Energy gap renormalization and diamagnetic susceptibility in quantum wires with different cross-sectional shape

Abstract

In this study, we investigate the effect of the cross-sectional shape on the energy gap renormalization and diamagnetic susceptibility in various quantum wires. To this end, we consider quantum wires with different cross-sectional shapes such as circular, square, hexagonal, and triangular. First, we employ the finite-element method and Arnoldi algorithm to solve the Schrödinger equation. Then, we calculate the energy levels, wavefunctions, binding energy, energy gap renormalization, and diamagnetic susceptibility. Our numerical results show that the binding energy decreases when the cross-sectional area is increased for all the quantum wires. Moreover, it is inferred that the cross-sectional shape is not important for large cross-sectional area when calculating the binding energy. Indeed, the main parameter is the cross-sectional area rather than the length of a side. The energy gap renormalization decreases with increasing cross-sectional area, regardless of the impurity concentration. We observe that the highest and lowest energy gap renormalization correspond to triangular and circular quantum wires, respectively. The absolute value of the diamagnetic susceptibility increases with increasing cross-sectional area for all the quantum wires investigated.

Appendix

To construct an approximate solution to Eq. (4), introduce the domain partition ${\Omega }={\Omega }_1 \cup {\Omega }_2$, such that ${\Omega }= {\prod }_{i=1}^n V_i$. Each element $V_i$, $i=1,2,\cdots ,n$ is a triangle with three vertices. The set of basis functions consists of continuous, piecewise-linear functions on ${\Omega }$ vanishing on the boundary $\partial {\Omega }$. Let $P_1, P_2, \cdots , P_n$ be the interior vertices, that is, those that do not lie on $\partial {\Omega }$. Then, the set of basis function is $\left\{ {\phi _1 ,\phi _2 ,\cdots } \right\}$, where

$$\begin{aligned} \phi _i \left( {P_j } \right) =\left\{ {{\begin{array}{l@{\quad }l} 1&{} {i=j} \\ 0&{} {i\ne j} \\ \end{array}} \qquad i,j=1,2,\cdots \cdots , m}. \right. \end{aligned}$$

(6)

The approximate solution $\psi$ can be represented as

$$\begin{aligned} \psi = \mathop \sum \limits _{j=1}^m c_j \phi _j , \quad \psi \left( {P_j} \right) = c_j, \quad j=1,2,\cdots , m. \end{aligned}$$

(7)

Inserting this $\psi$ into Eq. (4), multiplying Eq. (4) by $\phi _i$, and integrating by parts, one obtains

$$\begin{aligned}&\frac{\hbar ^{2}}{2m_1 }\sum \limits _{j=1}^m \int \limits _{{\Omega }_1} c_j \nabla \phi _j \nabla \phi _i \, \mathrm {d} x \, \mathrm {d} y+\frac{\hbar ^{2}}{2m_2 } \mathop \sum \limits _{j=1}^m \int \limits _{{\Omega }_2} c_j \nabla \phi _j \nabla \phi _i \, \mathrm {d} x \, \mathrm {d} y \nonumber \\&\qquad + \mathop \sum \limits _{j=1}^m c_j \left[ \int \limits _{{\Omega }_1} \left( {V_1 +V^\mathrm{int}} \right) \phi _j \phi _i \, \mathrm {d} x \, \mathrm {d} y \right. \nonumber \\&\qquad \left. + \int \limits _{{\Omega }_2} (V_2 +V^\mathrm{int})\phi _j \phi _i \, \mathrm {d} x \, \mathrm {d} y \right] \nonumber \\&\quad = E \mathop \sum \limits _{j=1}^m \int \limits _{\Omega } c_j \phi _j \phi _i \, \mathrm {d} x \, \mathrm {d} y. \end{aligned}$$

(8)

Using the following notations:

$$\begin{aligned} K_{ij}= & {} \frac{\hbar ^{2}}{2m_1 } \int \limits _{{\Omega }_1} \nabla \phi _j \nabla \phi _i \, \mathrm {d} x \, \mathrm {d} y+ \int \limits _{{\Omega }_1} V_1 \phi _j \phi _i \, \mathrm {d} x \, \mathrm {d} y \nonumber \\&+ \frac{\hbar ^{2}}{2m_2 } \int \limits _{{\Omega }_2} \nabla \phi _j \nabla \phi _i \, \mathrm {d} x \, \mathrm {d} y+ \int \limits _{{\Omega }_2} V_2 \phi _j \phi _i \, \mathrm {d} x \, \mathrm {d} y \nonumber \\&+\int \limits _{\Omega } V^\mathrm{int}\phi _j \phi _i \, \mathrm {d} x \, \mathrm {d} y \end{aligned}$$

(9)

and

$$\begin{aligned} M_{ij} = \int \limits _{\Omega } \phi _j \phi _i \, \mathrm {d} x \, \mathrm {d} y, \quad i,j=1,2,\cdots ,m, \end{aligned}$$

(10)

one obtains the generalized eigenvalue equation $KX=EMX$, where K is the stiffness matrix, M is the mass matrix, and $X=\left[ {c_1 ,c_2 ,\cdots ,c_m } \right] ^\mathrm{T}$ is the eigenvector corresponding to eigenvalue E. The generalized eigenvalue equation $KX=EMX$ is now solved by the Arnoldi procedure. In the following, we briefly present the Arnoldi algorithm in more detail. The Arnoldi method turns out to be one of the most successful algorithms for determining eigenvalues of large matrices [43, 45]. In numerical linear algebra, the Arnoldi method is an eigenvalue algorithm and an important example of an iterative procedure. The Arnoldi method finds the eigenvalues of general matrices. Here, we apply the algorithm to solve the general eigenvalue problem $KX=EMX$ derived from the finite-element discretization.

First, a shift $\sigma$ is determined if eigenvalues in the vicinity of a fixed parameter $\sigma$ are of interest. Since both K and M are positive, it is natural to take $\sigma =0$ and obtain the smallest eigenvalues. In other cases, one takes $\sigma$ near the eigenvalues which are sought. One subtracts $\sigma M$ from the eigenvalue equation to yield

$$\begin{aligned} \left( {K-\sigma M} \right) X=\left( {E-\sigma } \right) MX. \end{aligned}$$

(11)

This problem can be transformed into the following standard eigenproblem:

$$\begin{aligned} AX= {\Theta } X, \end{aligned}$$

(12)

where

$$\begin{aligned} A = \left( {K-\sigma M} \right) ^{-1}M, \quad \hbox {and } \qquad {\Theta } = \frac{1}{E-\sigma }. \end{aligned}$$

(13)

The largest eigenvalues ${\Theta }_i$ of the transformed matrix A now correspond to the eigenvalues $E_i =\sigma + \frac{1}{{\Theta }_i}$ of the original problem $KX=EMX$ closest to $\sigma$.

The Arnoldi algorithm computes an orthonormal basis V where A is represented by a tridiagonal matrix H, $AV_j =V_j H_{jj} +E_j$, such that $V_j$ and $E_j$ are matrices of size $m\times j$ and $H_{jj}$ is a matrix of size $j\times j$. The subscripts mean that $V_j$ and $E_j$ have j columns and $H_{jj}$ has j rows and columns. When no subscripts are used, the vectors and matrices have size m.

The basis V is built one column $v_j$ at a time. The first vector $v_1$ is chosen at random. In step j, the first j vectors are already computed and form the $m\times j$ matrix $V_j$. The next vector $v_{j+1}$ is calculated by first letting A operate on the newest vector $v_j$, then making the result orthogonal to all previous vectors.

This is formulated as $h_{j+1,j} v_{j+1} =Av_j -V_j h_j$, where the column vector $h_j$ consists of the Gram–Schmidt coefficients and $h_{j+1,j}$ is the normalization factor that gives $v_{j+1}$ unit length. Putting the corresponding relations from previous steps in front of this, one gets

$$\begin{aligned} AV_j =V_j H_{jj} +v_{j+1} h_{j+1,j} e_j^\mathrm{T} , \end{aligned}$$

(14)

where $H_{jj}$ is a $j \times j$ Hessenberg matrix with the vectors $h_j$ as columns. The second term on the right-hand side is nonzero only in the last column. The earlier normalization factors show up in the subdiagonal of $H_{jj}$.

The eigensolution of the small Hessenberg matrix $H_{jj}$ gives approximations to some of the eigenvalues and eigenvectors of the large matrix operator A. The eigenproblem

$$\begin{aligned} H_{jj} S_i ={\Theta }_i S_i , \quad i=1\cdots j \end{aligned}$$

(15)

can be solved inexpensively using the QR algorithm [34]. Then, $y_i =V_j S_i$ is an approximate eigenvector of A, and its residual is

$$\begin{aligned} r_i= & {} Ay_i -{\Theta }_i y_i =AV_j S_i -V_j S_i {\Theta }_i \nonumber \\= & {} \left( {AV_j -V_j H_{jj} } \right) S_i =v_{j+1} h_{j+1,j} S_{ij}. \end{aligned}$$

(16)

This residual has to be small in norm for ${\Theta }_i$ to be a good eigenvalue approximation. The norm of this residual is $r_i =\left| {h_{j+1,j} S_{ij} } \right|$. This norm is an error indicator and is used in termination conditions.

Indeed, the sequence of the k largest eigenvalues of $H_{jj}$ is monotonically increasing and bounded above by the k-th largest eigenvalue of A. The resulting algorithm is shown in the following:

• (1) Choose initial vector $v_1$ with $\Vert v_1 \Vert =1$;

• (2) For $j=0,1,2\cdots$ do:

  • (a) $w:=Av_j$;

  • (b) For $i=1,2,\cdots ,j$ do:

    $$\begin{aligned} \quad \quad h_{ij}:= & {} w_1 v_i , \\ \quad \quad w:= & {} w-h_{ij} v_{ii} \end{aligned}$$

  • (c) $h_{j+1,j} = \Vert w \Vert$;

  • (d) $v_{j+1} =w/h_{j+1,j}$;

  • (e) Solve projected eigenproblem $H_{jj} S={\Theta } S_i$

  • (f) If $Max \left| {h_{j+1,j} S_{ij} } \right| <tol$, stop;

    $$\begin{aligned} \quad i=1,2,\cdots j \end{aligned}$$

• (3) End for