In this paper, we derive some general conditions for resonant tunneling in the heterostructures composed of parallel layers. The electron motion in such a multi-layered structure is confined in the longitudinal direction (say, along the x-axis), which is perpendicular to the planes, and is free in the transverse direction. The three-dimensional Schrödinger equation of this structure can be separated into longitudinal and transverse parts, resulting in the reduced one-dimensional Schrödinger equation with respect to the longitudinal component of the wave function ψ(x) and the electron energy E. Thus, in the units as ℏ²/2m^* = 1, the one-dimensional stationary Schrödinger equation reads

where V(x) is a potential and the prime stands for the differentiation over the space x.

In general, resonant tunneling effects in heterostructures are studied when the potential profile V(x) is a compactly supported regular function. However, due to a rapid progress in fabricating micro- and nanodevices in recent years, the study of the electron transmission through heterostructures can additionally and effectively be investigated in the zero-thickness limit approximation when their width shrinks to zero. Within such an asymptotic approximation it is possible to produce the so-called point interaction models [see the book [1] for details and references], which are quite useful because they admit exact closed analytical solutions providing relatively simple situations, where an appropriate way of squeezing to zero can be chosen to be in relevance with a real structure. A whole body of literature [see, e.g., the works [2–11], to mention a few] has been published, where a number of interesting effects was discovered for the Schrödinger Equation 1 with the potentials in the form of distributions.

An example of such point interaction effects is the resonant tunneling through potentials squeezed to one point. This effect has been discovered and discussed in the works [12–23] for the potential V(x) = γδ′(x), δ′(x): = dδ(x)/dx, where δ(x) is Dirac's delta function and γ the potential strength. A non-zero transmission through this singular single-point potential (referred from now on to as the δ′-potential) has been shown to occur under certain conditions imposed on the strength γ, forming a discrete resonance set. It has been derived that the resonance set for the δ′-potential depends on the sequence of δ′-like regularizing functions. This leads to the conclusion about the existence of a hidden parameter in the Schrödinger Equation 1 with the δ′-potential.

The structures in which (regular) background potentials are perturbed by a point potential are also of significant importance. Thus, the perturbation of a harmonic oscillator by a so-called δ′-interaction [24] [the term adopted in the book [1] for which, at the point of singularity, the derivative ψ′(x) is continuous and ψ(x) discontinuous] as well as by a triple of δ′-interactions [25] has rigorously been defined as self-adjoint Hamiltonians. Another type of the point interactions, which were used as a perturbation of background potentials is the δ′-potential defined as the derivative of the delta function [the terms δ′-interaction and δ′-potential mean different point interactions, as suggested to distinguish these notions in Brasche and Nizhnik [26]]. This point interaction has been used as the perturbation of a constant electric field and a harmonic oscillator [27] as well as the perturbation of an infinite square well [28]. Three exactly solvable models with a one-dimensional V-shaped (conic) potential have been elaborated in the work [29]. These studies, which concern the perturbation of isotropic harmonic and conic potentials by a point interaction located at the origin have been extended to two and three dimensions in the papers [30–32]. All these systems are combined from a specific regular subsystem plus a point interaction centered inside this subsystem. In the work [33], the situation has been examined when a point interaction is found outside a background system at some finite distance.

In the present paper, we focus our attention on the perturbation of the background system with a general potential shape, which is characterized only by a transmission matrix. The perturbation is accomplished by a rectangle-like potential barrier/well located beyond the background system at an arbitrary distance. This potential geometry is applicable to heterostructures composed of two parallel layers described by corresponding potential profiles. A particular ubiquitous case of such systems is the double-barrier potential model that plays a key role in investigation of tunneling effects, beginning from the pioneering work [34, 35].

The motivation for this study is twofold. First, it is worthwhile to find the conditions for a perfect transmission across such a bilayer in the sense that there is no reflection of particles from the bilayer. Second, it would be of interest to investigate the effect of controlling the tunneling of particles with infinitesimally low energy through a background layer by “tuning” the parameters of an auxiliary rectangular well potential.

Transmission matrix for a single layer structure: Consider a potential profile V(x) of an arbitrary shape with the support on the interval x₁ < x < x₂, where x₁ and x₂ are arbitrary points on the x-axis. The transmission matrix for Equation 1 that connects the boundary conditions at x = x₁ and x = x₂ is defined through the matrix equation

where λ_ij ∈ ℝ and det Λ = 1.

For a structure consisting of one layer, it is supposed that V(x)≡h ∈ ℝ on some finite interval of length l and V(x)≡0 beyond this interval. When h>0 (h < 0), we are dealing with a barrier (a well), respectively. For this potential, the Λ-matrix (Equation 2) is calculated explicitly.

In the case when the potential V(x) in Equation 1 is a piecewise constant function, the Λ-matrix can be calculated explicitly. In particular, for the rectangle-like structure with the potential profile V(x) ≡ h ∈ ℝ defined on an interval x₁ < x < x₂ and V(x) ≡ 0 beyond this interval, the transmission matrix is

Reflection-transmission probabilities expressed in terms of transmission matrix elements: For positive energy solutions of Equation 1 and the left-to-right current, let us define the reflection-transmission coefficients R and T through the scattering setting

Using these expressions as boundary conditions at x = x₁ and x = x₂ in the matrix Equation 2, one can express the reflection and transmission probabilities in terms of the matrix elements λ_ij as follows

where

Hence, according to Equations 5, 6, the transmission probability T through the rectangular potential of height/depth h and width l reads

where q is defined in Equation 3. As follows from this formula, no reflection occurs in the trivial case (h ≡ 0), i.e., T = 1. The transmission of particles over the barrier/well with h ∈ ℝ\{0} is an oscillating function, while in the case of tunneling (k² ≤ h), the probability T is a monotonically decreasing function of height h and width l.

Point interactions realized from rectangular potentials: In general, a point approximation of the Schrödinger Equation 1 can be implemented by replacing the potential V(x) by ε^−νV(x/ε), where ν>0 is an appropriate number and ε a dimensionless parameter that tends to zero, so that a given one-dimensional system (for instance, a layered heterostructure) shrinks to a point as ε → 0.

In the particular case of a rectangular barrier with height h>0 and thickness l, a point approximation is materialized by replacing realistic values of h and l by ε^−νh and εl, respectively. Inserting the replaced values into the transmission probability (Equation 7), for the tunneling through the barrier (k² ≤ h), we obtain

It follows immediately from the comparison of the limit expressions (Equation 8) with the formula (Equation 7) that only the value ν = 1 turns out to be suitable for the point approximation of a rectangular barrier, whereas the values ν≠1 are not appropriate. The point interaction resulting from the approximation with ν = 1 is described by Dirac's delta function δ(x). It is called the δ-potential or δ-interaction. Its strength α is defined in Equation 8. The tunneling probability T is a monotonically decreasing function of this strength and therefore this is an adequate description of the tunneling behavior. The corresponding Λ-matrix connecting the two-sided boundary conditions for the wave function ψ(x) and its derivative ψ′(x) at x = ±0, is obtained immediately as a ε → 0 limit of the matrix (Equation 3):

The transmission probability across a well (k>0) with depth d>0 is determined by the same formula (Equation 7) where h is replaced by −d. In this case, instead of the monotonic behavior, the transmission T is described by an oscillating function with the amplitude values T = 1 at the points k2+dl=nπ, n ∈ ℕ. Setting in Equation 7 h = −d, we obtain the non-trivial point limit of the transmission of particles over a squeezed rectangular well for two values of the parameter: ν = 1 and ν = 2. However, the value ν = 2 from the physical point of view is adequate and in this case we have

As follows from this asymptotic representation, only the value ν = 2 turns out to be closer to the realistic oscillating behavior described by Equation 7 where h = −d < 0. The value ν = 1 is not appropriate because the transmission T for this value is a monotonically decreasing function. The other values ν and are not suitable because at these values the transmission is either perfect or completely reflective. However, unlike the barrier case, the non-zero transmission, being perfect, occurs only at some parameter values of the well forming a set of Lebesgue measure zero, namely

This set consists of the countable family of curves on the {d, l}-plane. The transmission matrix is

where I is the identity matrix.

In the present paper, we consider a heterostructure composed of two parallel layers separated by a distance r>0, one of which (called A-layer) has a rectangular potential profile (barrier or well) and for the other one (called B-layer) the potential profile is of an arbitrary shape. More precisely, the potential in Equation 1 for this bilayer structure is defined by the function

where y₁ < x₂ (so that r: = x₂ − y₁ > 0). Schematically, this potential profile is shown in Figure 1.

The background B-layer is given by the transmission matrix that connects the boundary conditions at x = x₂ and x = y₂:

Then the transmission matrix of the full bilayer system is the product Λ = CΛ₀ Λ _A, where the matrix Λ_A is given by Equation 3 with the A-layer parameters h and l, and the matrix

corresponds to the free space between the layers. It is obtained from the Λ_A-matrix by setting there h = 0. As a result of computing the product, the elements of the full Λ-matrix become

In the case if the A-layer is located behind the background layer, the transmission matrix is the product

The elements of this matrix are obtained from the matrix elements (Equation 16) by the following replacements:

There is no loss of generality if we consider only the situation when the A-layer is located in front of the background layer. For this case, the conditions P = 0 and Q = 0 (see Equations 5, 6) for perfect transmission result in the following two equations:

where

The conditions (Equations 19, 20) can be considered as equations with respect to the unknowns tan(ql) and tan(kr). Here, all the terms with tan(ql) are real independently whether q is real or imaginary. Notice also that the quantities u and v in general depend on k.

In the particular case when the B-layer is fully transparent, i.e., u = v = 0, then we have tan(ql) = 0. This means that the A-layer is also transparent. More precisely, tan(ql) = 0 if and only if u = v = 0 and the distance r in this case is arbitrary. To prove this, we insert c₁₁ = c₂₂ (u = 0) and c21=-k2c12 (v = 0) into Equations 19, 20 and assuming that tan(ql)≠0, these equations become

respectively, which are contradictory. Therefore tan(ql) = 0. Vice versa, assuming in Equations 19, 20 tan(ql) = 0, we obtain the equations

which can be satisfied only if u = v = 0.

In a general case if u² + v² ≠ 0, the solution of Equations 19, 20 with respect to sin(ql) [or cot(ql)] and tan(kr) reads

If the A-layer is located behind the background layer, the elements c₁₁ and c₂₂ must be rearranged in these equations, so that in Equation 25 we have to accomplish the replacement c₂₂→c₁₁ and u → −u.

The representation (Equations 24, 25) determines the relation between the whole set of system parameters: (i) the height/depth h and the thickness l of the rectangle-like A-layer, (ii) the characteristics of the B-layer provided by the elements c_ij(k) of the C-matrix (Equation 14), and (iii) the distance r between the layers, at which a given energy E = k² is transmitted across this bilayer structure perfectly (with zero reflection). It is important that this representation allows us to express explicitly (at any fixed energy E = k²) the parameters l and r as functions of h and the matrix elements c_ij(k). Indeed, from Equation 24, the width l is found as a function l = l(k; h; c_ij). Using this dependence in Equation 25, we find the function r = r(k; h; c_ij) of the same parameters.

Now it remains to prove the validity of the right-hand sides of both Equations 24, 25. Consider first the regime of tunneling through the A-layer (k² ≤ h). In this case, from Equation 24, we explicitly find the thickness of the A-layer:

while the other parameter r is calculated from Equation 25 where

Thus, we arrive at the following conclusion: For the perfect (zero reflection) transmission of particles with a given finite energy E = k² across the whole bilayer structure, in the regime as the particles tunnel through the A-barrier potential with height h > 0 (k² ≤ h), there exists a non-empty set of the system parameters defined as follows: The height h and the matrix C = C(k) can be arbitrary, but thickness l is uniquely defined by the expression (Equation 26) and the series of distances r is calculated according to Equation 25 where cot(ql) is given by Equation 27.

In the special case k² = h, the κ → 0 limit of Equations 26, 27 yields expressions

The solution with q>0 (h may be negative) corresponds to the situation when a particle passes over the barrier or well of the A-layer. In this case, Equation 24 is well-defined only if

Solving this inequality, we find the two critical values of h:

that determine the intervals

on which the values for the height/depth h are admissible. Thus, one can conclude: Contrary to the tunneling regime through the A-layer, for the perfect transmission of particles with a given finite energy E = k² over the A-barrier potential with height h>0 (k²>h), the values of barrier height must be bounded from below (h≥h₊). In the case of the transmission over the A-well potential with depth h < 0, the values of well depth must be bounded from above (h ≤ h₋). The countable sets of thickness l and spacing r are calculated according to Equations 24, 25, respectively.

Schematically, the regions of admissible values of height/depth h ∈ ℝ\{0} for the perfect transmission of particles with a given energy E = k² across the whole bilayer structure are represented in Figure 2 in the form a “band structure.” Here, the band h₋ < h < h₊ is a forbidden region for the perfect transmission.

In the case as the background B-layer is completely transparent (i.e., u = v = 0), the A-layer must be also transparent. This can happen only in the case of a well (h = −d < 0) when ql=E+dl=nπ, n ∈ {0} ∪ ℕ. Therefore, the perfect transmission through the full system can be realized at the energies

and an arbitrary distance r > 0.

For the squeezed A-barrier, the δ-point approximation (h → ε⁻¹h and l → εl) with any strength α = hl reduces (Equations 24, 25) to

Therefore, for the perfect transmission through the full system, the strength α and the wave number k are coupled by Equation 34, and the distance r>0 is calculated from Equation 35.

For the A-well squeezed by the replacements d → ε⁻²d and l → εl (see also Equation 10), Equation 24 reduces to tan(dl)=0, so that the A-well appears to be fully transparent only on the set dl=nπ, n ∈ ℕ. Therefore the perfect transmission across the full system can occur only if the background B-layer is also fully transparent, i.e., if u = v = 0. In this case, the distance r>0 is arbitrary.

One of special cases of the perfect transmission through the bilayer structure is the trivial situation when one of the layers is fully transparent. Then the other layer must also be fully transparent. This is a trivial case when the layers do not interact and the distance between the layers is arbitrary.

The other cases when tan(ql)≠0 in Equation 24 and u² + v² ≠ 0 are non-trivial because of back and fourth reflections between the A- and B-layers. Below we will consider the cases when the elements of the C-matrix as well as the quantities u(k) and v(k) are given explicitly.

Consider a background B-layer structure described by the potential with an arbitrary shape. In particular, it may be a multi-layered structure. Assume that the transmission matrix C for the B-layer is defined by Equation 14, where the elements c_ij are arbitrary functions of the wave number k. Suppose also that the limits of c_ij(k)'s as k → 0, which describe the transmission of quantum particles with infinitesimally low energy, are finite. Then, as follows from the notations for u and v given in Equation 21, and the expression

(see the second Equation 5), the zero-energy (k → 0) limit of tunneling through the B-layer is impossible if limk→0c21(k)≠0.

Let us consider the k → 0 limit of the λ₂₁-element given in Equation 16 for the full system, where the A-layer is considered as an auxiliary well, also called a prewell [36]. Assume that the parameters of this well are characterized by depth d and width l. Setting q=k2+d in λ₂₁, we obtain

Consider first the case when limk→0k-1c21(k) is finite. In this case, limk→0c22≠0 because detΛ = 1. The asymptotic equality (Equation 45) becomes finite if sin(dl)=0. In the case if limk→0k-1c21(k) is infinite, we need to provide the cancellation of divergences in Equation 45. Then sin(dl)≠0 and therefore we get the equation

for the A-layer located in front of the background B-layer. In the other case as the A-layer is located behind the B-layer, the condition (Equation 46) has the same form where ρ=limk→0(c11/c21).

Thus, the presence of an auxiliary well (either in front of a background B-layer or behind it) provides a non-zero k → 0 limit of tunneling through the full system at some values of the parameters d and l being a solution of Equation 46. This solution forms the resonance set

The set Ω includes the particular case as limk→0k-1c21(k) is finite because in this case |ρ| → ∞, reducing Ω to the set

Consider the influence of an auxiliary rectangular well on single- and double-layer structures. In the case of barriers, the tunneling through these structures with infinitesimally low energy E = k² becomes impossible. However, as follows from Equation 46, a non-zero tunneling occurs at some values of the well parameters d and l that form the discrete set Ω defined by Equation 47.

In summary, we have determined the two types of conditions, which are useful for the analysis of transmission across bilayer heterostructures. The potential profile of one (background) layer is assumed to have an arbitrary shape. It is only specified by a transmission matrix C with real elements for which detC = 1. The profile of the other layer is considered in the form of a rectangle, which may be either a barrier or a well.

The first type of conditions concerns the perfect (zero reflection) transmission across the bilayer structure. We have derived the system of two general Equations 24, 25 with respect to the parameters of the rectangular layer and the elements of the C-matrix. For a given energy E = k², height/depth h and the elements c_ij(k), one can explicitly calculate the A-layer thickness l from Equation 24 and then find the series of distances r using Equation 25. The only restriction on the A-layer parameters is that the well has to be sufficiently deep according to the inequalities (Equation 32).

The second type of conditions describes the general effect of the A-well on the transmission of particles with infinitesimally low energies E = k² across the whole bilayer structure with the B-layer of an arbitrary potential profile provided by the transmission C-matrix. Even in the case as the B-barrier is fully non-transparent for zero-energy particles, the presence of an auxiliary well always leads to a non-zero transparency of these particles at some parameter values of this well. A general Equation 46 for finding these resonant values has been derived. The particular cases where the transmission matrix C = C_B for the B-layer can be computed explicitly, have been considered. Here, these cases are specified by rectangular single- and double-layers. It is shown that in the case of barriers, a rectangular A-well makes the tunneling of zero-energy particles through these barrier structures possible on certain resonance sets. Notice that from the point of view of fabricating nanodevices, the transmission of quantum particles with extremely low energy in heterostructures is of big interest.