Transport Numerics

Coherent transport computation couples a finite scattering region to open leads. The leads set outgoing boundary conditions and inject propagating modes; replacing them by a closed finite box changes the problem. This page develops finite-matrix algorithms for a prescribed noninteracting model.

Helpful background. Tight-binding assembly fixes hoppings; Green-function numerics treats resolvent solves. Material response interpretation belongs to transport and response.

Retarded embedding and transmission

For central Hamiltonian $H_C$, retarded lead self-energies $\Sigma_\alpha^r(E)$ give

$$G^r(E)= \left[(E+i0^+)I-H_C-\Sigma_L^r-\Sigma_R^r\right]^{-1}, \qquad \Gamma_\alpha=i(\Sigma_\alpha^r-\Sigma_\alpha^{r\dagger}).$$

A passive retarded lead has $\Gamma_\alpha\geq0$. Under coherent elastic single-particle assumptions,

$$T(E)=\operatorname{Tr}\!\left[ \Gamma_LG^r\Gamma_RG^{r\dagger} \right].$$

At zero temperature the spin-resolved linear conductance is $G=(e^2/h)T(E_F)$. Multiply by two only when an independent spin degeneracy is actually present. At finite temperature integrate $T(E)$ against $-\partial f/\partial E$.

Compute the needed Green-function columns by solving linear systems; do not form a full inverse when only contact blocks are required. Factorization can be reused for multiple right-hand sides at the same energy.

Semi-infinite chain self-energy

For a lead with onsite energy $\epsilon$ and nearest-neighbor hopping magnitude $t$, its surface Green function satisfies

$$g^r(E)=\frac1{E+i0^+-\epsilon-t^2g^r(E)}.$$

Choose the root with negative imaginary part and $g(z)\sim1/z$ at large complex $z$. Inside the band,

$$g^r(E)=\frac{E-\epsilon-i\sqrt{4t^2-(E-\epsilon)^2}}{2t^2}.$$

For contact hopping $\tau$, $\Sigma^r=|\tau|^2g^r$. Outside the band, write $x=E-\epsilon$, assume $t>0$, and select

$$g^r(E)=\frac{x-\operatorname{sgn}(x)\sqrt{x^2-4t^2}}{2t^2} =\frac{2}{x+\operatorname{sgn}(x)\sqrt{x^2-4t^2}}, \qquad |x|>2t.$$

The second form avoids subtracting nearly equal quantities at large $|x|$. It has $g^r(E)\sim1/x$ on both sides of the band, whereas keeping the same subtraction sign at negative and positive energies selects the wrong real root on one side. The other root describes a growing exterior solution. Outside the lead band, $\Gamma=0$ and there is no propagating incoming channel from that lead.

At band edges the self-energy varies rapidly. Refine energy sampling there and do not infer smooth transport from a coarse interpolation.

An exactly matched chain provides a stronger test than checking only the sign of $\Gamma$. Take one central site of energy $\epsilon$, two identical leads of hopping magnitude $t$, and contact magnitudes $|\tau_L|=|\tau_R|=t$. Inside the band let $s=\sqrt{4t^2-x^2}$. Then each $\Gamma_\alpha=s$, the central denominator is $is$, and $T=s^2|1/(is)|^2=1$. Inserting more identical chain sites leaves this result unchanged. Test several energies strictly inside the band and both signs of $x$. At the exact band edge the group velocity vanishes and the expression becomes singular; distinguish the limiting value $T\to1$ from a propagating-mode calculation exactly at threshold.

Resonant-level benchmark

For a single level $\epsilon_d$ and energy-independent lead broadenings $\Gamma_L,\Gamma_R$, absorb any real self-energy shift into $\epsilon_d$. Then

$$T(E)= \frac{\Gamma_L\Gamma_R} {(E-\epsilon_d)^2+(\Gamma_L+\Gamma_R)^2/4}.$$

At resonance, $T_{\max}=4\Gamma_L\Gamma_R/(\Gamma_L+\Gamma_R)^2\leq1$, reaching one for equal couplings. This checks the factor of two between imaginary self-energy and broadening.

A finite numerical $\eta$ in $E+i\eta$ acts like additional absorption. It can reduce transmission and violate two-terminal current conservation unless treated as another physical reservoir. Decrease it until observables converge; do not interpret numerical broadening as a measured lifetime.

Conservation and convergence

For a lossless scattering matrix, verify $S^\dagger S=I$ and transmission plus reflection equals the number of incoming propagating channels. Compare Green-function and scattering-mode calculations on a small system.

Increase the central region when contacts or boundaries might alter the intended sample, but a larger region is not always a better approximation: it may represent a physically longer conductor. Keep the physical geometry fixed when refining a continuum discretization.

Energy quadrature must resolve narrow resonances and the thermal window. Disorder averages require independent realizations. Interactions, inelastic scattering, and superconducting particle–hole transport require additional structure and cannot be handled by reusing the normal-state formula without qualification.

Exercise

For $\Gamma_L=3\Gamma_R$, find the resonant transmission.

Solution

$T_{\max}=4(3)(1)/(3+1)^2=3/4$. A perfectly aligned level need not transmit perfectly when the contact couplings are asymmetric. This distinguishes resonance location from coupling-limited peak height.

References

• Datta, S., Electronic Transport in Mesoscopic Systems, Cambridge University Press (1995), doi:10.1017/CBO9780511805776.
• Groth, C. W., Wimmer, M., Akhmerov, A. R., and Waintal, X., “Kwant: a software package for quantum transport,” New Journal of Physics 16, 063065 (2014), doi:10.1088/1367-2630/16/6/063065.