Graphene: Quantum Transport via Evanescent Waves

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1 Graphene: Quantum Transport via Evanescent Waves Milan Holzäpfel 6 May 203 (slides from the talk with additional notes added in some places /7

2 Overview Quantum Transport: Landauer Formula Graphene: Introduction Eigenfunctions Transmission through a E = 0 'barrier' Conductivity Other Geometries Magnetic Field: Aharonov Bohm effect Summary and Literature 2/7

3 Landauer Formula ψ n,e : Eigenstate with energy E T: temperature, V: applied voltage, I: current I = e h de n T n (E[f L (E f R (E] Use T 0,V 0 : I = e ev T n (E F h I = GV G = e2 T n (E F h n multi-channel Landauer formula n T n (E = j out(ψ n,e j in (ψ n,e f L (E = f R (E ev Fano factor: n F = T n( T n n T n 3/7

4 2D: Conductance and Conductivity G: conductance, σ: conductivity 3D: G = σ A L A: cross section area, L: length 2D: G = σ W L W: width, L: length 2D: one complex variable z = x+iy instead of two real variables x, y ψ(z := ψ(rez,imz with wave function ψ(x,y 4/7

Graphene Castro Neto 2009, Rev Mod Phys 8 09, arxiv:0709.

5 Graphene Castro Neto 2009, Rev Mod Phys 8 09, arxiv: B A δ3 δ δ2 a a 2 What is G(E = 0 Low energy limit, valley K only: H K ( r = v F 0 p x ip y p x +ip y 0 p = i n(e E ψ = ψ A ψ B v F 0 6 m s 5/7

6 Schrödinger p = i Dirac 2D H = v F p σ Hamiltonian: ( 0 px ip = v y H = p2 F p x +ip y 0 2m Eigenstates: ( ψ( r = exp(i k r ψ( r = exp(i k r sexp(iφ k E = 2 2 k Eigenvalues: 2m E = s v F k s {,} Probability current: (= valley K of graphene, used from here on j( r = Re(ψ p m ψ = 2mi (ψ ψ ψ ψ j( r = ψ v F σψ 6/7

7 Dirac 2D Zero Energy Modes H = v F p σ = i v F ( 0 x i y x +i y 0 r R 2 (infinite plane and E 0: ψ( r = ( sexp(iφ k E = s v F k exp(i k r s {,} E = 0 : H ( ψ ψ 2 = 0 ( x +i y ψ = 0 ( x i y ψ 2 = 0 Solutions: ψ (x,y = ψ (x+iy ψ 2 (x,y = ψ 2 (x iy c.c. analytic analytic 7/7

8 Transport: Piecewise Wave Function Wide graphene stripe L Potential C Periodic boundary conditions R Fermi energy ψ L = [( e e ik x x iφ +r ( ] e e e ik x x ik yy iφ ψ C = ( ψ C ψ C 2 analytic c.c. analytic ψ R = t ( e iφ e ik x x+ik y y ψ L (x = 0,y = ψ C (x = 0,y ψ C (x,y = c e ik yy+k y x ψ C 2 (x,y = c 2 e ik yy k y x k v F = E = V 0 Evanescent waves ψ L (x = 0 = ψ C (x = 0 ψ C (x = L x = ψ R (x = L x 4 equations for t, r, c, c 2 8/7

9 Transport: Transmission Probability For the conductance, we want the transmission probability T: T = j out j in T = t 2 Solving the linear system gives: T = cos 2 (φ cosh 2 (k y L x sin 2 (φ cosh 2 (k y L x V 0 = E = v F k k y L x k F L x Energy of the plane wave: Use: Assume: k F := k k F L x Use φ 0 and calculate T again: T = 4 ψ C (x = 0,y ψ C(x = L x,y + ψc 2 (x = 0,y ψ2 C(x = L x,y 2 9/7

10 Transport: Conductivity Periodic boundary conditions: ψ(y = 0 = ψ(y = L y k y = 2π L y n with n Z Total transmission: k y L x k F L x n 0 T = n= n 0 cosh 2 (k y L x Conductance: G = e2 h T = e2 Ly }{{} hπ L x n= σ this might be (conductivity cosh 2 (k y L x L y L x cosh 2 (x dx L y 2πL x } {{ } =2 F = 3 0/7

11 Transport: Conductivity Conductivity at the Dirac point: 2 spin and 2 valley states (neglected factor 4: Measurement: σ = e2 hπ σ = 4e2 hπ Novoselov et al. 2005, Nature 438 p. 97, arxiv:cond mat/ e 2 h Quantum transport through evanescent waves! Conductivity σ > 0 at Dirac point confirmed, order of magnitude agrees /7

12 The Missing Pi Theory: σ = 4e2 hπ Graphene on Silicon Oxide substrate σ = 20kΩ Graphene with Substrate Etched Away 2/7 4e 2 h Novoselov et al. 2005, Nature 438 p. 97, arxiv:cond mat/ Coupling to substrate Charge inhomogeneities Mayorov et al 202, Nano Letters 2 p. 4629, arxiv: E mev n 0 8 cm 2 = µm 2

13 Corbino Geometry ψ C = ( ψ C ψ C 2 analytic c.c. analytic ψ C (z analytic and w(z analytic ψ C (w(z analytic use w(z to map ( geometry: 2πz w(z = R exp L y with L x = ( L 2π ln R2 y R Wide graphene stripe L Potential Boundary conditions: [ ( ψ L = e e ik x x iφ +r ( ψ ] L (x = 0,y = ( +r +r e e ik x x iφ e ik yy ψ R = t ( e iφ e ik x x+ik y y φ 0 current ψ R (x = L x,y = ( t te ik yy C Periodic boundary conditions R Fermi energy e ik y y 3/7

14 Corbino Geometry: Conductance ( 2πz w(z = R exp L y with L x = ( L 2π ln R2 y R T n = cosh 2 (k y L x = cosh 2 (n ln(r 2 /R current k y = 2π L y n with n+ 2 Z R 2 R R : G 2e2 h F 3 ln(r 2 /R R R 2 : G 8e2 h R R 2 F G h 8e 2 ψ(y = 0 = ψ(y = L y Berry s phase 4/7

15 Magnetic field B = ( 00 B z (x,y see ch. 2.3 from Katsnelson, 202 B = 0 B z = 2 current φ(x,y ψ,2 C = exp( qφ ψ,2 C ( 2πz w(z = R exp flux Φ L y ( H(Bψ C = 0 R2 φ(r 2 φ(r = Φ ln H(B = 0 ψ C R = 0 Boundary Conditions: 0 R ψ L (x = 0,y = ( +r ik e y y ψ R (x = L x,y = ( t te ik yy ψ C (x,y = e qφ(r e ikyy e k yx T = 4 ψ C (x = 0,y ψ C(x = L x,y + ψc 2 2 (x = 0,y ψ2 C(x = L x,y φ 0 e q(φ(r φ(r 2 e k yl x +r B 0 5/7

16 Aharonov Bohm effect ( 2πz w(z = R exp L y φ(r 2 φ(r = Φ ln ( R2 R T = 4 ψ C (x = 0,y ψ C(x = L x,y + ψc 2 (x = 0,y ψ2 C(x = L x,y 2 B = 0 current B 0 flux Φ e q(φ(r φ(r 2 e k yl x Final result: G = e2 h [ f (R 2 /R cos F = ( eφ 3 +f 2(R 2 /R cos ( eφ ] R 2 /R = 5: Effects of 5% and 42% 6/7

17 Summary Minimal conducitivity at E = 0 is σ = 4e2 hπ Analytic functions (= conformal maps for σ(e = 0 in different geometries Aharonov-Bohm effect in a Magnetic field at E = 0 predicted Literature This talk: Katsnelson: Graphene (202, Cambridge University Press (ch. 2.3 and 3 Landauer formula: Cuevas, Scheer: Molecular Electronics (200, World Scientific Publishing (ch. 4 Datta: Electronic Transport in Mesoscopic Systems, (995, Cambridge University Press Berry's Phase: Böhm: The Geometric Phase in Quantum Systems (2003, Springer (esp. ch. 2 together with ch. 2.4 from Katsnelson, 202; ch. 2 7/7 see ch. 3. of Katsnelson, 202 for 'intrinsic disorder' related to zitterbewegung (no (external disorder or scattering at impurities

Spin Filtering: how to write and read quantum information on mobile qubits

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