire at low electron density K. A. Matveev Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439, USA and Department