Superlattices with coupled degenerated spectrum

Since the experimental beginnings of superlattices some decades ago [1], there has been large interest in more complicated devices like those where longitudinal as well as transversal or lateral potentials [2] are designed at will. The confinement at small scales in two directions leads to important effects in the electronic spectra. Translational invariance in both directions is broken, making the physical phenomena richer among other aspects through the coupling of longitudinal normal modes. However, the study of superlattices with two-dimensional potentials has not been conducted with sufficient detail, since advances have been achieved principally for quasi–one-dimensional problems, for which the transversal directions effects have been simplified or uncoupled.

In this letter, superlattices or multilayers with coupled channels are described by sectionally constant potentials in the longitudinal direction and arbitrary lateral dependence, to allow the explicit analytic calculation of the scattering amplitudes. Three characteristic features are considered in detail: number of cells, number of partitions in each cell, and number of coupled normal modes or channels. Using the scattering matrix method, the relevant functions are expressed in terms of Chebyshev polynomials, leading to a transparent analytical and fast numerical tool. It is shown that quantum tunneling and interference phenomena in superlattices with interacting energy modes present features not reported before, along with sharp resonances and threshold effects. In particular, under analytically easily formulated conditions, superlattices with coupled modes lead to either zero transmission for a single mode, or simultaneous exact zero transmission in two or more distinct modes. This behaviour depends on the kind of "colliding" or degenerating eigenvalues, whether they are or not coupled. This last result could make it possible to determine experimentally the coupling strength between modes in a superlattice. The energy spectra are efficiently calculated in a transparent way using simplified symmetric functions.

The (unitary) scattering matrix S is introduced by $\Phi _{\textit{out}}=S\Phi_{\textit{in}}$, where $\Phi _{\textit{out}}$ and $\Phi _{\textit{in}}$ are the incoming and outgoing wave vectors on the left- and right-hand sides of the superlattice. The S-matrix is here given as

$$\begin{equation} S=\left(\begin{array}{c@{\quad}c} r & t^{\prime} \\ t & r^{\prime} \end{array}\right), \end{equation} \tag{ 1 }$$

where the reflection and transmission amplitudes r, t and $r^{\prime},t^{\prime}$, corresponding to the incidence from the left- and right-hand side, respectively, are to be found. The two-terminal conductance [3] is given by Landauer's formula as

$$\begin{equation*} G= \frac{2e^2}{h} N\bar T, \end{equation*}$$

in terms of the average transmission probability $\bar T$ and the number N of transverse channels in the conductor leads.

For electronic transport in superlattices, formed by sequences of n equal cells and effective mass m, Schrödinger's equation in each cell is given in terms of a two-dimensional external potential $V(x,y)$. It is assumed that the superlattice has transversal width w in the y-direction and that electron transport occurs in the longitudinal x-direction. For clarity, the potential is considered as infinite for y outside the interval $(0,w)$. It is not possible to uncouple the y-dependence, as is usual in infinite crystals, since there is no translational invariance.

A direct way to solve the problem is to expand the wave function in N transversal normal modes or channels as $\psi(x,y)= \sum_{j=1}^{N} \phi_{j}(x)\sin(k_j y)$, with $k_j=j\pi/w$, so that $\psi(x,0)=0=\psi(x,w)$. This simple procedure leads to a system of N differential equations

$$\begin{equation} \phi''(x)- u^2 \phi(x)=0, \end{equation} \tag{ 2 }$$

where $\phi=(\phi_1,\dots,\phi_N)^T$, $u^2=\frac{2m}{\hbar ^{2}}(V-E\textbf{1})+K^{2}$, is a symmetric $N\times N$ matrix, 1 is the $N\times N$ identity matrix, $K= \text{diag}(k_1,\dots,k_N)$, and coupling matrix elements

$$\begin{equation} V_{jm}(x)= \int_0^w V(x,y)\sin(k_j y)\sin(k_m y)\text{d}y. \end{equation} \tag{ 3 }$$

The next step is to divide each cell into a set of I distinct layers with constant matrices $u_i=u(\xi_i)$ for $\xi_i\in [x_i,x_{i+1}],\quad i=0,1,\dots,I-1$. Thus, the longitudinal potential matrix elements are approximated by sectionally constant functions with discontinuities at $x=x_i$ for $i=0,1,\dots,I$, where x₀ and x_I denote the boundary points of the cell. Continuity of the wave function and its derivative is imposed at all interfaces between layers.

Defining $f=(f_1,\dots,f_N)^T$ with $f_j=(\phi_j,\phi_j')^T$, it follows that

$$\begin{equation} f'=U_i f,\quad \text{with}\quad U_i=\left(\begin{array}{c@{\quad}c} 0 & \textbf{1}\\ {u_i^2} & 0 \end{array}\right),\quad x\in(x_i,x_{i+1}). \end{equation} \tag{ 4 }$$

The solution of this system is

$$\begin{equation} f(x)=W_{\textit{cell}}f(x'), \end{equation} \tag{ 5 }$$

with $x_0<x'<x_1,\,x_{I-1}<x<x_I$ and where

$$\begin{equation*} W_{\textit{cell}}=W_I(x-x_{I-1})\cdots W_2(x_2-x_1)W_1(x_1-x'), \end{equation*}$$

and $W_i(x)=\exp\{xU_{i}\}$. After expanding in power series,

$$\begin{equation} W_i=\left(\begin{array}{c@{\quad}c} \cosh(x u_i) & u_i^{-1}\sinh(x u_i)\\ u_i\sinh(x u_i)& \cosh(x u_i) \end{array}\right). \end{equation} \tag{ 6 }$$

The transfer matrices W_i and $W_{\textit{cell}}$ are symplectic. The remaining calculation can be made using the well-known [4] Sylvester formula for matrix functions, which includes the important possibility of degenerated spectra. The transfer matrix W of an arrangement of n adjacent equal cells is the n-th power of the cell matrix $W=W_{\textit{cell}}^n$.

The application of the Floquet theorem idea [5] to superlattices for eigenvectors leads with eq. (5) to

$$\begin{equation*} f(x+a)=\lambda f(x)= W_{\textit{cell}} f(x), \end{equation*}$$

for $a=x_I-x_1$, so that the multiplier λ is an eigenvalue of $W_{\textit{cell}}$; i.e., it must satisfy the characteristic equation

$$\begin{equation} p(\lambda)= \text{det}(W_{\textit{cell}} -\lambda\textbf{1})=0. \end{equation} \tag{ 7 }$$

Therefore, to obtain bounded wave vectors $f(x+na)$ for large superlattices, it is necessary that $\vert\lambda \vert\le 1$.

The Cayley-Hamilton theorem states that the characteristic polynomial $p(\lambda)$ of $W_{\textit{cell}}$ is nullified by its eigenvalues as well as by $W_{\textit{cell}}$ itself. To obtain a relation for the n-th power of $W_{\textit{cell}}$, the usual procedure (see, for example, [6]) is to start with the characteristic equation for $W_{\textit{cell}}$, which leads to the recurrence formula

$$\begin{equation} W_{\textit{cell}}^{2N+l}= e_{1}W_{\textit{cell}}^{2N-1+l} + \dots + e_{2N-1} W_{\textit{cell}}^{l+1}- W_{\textit{cell}}^{l}, \end{equation} \tag{ 8 }$$

with $l=0,1,\dots$. Here, e_m stands for the elementary homogeneous symmetric functions of the eigenvalues $\{\lambda_{i}\}$. Introducing a generating function, it follows that all powers of $W_{\textit{cell}}$ can be obtained recursively in terms of the first $(2N-1)$-th powers of $W_{\textit{cell}}$ as

$$\begin{equation} W_{\textit{cell}}^{2N+l}= \sum_{k=0}^{2N-1}(-)^k s_{l,k} W_{\textit{cell}} ^{2N-1-k}. \end{equation} \tag{ 9 }$$

The coefficients s_l,k can be explicitly obtained, and are given by the polynomials of degree $l+k+1$,

$$\begin{equation} s_{l,k}= \sum_{m=0}^{k} (-)^{m} e_m h_{l+k+1-m},\qquad k\le 2N, \end{equation} \tag{ 10 }$$

which are the Schur functions [6] for the partition $(l+1,1^k)$ of the integer $l+1+k$ as the sum of the integer l + 1 plus k times the unit. The h_i are the complete homogeneous symmetric functions of the eigenvalues.

Having the characteristic polynomial real coefficients, if λ is an eigenvalue, then $\lambda^*$ is also an eigenvalue, in general distinct. Further, if λ is an eigenvalue, then $1/\lambda$ is an eigenvalue, too, since the matrix $W_{\textit{cell}}$ is symplectic, $\text{det}(J)=(-1)^N$ and $p(\lambda)= \lambda^{2N} p(1/\lambda)$. This leads to the simplification $e_{j}=e_{2N-j},\quad j=1,\dots,N-1$, so that only half of the coefficients of the characteristic polynomial need to be computed. Notice that for $\vert\lambda\vert=1$, it follows that $\lambda^*$ and $1/\lambda$ coincide. The polynomial can be written as $p(\lambda)=\lambda^N \tilde{p}(\mu)$, where $\mu= \lambda + \lambda^{-1}$ and a reduced characteristic polynomial $\tilde{p}$, which is a polynomial of N-th degree in μ, halving the computations. The eigenvalues of the reduced characteristic equation, $\mu_i=\lambda_i+1/\lambda_i$, shall be called here the partial traces, since their sum is the full trace. Introducing $\lambda_i= \exp(i\phi_i)$, with $\phi_i=-\phi_{i+N}$ for $i=1,\dots,N$, the $\phi_i$ are complex numbers in general. Considering now the Waring-Louck [7] expression, we have

$$\begin{equation*} h_l= \sum_{i=1}^{2N}\frac{\lambda_i^{2N+l-1}}{\prod_{j\ne i}(\lambda_i-\lambda_j)}. \end{equation*}$$

Ordering the eigenvalues as $(\lambda_1,\dots, \lambda_N,\lambda_1^{-1},\dots, \lambda _N^{-1})$, h_l can be written after some rearrangements as

$$\begin{equation} h_l= \sum_{i=1}^{N}\frac{U_{N+l-1}(\mu_i/2)}{2^{N-1}\prod_{j\ne i}(\mu_i/2- \mu_j/2)}, \end{equation} \tag{ 11 }$$

where $U_{k-1}(\cos(\phi))= \sin(k\phi)/\sin(\phi)$ is the Chebyshev polynomial of the second kind of the $(k-1)$-th degree. The eigenvalues appear only through the partial traces. The functions h_l are a kind of Chebyshev polynomials in N variables. This is the generalization of the old result for a single channel leading to the Chebyshev polynomials of the second kind in one variable. Other functions of the eigenvalues can be similarly rewritten. Of importance here is the discriminant of the characteristic polynomial

$$\begin{equation*} \Delta=\prod_{i<j}(\lambda_i-\lambda_j)^2=\prod_{i}(\mu_i^2-4)\prod_{i<j}(\mu_i-\mu_j)^4. \end{equation*}$$

This expression vanishes for degenerated eigenvalues and is of great help to understand the spectrum. From the point of view of the theory of Lie algebras and groups, the important quantity is their rank, which determines how many independent algebra generators are involved.

The transfer matrix eigenvalues are algebraic functions of the traces of its powers. But since the traces are known analytic functions of the energy, by the implicit function theorem, the energy can be seen as a complex function $E(\lambda)$ of the eigenvalues. Since for sufficiently large energy the Schrödinger equation uncouples, generically, the transfer matrix eigenvalues are asymptotically complex with unit moduli. Topologically, depicted on the surface of a unit cylinder with the energy as axis, the eigenvalues $\lambda_i$ with unit moduli and their inverses will give N pairs of curve segments evolving in opposite senses and intersecting each other at angles 0 and π corresponding to degenerated eigenvalues $\lambda_{i+}=\lambda_{i-}=\pm1$. For at least two channels, beyond these crossings between inverse eigenvalues, there will also be crossings between curves of distinct eigenvalues, $\lambda_i$ and $\lambda_j$ for $i\ne j$, degenerating at other angles. This is a distinct type of degeneration. For non-zero coupling the curves leave the unit cylinder by increasing or decreasing the radii of the eigenvalues, following some branch of the energy Riemann surface. For zero coupling, after colliding, the eigenvalues remain on the unit cylinder. The eigenvalues collisions can take place in allowed spectral regions, with non-zero transmission, but also in forbiden regions for eigenvalues with non-unit moduli outside or inside the unit cylinder. In the last case, they would not be observed directly.

For a periodic lattice and a single channel some of the above facts had been recognized since the times of Bloch, Kronig-Penney and Kramers [8]. For a single channel the reduced characteristic equation is $\mu-e_1=0$ and leads to the old Kronig-Penney condition $\vert\text{Tr}(W_{\textit{cell}})\vert < 2$ for non-zero transmission as used by Esaki [1]. Using $s_{(l+1,1)}=-h_l$, the well-known text-book result [9] follows $W_{\textit{cell}}^{2+l}=U_{l+1}(\mu/2) W_{\textit{cell}}-U_{l}(\mu/2)$.

Consider now a cell formed by only two layers with potentials V₀ and V₁, of lengths a and b. To the left in fig. 1 the energy is shown in the vertical direction as a function of λ in the horizontal complex plane for $a=40\ \unicode{8491}$, $b=45\ \unicode{8491}$, and $V_0=0$ and $V_1=0.4\ \text{eV}$, as a reference a unit cylinder is also depicted. The two evolving curves correspond to the two roots $\lambda_\pm$ and their projection on the horizontal plane are a unit circle with two small straight real segments, which correspond to the vertical ellipses. To the right, the top figure shows the real part of the complex trace as a function of the eigenvalues, the cubic $x^3-zx^2+(1+y^2)x -y^2z=0$, for $x=\text{Re}\,\lambda$, $y=\text{Im}\,\lambda$ and $z=\text{Re}\,\mu$, intersecting the unit cylinder. The bottom right figure displays a detail with two ellipses joined to two parabolic segments embedded in the Riemann surface. The ellipses correspond to real λ and the parabolic segments to $\vert \lambda\vert= 1$. The trace of the whole transfer matrix (see blue curve) determines the energy values for which the transmission coefficient is unit, as shown in fig. 2 for 30 layers with $a=40\ \unicode{8491}$, $b=40\ \unicode{8491}$, $V_0=0.4\ \text{eV}=-V_1$. The eigenvalues are degenerated for $\vert\mu\vert=2$ and there is no transmission for those regions for which $\vert\mu\vert>2$ (see red curve). The figure on the right gives in black the forbidden spectrum for $V=(-3\ \text{eV},3\ \text{eV})=-V_0$ and energies $E=(-0.5\ \text{eV},1 \ \text{eV})$ (vertical coordinate).

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For coupled channels the problem is richer, not only because there are more partial traces, but also due to the presence of sharp resonances and cusps, among other irregularities [10], in the transmission coefficients. The channels will open according to the energy thresholds. Starting below the first energy threshold, the first partial trace to become smaller than two will determine the first branch of the Riemann surface. The subsequent decreasing partial traces will determine the next branches. The diverging Chebyshev polynomials of the traces larger than two are associated to forbidden energy bands. Following the eigenvalues behaviour mentioned above, as the energy increases, all partial traces shall oscillate around the strip ±2 giving rise to N curves with alternating forbidden and allowed energy regions. These curves determine the regions where transmission in a given channel is zero, perfect or only partial. Thus, the new trace conditions are $\vert\mu_i\vert< 2$ for non-zero transmission. For $\vert\mu_i\vert=2$, the eigenvalues are degenerated. Further, two or more partial traces could coincide at certain energies, due to degenerated eigenvalues but with $\lambda_{i+}\ne \lambda_{i-}$. Between two consecutive degenerated eigenvalues of this new kind, their moduli would be not equal to one, leading to vanishing transmission coefficients over an energy interval. This fact could be of experimental interest, as far as, for example, the length of the interval would depend on the coupling strength.

For two channels the reduced equation is $\mu^2 -e_1\mu +e_2-2=0$ with $e_1=\text{Tr}(W _{\textit{cell}})$ and $e_2=((\text{Tr}(W_{\textit{cell}}))^2-\text{Tr}(W _{\textit{cell}}^2))/2$. Calling $\mu_{1,2}$ the solutions, the discriminant here is $(\mu_1-\mu_2)^2=e_1^2-4e_2+8$ and vanishes for degenerated eigenvalues of distinct channels, $\lambda_{1,2}$, so that $\mu_{1,2}=\text{Tr}(W_{\textit{cell}})/2$. When the discriminant is negative, the partial traces shall be complex, so that the transfer matrix eigenvalues will have moduli distinct of unit and, therefore, both transmission coefficents will be zero over the corresponding energy region.

The generalized Kronig-Penney condition is $\vert\mu_{1,2}\vert<2$ for non-zero transmission and $\vert\mu_{1,2}\vert>2$ for zero transmission for sufficiently large superlattices. Now

$$\begin{equation*} h_l= \frac{U_{l+1}(\mu_1/2)- U_{l+1}(\mu_2/2)}{\mu_1 -\mu_2}, \end{equation*}$$

and the generalization of the one-channel formula is

$$\begin{equation*} W_{\textit{cell}}^{l+4}= s_{l,0} W_{\textit{cell}}^{3}- s_{l,1} W_{\textit{cell}}^{2}+ s_{l,2} W_{\textit{cell}}- s_{l,3}, \end{equation*}$$

where now $s_{l,0}=h_{l+1}$, $s_{l, 1}=-h_{l+2}+e_1 h_{l+1}~s_{l,2}= e_1h_{l} - h_{l-1}$, $s_{l,3}= h_l$.

As an example consider a superlattice of width w with cells formed by two sections (see fig. 3). The first section has length d₁ with a part of width w₁ < w at zero potential and a remaining part with a potential barrier of height V₀ and width $w-w_1$. The second section has length d₂ and zero potential. The potential matrix elements are $V_{11}= V_0(1 - w_1/w) +V_0 \sin(2\pi w_1/w)/2\pi$, $V_{22}= V_0(1 - w_1/w) +V_0 \sin(4\pi w_1/w)/4\pi$, $V_{12} = V_0 [\sin(3\pi w_1/w) - 3\sin(\pi w_1/w)]/3\pi$. There is zero coupling for w₁ zero (full wide barrier) or equal to w (no barrier). In fig. 4, to the left, the energy is shown as function of the eigenvalues for $V_0=1\ \text{eV}$, $w=80\ \unicode{8491}$, $w_1=40\ \unicode{8491}$, $d_1=40\ \unicode{8491}$ and $d_2=20\ \unicode{8491}$. At the top right the projection of the energy on the eigenvalues plane is given for $w_1=40\ \unicode{8491}$ and 35 Å, respectively. The circles correspond to allowed transmission, whereas the curves outside the unit circles correspond to forbidden regions. The bottom figure to the right shows a section of the Riemann surface containing a joining of surface pieces arising from both eigenvalues.

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Figure 5 shows the coefficients T₁₁, $-T_{22}$ (black continuous and dotted curves) and $\vert\mu_{1,2}\vert/2$ (thicker curves) for two coupled channels, one-hundred layers with $V_0=0.3\ \text{eV}$, $w=100\ \unicode{8491}$, $w_1=50\ \unicode{8491}$, $d_1=45\ \unicode{8491}$ and $d_2=40\ \unicode{8491}$. There is no transmission simultaneously for both channels for avoided crossings of the partial traces. In fig. 6 the coefficients T₁₁, $-T_{22}$ (black continuous and dotted curves) and $\vert\mu_{1,2}\vert/2$ (thicker curves) for two coupled channels, are again shown but now, to study the effect of the material parameters on channels coupling, four distinct values of the barriers widths are given: on the top, left, for $w_1=10\ \unicode{8491}$; on the top, right, $w_1=40\ \unicode{8491}$; on the bottom, left, for $w_1=60\ \unicode{8491}$ and on the bottom, right, for $w_1=90\ \unicode{8491}$. A strong dependence on the material parameters is observed and which still needs to be studied thoroughly.

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A more detailed analysis of the spectrum and explicit analytic relations for three and four channels, as well as calculations for superlattices under constant electromagnetic fields, shall be given somewhere else. The obtained results are related to the stability studies of Lyapunov, Krein and others [11]. It seems that similar methods developed for superlattices are also applicable for graphene superlattices [12].