Quantum Transport and Mesoscopic Physics

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1 Introduction

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Recommended Textbooks

J.M. Ziman Principles of the theory of solids, 2nd Edition
A. Atland and B. Simons, Condensed Matter Field Theory
S.M. Girvin and Kun Kang, Modern Condensed Matter Physics

Presentation Topics

Coulomb blockade.
Kondo effect
Orthogonality catastrophe and quantum speed limit
Mott transition
Spin Hall effect
Composite fermions and the fractional quantum Hall effect
Charge density waves
Strongly coupled large polarons

2 Brownian Motion

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2.1 Stationary 1D Brownian Motion

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Definition

A quantum particle exhibits coherent motion iff its lifetime

t_{ℓ}

is much greater than

\frac{ℏ}{E}

.

t_{ℓ} >> \frac{ℏ}{E}

Definition

A quantum particle exhibits incoherent motion iff its lifetime

t_{ℓ}

less than

\frac{ℏ}{E}

.

t_{ℓ} \leq \frac{ℏ}{E}

Definition

A stationary 1D Brownian particle is a system of a stationary 1D particle experiencing many fast collisions with momentum

p_{i}

integrated over a small time step

Δ t

. Defined the force experienced by the particle

f (t)

as a function of time.

f (t) = \frac{2}{Δ t} \sum_{i} p_{i}

Definition

The ensemble average denoted

⟨ x ⟩

of a random variable

x

is the integral of the variable weighted by the probability density function

P_{x} (x)

across all possible values.

⟨ x ⟩ = \int_{- \infty}^{\infty} x P_{x} d x

Definition

The momentum diffusion coefficient is defined

D = 2 ν ⟨ p_{i}^{2} ⟩ = 2 ν m_{m o l} k_{B} T

where

ν

is the average frequency of collisions and

m^{*}

is the effective mass of the particles.

Result

⟨ f ⟩ = 0

Result

⟨ f^{2} ⟩ = \frac{4}{(Δ t)^{2}} \sum_{i} ⟨ p_{i}^{2} ⟩ = \frac{4}{Δ t} ν ⟨ p_{i}^{2} ⟩ = \frac{2 D}{Δ t}

Result

⟨ f^{2 n + 1} ⟩ = 0

Result

⟨ f^{2 n} ⟩ = ⟨ f^{2} ⟩^{n} (2 n - 1)!! = {(\frac{4}{Δ t} ν ⟨ p_{i}^{2} ⟩)}^{n} (2 n - 1)!! = {(\frac{2 D}{Δ t})}^{n} (2 n - 1)!!

Definition

A Gaussian Distribution with variance

σ = ⟨ f^{2} ⟩

is a distribution of the form:

P_{f} (x) = \frac{1}{\sqrt{2 π σ}} e^{- x^{2} / 2 σ}

Definition

A probability density function of a variable is a function that when integrated over a region of possible values represents the probability of the variable being in that region.

Result

The probability density function of

f (t)

at a particular time is Gaussian with

σ = ⟨ f^{2} ⟩ = \frac{2 D}{Δ t}

.

Definition

A functional probability density function is a probability density function that is integrated over different functions.

Result

The functional probability density of

f (t)

is

P (f (t)) = C exp [- \frac{1}{4 D} \int_{- \infty}^{\infty} f^{2} (t) d t]

2.2 1D Brownian Motion with Friction

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Definition

The friction coefficient is a constant that describes the amount of friction experienced by a 1D Brownian particle.

Γ = \frac{m_{m o l} ν}{M}

Definition

A 1D Brownian Particle is a system of a 1D particle of mass

m_{p}

moving at velocity

v_{p}

which is experiencing many fast collisions. Define the force experienced by the particle

f_{p} (t)

in terms of the force experienced by a stationary 1D Brownian particle

f (t)

and the friction coefficient

Γ

.

f_{v} (t) = f (t) - 2 Γ m_{p} v_{p}

Definition

The position diffusion coefficient denoted

D

is defined in terms of the noise intensity

D

, the friction coefficient

Γ

and the mass of the particle

m_{p}

.

D = \frac{D}{(2 Γ m_{p})^{2}} = \frac{k_{B} T}{2 Γ m_{p}}

Result

The motion

q (t)

and the momentum

p (t)

of a 1D Brownian particle is described by the following equations.

p (t) = \int_{0}^{t} e^{- 2 Γ (t - τ) f (τ) d τ}

q (t) = \frac{1}{m_{p}} \int_{0}^{t} p (τ) d τ = \frac{1}{m_{p}} \int_{0}^{τ} f (τ) \frac{1 - e^{- 2 Γ (t - τ)}}{2 Γ} d τ

Result

⟨ q ⟩ = 0

Result

For

t >> \frac{1}{Γ}

,

⟨ q^{2} ⟩ = ⟨ (q (t) - q (0))^{2} ⟩ = 2 D [t - \frac{2}{2 Γ} (1 - e^{- 2 Γ t}) + \frac{1}{4 Γ} (1 - e^{- 4 Γ t})] \approx 2 D t

2.3 Lagevin and Fokker-Plank Equations

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Definition

A 1D Brownian Particle in a Potential is a system of a 1D Brownian particle experiencing a potential

U (p)

. The system can be written in terms of position

q

and momentum

p

or as a time dependent force function

f_{U} (t)

\dot{q} = \frac{p}{m_{p}}, \dot{p} = f (t) - 2 Γ p - U^{'} (q)

f_{U} (t) = f (t) - 2 Γ m_{p} v_{p} - U^{'}

Theorem

The Lagevin and Fokker-Plank Equations describe the probability density function

ω (q, p, t, q (0), p (0), 0)

of a 1D Brownian particle in a potential

U (q)

.

ω (q, p, 0, q (0), p (0), 0) = δ (q - q (0)) δ (p - p (0))

\partial_{t} ω (q, p, t) = - \partial_{q} (\frac{p}{m_{p}} ω (q, p, t)) - \partial_{p} (- 2 Γ p - U^{'} (q) + D \partial_{p}^{2}) ω (q, p, t)

Definition

The Diffusion Equation describes a Brownian particle where there is no potential or drift, that is

Γ = 0

and

U (q) = 0

.

\frac{\partial n}{\partial t} - D \nabla_{x}^{2} n = 0

Result

The solution to the diffusion equation is of the form

\partial_{t} ω = D \partial_{x}^{2} ω

ω = \frac{C}{\sqrt{t}} e^{- x^{2} / 4 D t}

Definition

The time average denoted

⟨ x ⟩

of a random variable

x (t)

is the time integral of the variable over a very long period.

⟨ x ⟩ = lim_{T \to \infty} \frac{1}{T} \int_{0}^{T} x (t) d t

Definition

A system is ergodic iff the ensemble average and the time average are equivalent.

2.4 Harmonic Oscillator Diffusion

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Definition

The Brownian harmonic oscillator is a system describing a Brownian particle in a harmonic oscillator potential

U (q)

U (q) = \frac{1}{2} m ω_{0}^{2} q^{2}

\partial_{t} ω (q, p, t) = - \partial_{q} (\frac{p}{m_{p}} ω (q, p, t)) - \partial_{p} (- 2 Γ p ω (q, p, t) - m ω_{0}^{2} q ω (q, p, t) + D \partial_{p}^{2}) ω (q, p, t)

Result

For a Brownian harmonic oscillator,

⟨ \ddot{q} ⟩ = - ω_{0}^{2} ⟨ q ⟩ - \frac{2 Γ}{m} ⟨ \ddot{q} ⟩

Definition

The time correlation function

Q_{q q} (t)

of a variable

q (t)

is defined

Q_{q q} (t) = ⟨ q (t) q (0) ⟩

.

Definition

The power spectrium

S_{q q} (ω)

of a variable

q (t)

is the Fourier transform of the time correlation function

S_{q q} (ω) = \int_{\infty}^{\infty} e^{i ω t} ⟨ q (t) q (0) ⟩ d t

,

Result

For a real variable

q (t)

the power spectrum is

S_{q q} (ω) = 2 Re (\int_{- \infty}^{\infty} e^{i ω t} ⟨ q (t) q (0) ⟩ d t)

Result

The stationary solution to the Brownian Harmonic oscillator is of the form

ω (q, p, t) = C e^{- (\frac{p^{2}}{2 m} + U (q)) / α}

Result

The power spectrum of

q

for the Brownian harmonic oscillator is

S_{q} q (ω) = \frac{2 Γ ω_{0}^{2}}{(ω^{2} - ω_{0}^{2})^{2} + 4 Γ^{2} ω^{2}}

2.5 Escape and Activation via Tunneling

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Definition

The probability current

j (q)

is the rate at which probability to crossing point

q_{B}

.

j (q_{B}) = \int \frac{p}{m} ω (q_{B}, p, t)

Definition

The escape rate is the reciprocal of the average lifetime

τ

and can be written in terms of probability current.

W_{e s c} = \frac{j}{N} = \frac{1}{τ}

Definition

The number of trapped particles

N

is defined

N = \int_{- \infty}^{\infty} d p \int_{- \infty}^{q_{B}} d q ω (q, p, t)

Result

The time derivative of the number of particles

N

is

\frac{\partial N}{\partial t} = - W_{e s c} N

and the decay over time is

N (t) = N (0) e^{- W_{e s c} t}

2.6 Escape and Activation via Tunneling

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Definition

The probability current

j (q)

is the rate at which probability to crossing point

q_{B}

.

j (q_{B}) = \int \frac{p}{m} ω (q_{B}, p, t)

Definition

The escape rate is the reciprocal of the average lifetime

τ

and can be written in terms of probability current.

W_{e s c} = \frac{j}{N} = \frac{1}{τ}

Definition

The number of trapped particles

N

is defined

N = \int_{- \infty}^{\infty} d p \int_{- \infty}^{q_{B}} d q ω (q, p, t)

Result

The time derivative of the number of particles

N

is

\frac{\partial N}{\partial t} = - W_{e s c} N

and the decay over time is

N (t) = N (0) e^{- W_{e s c} t}

3 Electron Transport

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3.1 Bloch's Theorem and Berry Phase

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Theorem

Bloch's Theorem states that the solution any system with a periodic potential can be represented with a periodic function

u_{k, n} (r)

and

e^{i k \cdot r}

, that is

H Ψ_{k, n} (r) = E_{k, n} Ψ_{k, n} (r), Ψ_{k, n} (r) = e^{i k \cdot r} u_{k, n} (r)

Definition

The Berry connection denoted

A_{n, k} (r)

is defined as

A_{n, k} (r) = i ⟨ u_{k, n} (r) | \nabla_{k} u_{k, n} (r) ⟩

Definition

The Berry phase denoted

γ_{n, k}

is defined as

γ_{n, k} = \int A_{n, k} (r) \cdot d r

Result

For the band wavefunction

Φ_{n} (r) = \int c (k) Ψ_{k, n} (k) Ψ_{k, n} (r) d k

, written in terms of eigenstates

Ψ_{k, n}

,

r = i \partial_{k} + i ⟨ u_{k, n} (r) | \nabla_{k} u_{k, n} (r) ⟩ = i \partial_{k} + A_{n} (r)

⟨ r ⟩ = {(\frac{(2 π)^{d}}{V})}^{2} \int c^{*} (k) r c (k) d k = {(\frac{(2 π)^{d}}{V})}^{2} \int c^{*} (k) (i \partial_{k} + A_{n, k} (r)) c (k) d k

⟨ \dot{r} ⟩ = \frac{\partial E_{k, n}}{\partial k}

3.2 Anomalous Quantum Hall Effect

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Definition

The anomalous quantum hall effect Hamiltonian

H

describes the behavior of many electrons in a lattice

H_{0}

experiences an electric field

E

pointing in the x direction.

H = H_{0} - e_{q} E x

Definition

The Berry curvature denoted

Ω_{n, k} (r)

is defined as

Ω_{n, k} (r) = \nabla_{r} \times A_{n, k} (r)

Result

The average velocities for 2D electrons experiencing an electric field in the x direction are

⟨ v_{x} ⟩ = \frac{1}{ℏ} \frac{\partial E_{n k}}{\partial k_{x}}

⟨ v_{y} ⟩ = \frac{1}{ℏ} \frac{\partial E_{n k}}{\partial k_{x}} + \frac{q E}{ℏ} {(Ω_{n, k} (r))}_{z}

Result

current

Definition

Churn number

Result

conductivity

Boltzmann Kinetic Equation

4 Scattering and Conductivity

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Impurity Scattering

Electric Conductivity

Thermal Conductivity

Magnetoconductivity

Cyclotron Resonance

5 Low-dimensional Systems

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Landauer 1D Conductivity

2D Electron Systems

Quantum Hall Effect

6 Localization

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Weak Localization

Anderson Localization

Density Matrix and the Quantum Kinetic Equation

7 Phonon

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Electron-phonon Interaction

Polaronic Effect

Ohmic Dissipation

The Orthogonality Catastrophe

Holstein Polarons

8 Topological Materials

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Variable-range Hopping

The Coulomb Gap

The Berry Phase

Group Velocity in Topologically Nontrivial Solids

The Kitaev Chain

Lattice

Quantum Transport and Mesoscopic Physics 60ZR Edit Save Copy Link Copy Anchor

1 Introduction Z4MT Edit Save Copy Link Copy Anchor

2 Brownian Motion 0ZE2 Edit Save Copy Link Copy Anchor

2.1 Stationary 1D Brownian Motion DFZJ Edit Save Copy Link Copy Anchor

2.2 1D Brownian Motion with Friction 9ET5 Edit Save Copy Link Copy Anchor

2.3 Lagevin and Fokker-Plank Equations 26RC Edit Save Copy Link Copy Anchor

2.4 Harmonic Oscillator Diffusion K69N Edit Save Copy Link Copy Anchor

2.5 Escape and Activation via Tunneling 8553 Edit Save Copy Link Copy Anchor

2.6 Escape and Activation via Tunneling 8553 Edit Save Copy Link Copy Anchor

3 Electron Transport ZKM7 Edit Save Copy Link Copy Anchor

3.1 Bloch's Theorem and Berry Phase 45C9 Edit Save Copy Link Copy Anchor

3.2 Anomalous Quantum Hall Effect 63P4 Edit Save Copy Link Copy Anchor

4 Scattering and Conductivity REZC Edit Save Copy Link Copy Anchor

5 Low-dimensional Systems AW5M Edit Save Copy Link Copy Anchor

6 Localization D3P7 Edit Save Copy Link Copy Anchor

7 Phonon 1EDE Edit Save Copy Link Copy Anchor

8 Topological Materials NMJH Edit Save Copy Link Copy Anchor

▾

Quantum Transport and Mesoscopic Physics

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1 Introduction

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2 Brownian Motion

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2.1 Stationary 1D Brownian Motion

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2.2 1D Brownian Motion with Friction

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2.3 Lagevin and Fokker-Plank Equations

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2.4 Harmonic Oscillator Diffusion

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2.5 Escape and Activation via Tunneling

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2.6 Escape and Activation via Tunneling

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3 Electron Transport

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3.1 Bloch's Theorem and Berry Phase

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3.2 Anomalous Quantum Hall Effect

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4 Scattering and Conductivity

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5 Low-dimensional Systems

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6 Localization

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7 Phonon

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8 Topological Materials

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