Advanced Electronic Structure Calculations for Nanoelectronics

Abstract

This paper describes our work over the past few years to use tools from quantum chemistry to describe electronic structure of nanoelectronic devices. These devices, dubbed “artificial atoms,” comprise a few electrons, confined by semiconductor heterostructures, impurities, and patterned electrodes, and are of intense interest due to potential applications in quantum information processing, quantum sensing, and extreme-scale classical logic. We detail two approaches we have employed: finite-element and Gaussian basis sets, exploring the interesting complications that arise when techniques that were intended to apply to atomic systems are instead used for artificial, solid-state devices.

18.8 Appendix: Efficient Gaussian Approximation of the Coulomb Kernel

Let us consider a family of generalized Gaussian functions of the form

$$\begin{aligned} g(\mathbf {x};\mathbf {A},\mathbf {b}) = \exp \left( - \mathbf {x}^T \mathbf {A} \mathbf {x} + i \mathbf {b}^T \mathbf {x}\right) , \end{aligned}$$

(18.51)

for a positive semidefinite $\mathbf {A}$. In applications to quantum mechanical simulations of semiconductor qubits, these functions are used as a basis for electronic wavefunctions. They appear in various Hamiltonian matrix elements together with the Coulomb kernel that mediates the electrostatic interaction between electrons and nuclei. Up to normalization factors, the 1-body matrix elements describing electron–nuclear interactions have the form

$$\begin{aligned} V_1 \propto \int \frac{g(\mathbf {x} - \mathbf {x}_1;\mathbf {A}_1,\mathbf {b}_1) g(\mathbf {x} - \mathbf {x}_2;\mathbf {A}_2,\mathbf {b}_2) }{|\mathbf {x}|} d\mathbf {x}, \end{aligned}$$

(18.52)

and the 2-body matrix elements describing electron–electron interactions have the form

$$\begin{aligned} V_2 \propto \int \frac{g(\mathbf {x} - \mathbf {x}_1;\mathbf {A}_1,\mathbf {b}_1) g(\mathbf {x} - \mathbf {x}_2;\mathbf {A}_2,\mathbf {b}_2) g(\mathbf {y} - \mathbf {x}_3;\mathbf {A}_3,\mathbf {b}_3) g(\mathbf {y} - \mathbf {x}_4;\mathbf {A}_4,\mathbf {b}_4) }{|\mathbf {x} - \mathbf {y}|} d\mathbf {x} d\mathbf {y} . \end{aligned}$$

(18.53)

An important restriction to this problem is the existence of a minimum length scale, which we define through bounds on coefficients,

$$\begin{aligned} \Vert \mathbf {A} \Vert _2 \le a_{\max } \ \ \ \mathrm {and} \ \ \ \Vert \mathbf {b} \Vert _2 \le b_{\max } . \end{aligned}$$

(18.54)

We would like an efficient approximation to $V_1$ and $V_2$ that achieves a target absolute accuracy $\epsilon$, which is set by the energy scale of interest.

Gaussian functions possess a great deal of simplifying mathematical structure, especially if they are isotropic ($\mathbf {A} = a \mathbf {I}$). The products of Gaussians that appear in $V_1$ and $V_2$ can be rewritten as a single Gaussian,

$$\begin{aligned} g(\mathbf {x} - \mathbf {x}_1;\mathbf {A}_1,\mathbf {b}_1) g(\mathbf {x} - \mathbf {x}_2;\mathbf {A}_2,\mathbf {b}_2) = c \, g(\mathbf {x} - \mathbf {x}_3;\mathbf {A}_3,\mathbf {b}_3) , \end{aligned}$$

(18.55)

with a new set of parameters,

$$\begin{aligned} \mathbf {A}_3 = \mathbf {A}_1 + \mathbf {A}_2 , \ \ \ \mathbf {b}_3 = \mathbf {b}_1 + \mathbf {b}_2, \ \ \ \mathbf {x}_3 = \mathbf {A}_3^{-1} (\mathbf {A}_1 \mathbf {x}_1 + \mathbf {A}_2 \mathbf {x}_2), \end{aligned}$$

(18.56)

and a complex coefficient with a magnitude that is less than one,

$$\begin{aligned} c = \exp ( \mathbf {x}_1^T \mathbf {A}_1 \mathbf {x}_1 + \mathbf {x}_2^T \mathbf {A}_2 \mathbf {x}_2 - \mathbf {x}_3^T \mathbf {A}_3 \mathbf {x}_3 + i \mathbf {b}_1^T \mathbf {x}_1 + i \mathbf {b}_2^T \mathbf {x}_2 - i \mathbf {b}_3^T \mathbf {x}_3) . \end{aligned}$$

(18.57)

The restricted family of Gaussians defined by (18.54) is not closed under products. Instead, the minimum length scales are reduced with narrower Gaussians and more rapid sinusoidal oscillations,

$$\begin{aligned} \Vert \mathbf {A}_1 + \mathbf {A}_2 \Vert _2 \le 2 a_{\max } \ \ \ \mathrm {and} \ \ \ \Vert \mathbf {b}_1 + \mathbf {b}_2 \Vert _2 \le 2 b_{\max } . \end{aligned}$$

(18.58)

However, this enables a reduction in form for $V_1$ and $V_2$ to

$$\begin{aligned} V_1&\propto \int \frac{g(\mathbf {x} - \mathbf {x}_1;\mathbf {A}_1,\mathbf {b}_1)}{|\mathbf {x}|} d\mathbf {x}, \end{aligned}$$

(18.59)

$$\begin{aligned} V_2&\propto \int \frac{g(\mathbf {x} - \mathbf {x}_1;\mathbf {A}_1,\mathbf {b}_1) g(\mathbf {y} - \mathbf {x}_2;\mathbf {A}_2,\mathbf {b}_2)}{|\mathbf {x} - \mathbf {y}|} d\mathbf {x} d\mathbf {y} . \end{aligned}$$

(18.60)

Unlike in the isotropic case, these integrals cannot be reduced to a finite number of standard special function evaluations. However, if the Coulomb kernel is approximated as a linear combination of n Gaussians,

$$\begin{aligned} \frac{1}{r} \approx \sum _{i=1}^n c_i \exp (-d_i r^2), \end{aligned}$$

(18.61)

then these integrals can be reduced to special functions.

It is not obvious that (18.61) represents a sensible approximation strategy. It is effectively a quadrature of an exact integral decomposition,

$$\begin{aligned} \frac{1}{r} = \frac{2}{\sqrt{\pi }} \int \limits _0^\infty \exp (-s^2 r^2) d s , \end{aligned}$$

(18.62)

but it lacks both the singularity and long tail of the Coulomb kernel. Because our goal is absolute accuracy rather than relative accuracy, we can safely cut off the small tails at very large separations. Because of the minimum length scale imposed by (18.54), we can safely remove the singularity. To better understand these approximations and their errors, we decompose a restricted generalized Gaussian into the convolution between a narrow normalized isotropic Gaussian kernel ($D \ge 2 a_{\max }$) and an unrestricted generalized Gaussian,

$$\begin{aligned} g(\mathbf {x};\mathbf {A}_1,\mathbf {b}_1) = C \left( \frac{D}{\pi } \right) ^{3/2} \int \exp (-D |\mathbf {x} - \mathbf {y}|^2) g(\mathbf {y};\mathbf {A}_2,\mathbf {b}_2) d \mathbf {y} \end{aligned}$$

(18.63)

with transformed parameters

$$\begin{aligned} \mathbf {A}_2 = \mathbf {A}_1 (\mathbf {I} - D^{-1} \mathbf {A}_1)^{-1} , \ \ \ \mathbf {b}_2 = \mathbf {A}_2 \mathbf {A}_1^{-1} \mathbf {b}_1, \end{aligned}$$

(18.64)

and a real coefficient that is always greater than one,

$$\begin{aligned} C = \sqrt{\mathrm {det}(\mathbf {A}_1^{-1} \mathbf {A}_2)} \exp \left( \frac{\mathbf {b}_1^T \mathbf {b}_2}{4 D} \right) . \end{aligned}$$

(18.65)

The normalized convolution kernel preserves the integral of positive functions, and the determinant in C accounts for the change in normalization between Gaussians of different widths in (18.63). The remaining amplification factor accounts for the suppression in amplitude when convolving oscillating functions with a Gaussian kernel, and it can take a maximum value of

$$\begin{aligned} F(D) \equiv \exp \left( \frac{b_{\max }^2 }{D - 2 a_{\max }}\right) . \end{aligned}$$

(18.66)

This factor will linearly amplify the error in (18.61) as it pertains to approximating $V_1$ and $V_2$. If $b_{\max } > 0$, then we must choose a D that is appreciably larger than $2 a_{\max }$ to suppress the amplification of our error bounds.

The key simplification to our kernel approximation is the use of (18.63) in (18.59) and (18.60). We split the narrow isotropic Gaussian kernel off from one of the generalized Gaussians and merge it with the Coulomb kernel to form a new translationally invariant kernel with no singularity,

$$\begin{aligned} \left( \frac{D}{\pi } \right) ^{3/2} \int \frac{1}{|\mathbf {x} - \mathbf {z}|} \exp (-D |\mathbf {z} - \mathbf {y}|^2) d\mathbf {z} = \frac{\mathrm {erf}( \sqrt{D} |\mathbf {x} - \mathbf {y}|)}{|\mathbf {x} - \mathbf {y}|} . \end{aligned}$$

(18.67)

The convolution of the normalized Gaussian kernel with the Gaussian kernels in (18.61) is also a Gaussian kernel,

$$\begin{aligned}&\left( \frac{D}{\pi } \right) ^{3/2} \int \exp (- d_i |\mathbf {x} - \mathbf {z}|^2) \exp (-D |\mathbf {z} - \mathbf {y}|^2) d\mathbf {z} = \left( \frac{D}{D+d_i}\right) ^{3/2} \\ \nonumber&\quad \exp \left( - \frac{D d_i}{D + d_i} |\mathbf {x} - \mathbf {y}|^2 \right) . \end{aligned}$$

(18.68)

It is then straightforward to minimize the maximum pointwise error between these nonsingular kernels,

$$\begin{aligned} \epsilon _n(D) = \min _{c_i , d_i} \max _{r \ge 0} \left| \frac{\mathrm {erf}(\sqrt{D}r)}{r} - \sum _{i=1}^n c_i \left( \frac{D}{D+d_i}\right) ^{3/2} \exp \left( - \frac{D d_i}{D + d_i} r^2 \right) \right| . \end{aligned}$$

(18.69)

This problem is scale invariant, so we can optimize for the specific case of $D = 1$,

$$\begin{aligned} \epsilon _n^0 = \min _{c^0_i , d^0_i} \max _{r \ge 0} \left| \frac{\mathrm {erf}(r)}{r} - \sum _{i=1}^n c^0_i \left( \frac{1}{1+d^0_i}\right) ^{3/2} \exp \left( - \frac{d^0_i}{1 + d^0_i} r^2 \right) \right| , \end{aligned}$$

(18.70)

and rescale the solutions,

$$\begin{aligned} \epsilon _n(D) = \sqrt{D} \epsilon _n^0, \ \ \ c_i = \sqrt{D} c_i^0, \ \ \ d_i = D d_i^0 . \end{aligned}$$

(18.71)

The error in the approximate $V_1$ and $V_2$ is proportional to $\epsilon _n(D) F(D)$, and we can minimize this error estimate over the choice of D,

$$\begin{aligned} D_{\mathrm {opt}} = 2 a_{\max } + \frac{1}{2} b_{\max }^2 \left( 1 + \sqrt{1 + \frac{8 a_{\max }}{b_{\max }^2}}\right) . \end{aligned}$$

(18.72)

which reduces the error estimate to

$$\begin{aligned} \epsilon _n(D_{\mathrm {opt}}) F(D_{\mathrm {opt}}) \le e \sqrt{D_{\mathrm {opt}}} \epsilon _n^0 . \end{aligned}$$

(18.73)

Thus a well-chosen minimum length scale ($D_{\mathrm {opt}}$) mitigates the amplification of errors resulting from oscillations in the generalized Gaussians. All that remains is the calculation and tabulation of minimax solutions.

We can simplify the minimax problem by redefining variables to get

$$\begin{aligned} \epsilon _n^0 = \min _{\alpha _i , \beta _i} \max _{r \ge 0} \left| \frac{\mathrm {erf}(r)}{r} - \sum _{i=1}^n \alpha _i \exp \left( - \beta _i r^2 \right) \right| . \end{aligned}$$

(18.74)

We solve this nonlinear minimax problem by repeatedly solving a linearized minimax problem and updating the nonlinear solution a small amount. This strategy is simple to implement, but it is fragile and requires good initial guesses to be within the basin of convergence of the global minimizer. We solve for a solution update that satisfies equioscillation of the residual function,

$$\begin{aligned} \frac{\mathrm {erf}(r_i)}{r_i} - \sum _{j=1}^n \alpha _j \exp \left( - \beta _j r_i^2 \right) + \sum _{j=1}^n \left[ \frac{\alpha _j \delta \beta _j}{\beta _j} \right] \beta _j r_i^2 \exp \left( - \beta _j r_i^2 \right) = (-1)^i \epsilon , \end{aligned}$$

(18.75)

at an ordered set of points $r_i$, where $\alpha _i$ and the quantity in brackets are the basic variables in the linear minimax problem. We then choose a new set of $r_i$ corresponding to the local extrema of the residual error,

$$\begin{aligned} \frac{2}{\sqrt{\pi }} \frac{\exp (-r_i^2)}{r_i} - \frac{\mathrm {erf}(r_i)}{r_i^2} + \sum _{j=1}^n 2 \alpha _j \beta _j r_i \exp \left( - \beta _j r_i^2 \right) \nonumber \\ + \sum _{j=1}^n \left[ \frac{\alpha _j \delta \beta _j}{\beta _j} \right] 2 \beta _j r_i ( 1 - \beta _j r_i^2) \exp \left( - \beta _j r_i^2 \right) = 0, \end{aligned}$$

(18.76)

and iterate the process until convergence. We accurately solve the linearized minimax problem within an inner loop before performing a nonlinear update in an outer loop to improve stability.

We construct minimax solutions for $1 \le n \le 84$, which is as far as double-precision arithmetic will allow us to go (the stability of the LAPACK linear solver for ill-conditioned matrices is the limiting factor). Because the Gaussian approximation cannot describe the Coulomb tail, the range over which the Gaussians have numerical support must be $1/\epsilon$ to attain an accuracy of $\epsilon$. On a logarithmic scale, the Gaussian widths in the kernel approximation are roughly uniformly distributed. Naïvely, we expect an exponential reduction of error in the density of these Gaussians, which suggests a scaling of

$$\begin{aligned} \epsilon _n^0 \propto \exp (- n / \ln (1/\epsilon _n^0)) \propto \exp (- \sqrt{n}) . \end{aligned}$$

(18.77)

Fig. 18.2

Maximum error in Coulomb kernel approximation by n Gaussians

As shown in Fig. 18.2, the observed error fits this form very well. This result demonstrates that there is still much to be gained in approximation efficiency by enforcing an upper bound on the length scales relevant to simulations utilizing approximate Coulomb kernels. A length-scale upper bound would enable an exponential reduction of approximation error in the number of Gaussians.