This is a part of research I am current working with Ewan Lister under supervision of Dr. Baruch Feldman at the University of Washington.

This post covers an overview of the research in informal language.

This research had been presented in UW Undergraduate Research Symposium 2023. The abstract for the presentation can be found at this link

For investigations into nanoscopic properties of matter, computational simulation is a key tool. In particular, simulations of the electronic configuration and conductance of nanoscale devices facilitate the continued miniaturization of semiconductor devices used in integrated circuits and computer processors. However, due to the importance of quantum mechanics at these scales, accurate calculations can be highly costly. In our research we consider improvements to one such parallelizable electronic transport code, TRANSEC. We seek to better understand how a Monte Carlo technique, combined with a special polynomial expansion, may improve the scaling of TRANSEC’s computing time with the size of the simulation. We apply the Monte Carlo technique within a simplified tight-binding model of a TRANSEC calculation. We test how the Monte Carlo technique facilitates the calculation of the electronic transmission probability, given an initial Hamiltonian energy matrix along with absorbing boundary conditions (referred to as complex absorbing potentials, or CAPs). We expect that the results of this study may provide algorithms which allow for faster calculation times when integrated at scale into TRANSEC. Improvements to computing time for determining parameters such as nanoscopic conduction helps to advance computer modeling of nanoscopic structures (such as transistors, interconnect, or molecular electronics), which could benefit semiconductor technology.

Background

We consider electronic conductance from the first principal level which can formulated as

where $f$ is a Fermi Distribution, $h$ is Plack’s constant, e is electronic charge. $T(E)$ is a transition probability, which is our main interest. $T(E) \sim |G^r(E)|^2$ defined by

In this project, we are interested in $T(E)$.

Dr. Feldman, our research mentor, employed a real space grid to solve for $G^r(E)$ and calculate $T(E)$. The method exhibits several advantages, including, but not limited to, being iterative, parallelizable, and straightforward.

Methodology

At the symposium, we presented the approximation of $T(E)$ using the Monte Carlo method, Faber Polynomial Expansion, and the Iterative method. We applied the polynomial expansion and the iterative method separately.

Firstly, note that, from literature,

By defining, $ S = \sqrt{\Gamma_R} G^r(E) \sqrt{\Gamma_L}$, we can reformulate above as,

The idea to apply Monte is the trace trick. Let $\psi_k = e^{\theta_k i}$ be a wave function of unit amplitude with random phase, $\theta_k$, where $n$ is a number of sample. Consider that $E[\psi_k] = 0$ and $Cov(\psi_k) = I$ Then, by trace trick, $E[\psi_k^T S^T S \psi_k] = 0^T S^T S 0 + tr( S^{\dagger} S I ) = T(E)$. Following from the Law of Large Number,

In Bracket notation,

With the approximation, in addition to that one multiplication factor does not longer scale with the dimension of S (but a sample of monte carlo), each $\braket{\psi_k | S^{\dagger} S | \psi_k}$ can be computed in parallel.

Now, we consider the computation of retarded Green function, $G^r(E) = [E1 - H_op - i \eta]^{-1}$.

Faber polynomial

We employ the Faber Polynomial Expansion of a matrix function to approximate its inverse. The Faber Polynomial is generated by,

where $F_n(z, W)$ is a generalized Faber Polynomial, and \Phi(s) = s + %\alpha_0 + \alpha_1 s^{-1} + \alpha_2 s^{-2} + … is a conformal mapping from an exterior of a disk to an exterior of a simply-connected domain %containing $z$. Consider by taking $W(s) = 1$, $\phi(s) = E \mathbf{1}$, $z = H_{op} - \eta i$, we can formulate,

And by truncating the summation term, we approximate $G^r(E)$.

Iterative Methods

As an alternative to polynomial expansion, we can approximate $G^r(E)$ using an iterative method. Leveraging the benefits of the Monte Carlo technique, we can express it as follows:

Suppose that $y = S \ket{\Psi_k}$. We want to compute the square of the norm of $y$. Thus, we can formulate,

That is, we obtained a system of linear equation, i.e. $Ax = b$. For large dimension of $A$, iterative methods can be used to numerically solve for $y$.

We previously use Quasi-Minimal Residual method to solve for $y$.

By combining either Polynomial Expansion or iterative methods to the Monte Carlo method, we can efficiently approximate $T(E)$.

Current Progress

Alhough iterative methods theoretically solve for $y$ rapidly, practical implementation is impeded by numerical floating point errors, which hinder the convergence of $y$ towards the actual solution. Regarding the Faber Polynomial, we employed conformal mapping in the shape of an ellipse, a tuning process that is challenging to adjust to meet the necessary requirements for polynomial convergence.

Reference

Roi Baer et al. “Ab initio study of the alternating current impedance of a molecular junction”. In: The Journal of Chemical Physics 120.7 (Feb. 2004), pp. 3387–3396. issn: 0021-9606. doi: 10.1063/1.1640611. eprint: https://pubs.aip.org/aip/jcp/article- pdf/120/7/3387/10858645/3387_1_online.pdf. url: https://doi.org/10.1063/1.1640611.

Baruch Feldman et al. “Real-space method for highly parallelizable electronic transport calculations”. In: Phys. Rev. B 90 (3 July 2014), p. 035445. doi: 10.1103/PhysRevB.90.035445. url: https://link.aps.org/doi/10.1103/PhysRevB.90.035445.

Youhong Huang, Donald J. Kouri, and David K. Hoffman. “General, energy-separable Faber polynomial representation of operator functions: Theory and application in quantum scattering”. In: The Journal of Chemical Physics 101.12 (Dec. 1994), pp. 10493–10506. issn: 0021-9606. doi: 10.1063/1.468481. eprint: https://pubs.aip.org/aip/jcp/article- pdf/101/12/10493/15360236/10493_1_online.pdf. url: https://doi.org/10.1063/1.468481.

Toshiaki Iitaka et al. “Calculating the linear response functions of noninteracting electrons with a time-dependent Schr ̈odinger equation”. English. In: Physical Review E 56.1 SUPPL. B (July 1997), pp. 1222–1229. issn: 2470-0045. doi: 10.1103/physreve.56.1222. Wanchaloem Wunkaew, Ewan Lister, Baruch Feldman (University of Washington, Seattle, WA)Improving Electronic Conductance Calculations with a Monte Carlo ApproachMay 19th, 2023 19 / 20