MIT OpenCourseWare http://ocw.mit.edu 8.512 Theory of Solids IISpring 2009For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.1 Lecture 6: Scaling Theory of Localization 6.1 Notion of dimensionless conductance. G g = where G is (resistance)−1 e2/π� G = σLd−2 L = dimension of system Recall Einstein’s formula which relates conductivity to diffusion: N (0) σ = 2e 2 · D Ω where N (0) = # of states/energy Ω = unit volume D = diffusion coefficient Way to derive above equation is to calculate charge current density in two ways. j = −eD� · n · · · current as diffusion of charge carriers j = −σ�V · · · V is local electrical potential dV = −σ · �n dn Now, the local chemical potential is linearly related to V , i.e. µ = eV + constant dn dn 2N (0) ∴ = = e = e · · · · 2 is for spin. dv dµ Ω σ Ω ∴ j = − · �n (1) e 2 N ( 0) Comparing this to Einstein’s formula, one gets N (0) σ = 2e 2 · D (2) Ω 2e2 �D ⇒ G = e 2 2N (0) · D · Ld−2 = · · N (0) Ω � L22 6.2 Thouless Energy The last equation suggests to define a quantity called Thouless Energy, ET �D � ET = = (3) L2 τT Now, 1 N (0) ∼ · · · Δ is level spacing Δ 2e2 ET ET ⇒ G ∼ ⇒ g ∼ � Δ Δ Physical interpretation Assume the one box pro blem is solved and we want to study the behavior of the system as it gets bigger. Each box has its own distribution of energy levels. As the two boxes are in contact, the wavefunctions that were initially localized in separate boxes would mix. For weak coupling, one can do perturbation theory if mixing is smaller than typical level spacing Δ. For weak mixing, the wavefunctions resemble the initial states and so one gets localized distribution. For strong coupling, the final state would be a complicated mixture of all initial states and we get extended states. Physically one can think of diffusion as lifetime to leak particle out of box. And so it’s reasonable to think of Thouless energy as effective coupling. Thus the ratio ET /Δ, which also represents the dimensionless conductance ’g,’ characterizes the further scaling prop erties.� � � � � 3 6.3 Sensitivity of energy eigenvalues to boundary conditions and it s relation to ET . Here we derive another form of ET . Usually one uses periodic boundary conditions ψα(x + L) = ψα(x) · · · ψα is an eigenstate of the Hamiltonian. One can as well use the twisted boundary condition iφψα(x)ψα(x + L) = e · · · φ is same for all α. Such boundary conditions can arise if a magnetic field is present. Consider applying a uniform magnetic field (mag netic field is confined in flux tube) along the z direction. And we study the behavior of the eigenstates sitting on a cylindrical surface whose axis is parallel to the magnetic field. 1 � eA �2 + ψα + V (r)ψα = Eαψα2m i c x ψ → ψ� exp −i A · d� o 1 � � ⇒ −�2 α(r) + v(r)ψ� α(r)ψ� α(r) = Eαψ� 2m Thus the gauge transformation has removed A from the Schrodinger equation, but now ψα(r) has different boundary conditions.� 4 �� L � ψ�(x + L) = ψ�(x) exp i A · d� o = ψ�(x) exp [iφB] where φB is magnetic flux. We will compute ΔEα using perturbation theory. e φ H = VxA = Vx c L To first order in H�, ΔEα = 0 as < H� >∼< Vx >= 0. For perturbation theory second order in A, we have two terms: the diamagnetic term e2 A2 c2 2m which gives a constant shift to all energy levels 1 1 = φ2 2 m The paramagnetic t erm gives, to second order in A (or φ), � | < β|VxA|α > |2 φ2 � | < β| Vx|α > |2 = Eα − Eβ L2 Eα − Eββ±α β�=α ∂2Eα 1 2 � | < β|Vx|α > |2 ∴ = + (4) 2 L2∂φ2 mL Eα − Eβα�=β Thus one gets the same matrix element < β|Vx|α > that appeared in derivation of Kubo formula. Typical variation in energy levels can look like this. As first order shift is zero, all curves have zero slope at α = 0. All low lying states don’t undergo much variation as they occur in pot ential valleys and hence localized. One can do a sanity check on Eq.(4). On general grounds, there is equal probability for any level to go up or down and so average variation in Eα’s should be zero. By using Thomas-Reich-Kunz F -sum rule, one can indeed show that � ∂2Eα = 0 2∂φα� � � � � � � 5 Now, | < β|Vx|α > |2 ∼ V2 . Fluctuation of such terms is dominated by the term with maximum weight, i.e., by smallest Eα − Eβ � | < β|Vx|α > |2 V2 variance ∼ Eα − Eβ Δ β�=α This is the average fluctuation in the curvature. Thus typically, � ∂2Eα � 1 V2 ∼ Et � ∂φ2 � L2 Δ 1 V2 1 Et ∼ ∼ 2 V2 · N(0) L2 Δ LNow, N2(0) N(0) 2πσ = 2e · V2 = 2e 2 D Ω Ω By comparing Boltzmann & Einstein f ormula for σ D ⇒ D = πV2N(0) ⇒ Et = ∼ ET2 ELThus for large ET /Δ, one gets extended states and hence larg e conductance, while for small T /Δ, one gets localized states and hence small conductance. Thus we have established the relevance of Thouless energy for conductance.� 6 6.4 Application t o 1 − Δ System One expects Ohm’s low behavior for small length. 1 g(L) ∼ as L increases, g(L) decreases L as ET g(L) < 1 ⇒ < 1 Δ and so the states are localized. ⇒ For large L, g(L) ∼ e −L/ξ · · · ξ is the localization length τDue to inelastic scattering t he eigenvalues b ecome ill-defined because of dephasing. Let us define a quantity τφ as the time over which a state lo oses the information about its phase. φ for various scattering mechanisms can be given by T2 T31 1 ∼ ∼ ω2τee �F τe−phonon D Let us define Lφ as the length over which the dephasing occurs. Lφ = Doτφ So our argument wor ks if Lφ >> width of