Mesoscopic Transport

Mesoscopic transport studies charge, spin, heat, and quasiparticle flow through devices small enough that discreteness, phase coherence, Coulomb blockade, or quantum statistics matter. From the open-system point of view, the central idea is simple: a small quantum system is coupled to large reservoirs, and the observable current is generated by exchange events across the system-reservoir boundaries.

A single-level dot provides a concrete route from reservoir occupations to tunneling events, mean current, and current fluctuations. For confinement and charge stability, see Quantum Dots and Coulomb Blockade. Coherent interference in larger conductors is developed in Quantum Coherence in Conductors.

Leads as Reservoirs

A common microscopic model splits the Hamiltonian into

$$H = H_S + \sum_\alpha H_\alpha + H_T,$$

where $H_S$ describes the central device, $H_\alpha$ are leads, and $H_T$ tunnels particles between them. For noninteracting electronic leads,

$$H_\alpha = \sum_k \epsilon_{k\alpha} c_{k\alpha}^\dagger c_{k\alpha}.$$

Each lead is usually assumed to be in a grand-canonical state,

$$\rho_\alpha \propto \exp \left[ - \beta_\alpha \left( H_\alpha-\mu_\alpha N_\alpha \right) \right],$$

with temperature $T_\alpha$ and chemical potential $\mu_\alpha$. The Fermi function is

$$f_\alpha(E) = \frac{1} {e^{\beta_\alpha(E-\mu_\alpha)}+1}.$$

The Markovian lead approximation assumes that the leads remain large, rapidly relaxing reservoirs even as particles enter and leave the small device.

Single-Level Quantum Dot

The simplest sequential-tunneling model is a spinless single-level quantum dot with energy $\epsilon$. A single spinless orbital already has only two occupation states by the Pauli principle:

$$\lvert 0\rangle, \qquad \lvert 1\rangle=d^\dagger\lvert 0\rangle.$$

If a physical dot has spin or additional orbitals, restricting to these two states requires those alternatives to be energetically excluded or polarized. The sequential-tunneling approximation neglects coherent level broadening and higher-order tunneling. Here $\Gamma_\alpha$ is a rate, so the corresponding energy width is $\hbar\Gamma_\alpha$. Lead $\alpha$ fills the dot with rate

$$\Gamma_\alpha^+ = \Gamma_\alpha f_\alpha(\epsilon),$$

and empties it with rate

$$\Gamma_\alpha^- = \Gamma_\alpha \left[ 1-f_\alpha(\epsilon) \right].$$

The probabilities obey the rate equation

$$\dot p_1 = \sum_\alpha \Gamma_\alpha^+ p_0 - \sum_\alpha \Gamma_\alpha^- p_1, \qquad p_0=1-p_1.$$

This is a classical master equation for populations, but it has a quantum origin: the rates come from tunneling between a discrete quantum state and fermionic reservoirs. For the general population-equation structure, see Pauli Rate Equations.

Detailed Balance with Chemical Potential

For one equilibrium lead,

$$\frac{\Gamma_\alpha^+} {\Gamma_\alpha^-} = \frac{f_\alpha(\epsilon)} {1-f_\alpha(\epsilon)} = e^{-\beta_\alpha(\epsilon-\mu_\alpha)}.$$

This is local detailed balance. Adding one electron to the dot changes the reservoir particle number and energy, so the thermodynamic exponent contains $\epsilon-\mu_\alpha$, not only $\epsilon$.

When all leads have the same temperature and chemical potential, the dot relaxes to the grand-canonical occupation

$$p_1^{\mathrm{eq}} = f(\epsilon).$$

Transport appears when different reservoirs impose different chemical potentials or temperatures. Chemical Potential distinguishes the thermodynamic, electrochemical, and voltage conventions behind those reservoir parameters.

Current

Let $e>0$ denote the magnitude of the electron charge. Define $I_\alpha$ as $e$ times the electron-number current from lead $\alpha$ into the dot; conventional electric current carried by negative electrons has the opposite sign:

$$I_\alpha = e \left( \Gamma_\alpha^+ p_0 - \Gamma_\alpha^- p_1 \right).$$

At steady state, define

$$\Gamma^+ = \sum_\alpha\Gamma_\alpha^+, \qquad \Gamma^- = \sum_\alpha\Gamma_\alpha^-.$$

Then

$$p_1^{\mathrm{ss}} = \frac{\Gamma^+} {\Gamma^+ + \Gamma^-}, \qquad p_0^{\mathrm{ss}} = \frac{\Gamma^-} {\Gamma^+ + \Gamma^-}.$$

For two leads, the steady current from the left lead is

$$I_L^{\mathrm{ss}} = e\, \frac{ \Gamma_L^+\Gamma_R^- - \Gamma_L^-\Gamma_R^+ }{ \Gamma^+ + \Gamma^- }.$$

The numerator is a competition between left-to-right and right-to-left cycles. At equilibrium the two terms cancel by detailed balance.

High-Bias Limit

In the ideal high-bias sequential-tunneling limit,

$$f_L(\epsilon)\simeq1, \qquad f_R(\epsilon)\simeq0.$$

Then the dot is filled from the left and emptied to the right:

$$\Gamma_L^+=\Gamma_L, \qquad \Gamma_R^-=\Gamma_R,$$

with the reverse rates negligible. The steady current is

$$I = e \frac{\Gamma_L\Gamma_R} {\Gamma_L+\Gamma_R}.$$

The smaller rate is the bottleneck. If $\Gamma_L\gg\Gamma_R$, the dot is almost always occupied and the right barrier controls the current. If $\Gamma_R\gg\Gamma_L$, the dot is almost always empty and the left barrier controls the current.

Counting Statistics

Counting statistics keeps track of the number of transferred particles, not just the mean current. For the single-level dot, let $\chi$ count electrons entering the right lead. In the basis $(p_0,p_1)^T$, a counting-field rate matrix is

$$W(\chi) = \begin{pmatrix} -\Gamma^+ & \Gamma_L^-+\Gamma_R^-e^{i\chi} \\ \Gamma_L^+ + \Gamma_R^+e^{-i\chi} & -\Gamma^- \end{pmatrix}.$$

The characteristic function $Z(\chi,t)=\langle e^{i\chi N_R}\rangle$ has the long-time form

$$Z(\chi,t) \sim e^{\lambda_0(\chi)t},$$

where $\lambda_0(\chi)$ is the eigenvalue of $W(\chi)$ that approaches zero at $\chi=0$. Current cumulants are obtained by differentiating $\lambda_0(\chi)$ with respect to $i\chi$.

This is the transport version of a quantum-jump unraveling: jumps across a selected barrier are counted, and the statistics of those jumps carries information beyond the average current.

Shot Noise and Fano Factor

Current noise reveals correlations between transfer events. For a stationary unidirectional current, define the low-frequency noise normalization by $S=2e^2\lim_{t\to\infty}\operatorname{Var}N_R(t)/t$ and $I=e\lim_{t\to\infty}\langle N_R(t)\rangle/t>0$. Then

$$S = 2eI F,$$

where $F$ is the Fano factor. Independent Poisson transfer events have $F=1$. Sequential tunneling through a single dot in the high-bias limit gives

$$F = \frac{ \Gamma_L^2+\Gamma_R^2 }{ (\Gamma_L+\Gamma_R)^2 }.$$

For symmetric barriers, $F=1/2$: the restriction to zero or one electron anticorrelates successive transfers because the dot must empty before it can be filled again. Here Pauli exclusion of the single spinless orbital is sufficient; a larger dot can impose the same restriction through Coulomb blockade. Super-Poissonian noise can occur when transport switches between slow and fast configurations, when avalanches occur, or when internal dynamics bunches transfer events.

In coherent scattering language, a quantum point contact with transmission eigenvalues $T_n$ has conductance proportional to $\sum_n T_n$ and zero-temperature shot noise proportional to

$$\sum_n T_n(1-T_n).$$

Fully open and fully closed channels do not produce partition noise. Partially transmitting channels do.

Relation to Master Equations

Mesoscopic transport uses several levels of description:

Regime	Common method	Open-system interpretation
coherent noninteracting conductor	scattering matrix or Landauer-Büttiker theory	reservoirs set incoming occupations
weak tunneling and Coulomb blockade	rate equation or quantum master equation	jumps between charge states
coherent dot dynamics with weak leads	Redfield or Lindblad-type equation	lead-induced transitions and dephasing
strong coupling or structured contacts	nonequilibrium Green functions or reaction-coordinate methods	reservoir memory and level broadening matter
monitored transport	counting statistics or quantum trajectories	individual transfers are records

No one description dominates every regime. A rate equation can be excellent in sequential tunneling and wrong in coherent cotunneling. A Lindblad equation can preserve positivity while using rates that are not appropriate for a biased conductor. A scattering approach can compute coherent conductance while hiding internal relaxation and measurement backaction.

Measurement and Backaction

Mesoscopic devices can also measure other quantum systems. A quantum point contact or single-electron transistor coupled to a charge qubit turns qubit state information into a current signal. The same coupling causes dephasing because different qubit states imprint different phases, transmission probabilities, or tunneling rates on the detector.

As in optical or circuit-QED readout, outgoing degrees of freedom carry information about the measured system. The observed current can condition its state, while inaccessible information contributes to dephasing.

The correct trajectory description depends on detector bandwidth and whether individual tunneling events are resolved. When events are resolved, jump trajectories are natural. When only a noisy current is measured, a diffusive approximation may be better.

Thermodynamics of Transport

Each transferred particle carries energy and charge. When $\Delta E_\alpha$ and $\Delta N_\alpha$ denote changes of lead $\alpha$ itself, its entropy change in units of $k_B$ is

$$\beta_\alpha \left( \Delta E_\alpha-\mu_\alpha\Delta N_\alpha \right).$$

A filling event has $\Delta E_\alpha=-\epsilon$ and $\Delta N_\alpha=-1$, hence reservoir entropy change $-\beta_\alpha(\epsilon-\mu_\alpha)$. Using changes of the dot instead reverses that sign. This is why transport fluctuation theorems involve affinities built from temperature and chemical-potential differences. For isothermal charge transport, the bias contribution is controlled by

$$\beta eV,$$

where $eV=\mu_L-\mu_R$ for a two-terminal conductor. See Fluctuation Theorems and Entropy Production.

Exercises

Local Detailed Balance

Show that

$$\frac{\Gamma_\alpha^+} {\Gamma_\alpha^-} = e^{-\beta_\alpha(\epsilon-\mu_\alpha)}$$

for the single-level dot rates $\Gamma_\alpha^+=\Gamma_\alpha f_\alpha(\epsilon)$ and $\Gamma_\alpha^-=\Gamma_\alpha[1-f_\alpha(\epsilon)]$.

Solution

Using

$$f_\alpha(\epsilon) = \frac{1} {e^{\beta_\alpha(\epsilon-\mu_\alpha)}+1},$$

we have

$$1-f_\alpha(\epsilon) = \frac{ e^{\beta_\alpha(\epsilon-\mu_\alpha)} }{ e^{\beta_\alpha(\epsilon-\mu_\alpha)}+1 }.$$

Therefore

$$\frac{f_\alpha(\epsilon)} {1-f_\alpha(\epsilon)} = e^{-\beta_\alpha(\epsilon-\mu_\alpha)}.$$

The factors $\Gamma_\alpha$ cancel.

Steady Current

Derive the steady current for the two-lead single-level dot.

Solution

The steady occupation is

$$p_1^{\mathrm{ss}} = \frac{\Gamma^+} {\Gamma^+ + \Gamma^-}, \qquad p_0^{\mathrm{ss}} = \frac{\Gamma^-} {\Gamma^+ + \Gamma^-}.$$

The left current into the dot is

$$I_L = e \left( \Gamma_L^+p_0 - \Gamma_L^-p_1 \right).$$

Substituting the steady probabilities gives

$$I_L^{\mathrm{ss}} = e \frac{ \Gamma_L^+\Gamma^- - \Gamma_L^-\Gamma^+ }{ \Gamma^+ + \Gamma^- }.$$

Using $\Gamma^\pm=\Gamma_L^\pm+\Gamma_R^\pm$, the same-lead products cancel:

$$I_L^{\mathrm{ss}} = e \frac{ \Gamma_L^+\Gamma_R^- - \Gamma_L^-\Gamma_R^+ }{ \Gamma^+ + \Gamma^- }.$$

High-Bias Occupation

In the high-bias limit $\Gamma_L^+=\Gamma_L$ and $\Gamma_R^-=\Gamma_R$, with reverse rates zero. Find $p_1^{\mathrm{ss}}$ and $I$.

Solution

Here

$$\Gamma^+=\Gamma_L, \qquad \Gamma^-=\Gamma_R.$$

Thus

$$p_1^{\mathrm{ss}} = \frac{\Gamma_L} {\Gamma_L+\Gamma_R}.$$

The right-going current is the rate at which the occupied dot empties into the right lead:

$$I = e\Gamma_R p_1^{\mathrm{ss}} = e \frac{\Gamma_L\Gamma_R} {\Gamma_L+\Gamma_R}.$$

Counting-Field Signs

In the counting matrix $W(\chi)$ above, why does the right-emptying rate carry $e^{i\chi}$ while the right-filling rate carries $e^{-i\chi}$?

Solution

The convention was to count electrons entering the right lead. When the dot empties to the right, one electron is added to the counted lead, so the jump is weighted by $e^{i\chi}$. When an electron enters the dot from the right lead, the counted lead loses one electron, so the jump is weighted by $e^{-i\chi}$. Reversing the current convention would reverse these signs.

References

• D. A. Bagrets and Yu. V. Nazarov, “Full counting statistics of charge transfer in Coulomb blockade systems,” Physical Review B 67, 085316 (2003).
• S. Datta, Electronic Transport in Mesoscopic Systems, Cambridge University Press (1995).
• M. Esposito, U. Harbola, and S. Mukamel, “Nonequilibrium fluctuations, fluctuation theorems, and counting statistics in quantum systems,” Reviews of Modern Physics 81, 1665-1702 (2009).
• S. A. Gurvitz and Ya. S. Prager, “Microscopic derivation of rate equations for quantum transport,” Physical Review B 53, 15932-15943 (1996).
• L. S. Levitov and G. B. Lesovik, “Charge distribution in quantum shot noise,” JETP Letters 58, 230-235 (1993).
• Y. V. Nazarov and Y. M. Blanter, Quantum Transport: Introduction to Nanoscience, Cambridge University Press (2009).
• Ya. M. Blanter and M. Büttiker, “Shot noise in mesoscopic conductors,” Physics Reports 336, 1–166 (2000). arXiv:cond-mat/9910158.