# Multi-Band Model Hamiltonian for Tunnel Devices with Van Der Waals Heterojunctions

Owen Loison, Adel M'Foukh, Davide Romanin, Marco Pala, Philippe Dollfus

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Owen Loison, Adel M'Foukh, Davide Romanin, Marco Pala, Philippe Dollfus. Multi-Band Model Hamiltonian for Tunnel Devices with Van Der Waals Heterojunctions. 2025 International Conference on Simulation of Semiconductor Processes and Devices (SISPAD), Sep 2025, Grenoble, France. pp.1-4, 10.1109/SISPAD66650.2025.11186327. [hal-05411859]

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# Multi-band model Hamiltonian for tunnel devices with van der Waals heterojunctions

### Owen Loison

Université Paris-Saclay, CNRS
Centre de Nanosciences et de Nanotechnologies
91120, Palaiseau, France
owen.loison@universite-paris-saclay.fr

### Adel M'Foukh

Université Paris-Saclay, CNRS
Centre de Nanosciences et de Nanotechnologies
91120, Palaiseau, France
adel.mfoukh@cnrs.fr

### Davide Romanin

Université Paris-Saclay, CNRS
Centre de Nanosciences et de Nanotechnologies
91120, Palaiseau, France
davide.romanin@cnrs.fr

### Marco Pala

DPIA
University of Udine
Udine, Italy
marco.pala@uniud.it

### Philippe Dollfus

CNRS, Université Paris-Saclay
Centre de Nanosciences et de Nanotechnologies
91120, Palaiseau, France
philippe.dollfus@cnrs.fr

**Abstract**—We study the electronic properties of a van der Waals (vdW) heterostructure made of SnSe₂ and WSe₂ by *ab-initio* methods in order to simulate an Esaki diode built with these materials. We use Density Functional Theory (DFT) to parametrize a k·p model in order to achieve the transport calculations. Furthermore we introduce a tunneling coefficient parametrized by DFT. This coefficient depends on the electric field in the device in order to precisely describe the quantum tunneling happening in the heterostructure. This model Hamiltonian is then used to compute current-voltage characteristics for our Esaki diode via Non-Equilibrium Green's Functions (NEGF) calculations.

**Index Terms**—quantum transport, NEGF, 2D materials, broken gap.

## I. INTRODUCTION

Among numerous possibilities for adjusting the electronic properties of 2D materials, vdW heterostructures of Transition Metal Dichalcogenides (TMDs) may provide staggered and broken-gap heterojunctions that can enhance on-state currents of tunnel devices [1] [2]. The use of *ab initio* Hamiltonians to simulate the transport properties of these systems can be challenging in the case of supercells with a large number of atoms arising either from lattice mismatch or from arbitrary orientation of the two lattices. In this work, we propose a model Hamiltonian based on a multiband envelope function model calibrated against DFT and apply it to an Esaki diode [3] based on a vertical SnSe₂/WSe₂ heterostructure. Due to the importance of vertical tunneling between the two materials, we propose a field-dependent expression for the tunneling coupling, which can be assessed with DFT simulations.

## II. TRANSPORT MODEL

To compute the electronic properties of this heterostructure, we used a plane-wave DFT approach, as implemented in Quantum ESPRESSO [4], with Perdew-Burke-Ernzerhof [5] functionals, a 6 × 6 × 1 Monkhorst-Pack grid of *k*-points and spin-orbit taken into account. Van der Waals correction were

considered in the calculations with Grimme-D3 [6]. The unit cell of each isolated material has been relaxed and resulted in a lattice parameter of 3.32 Å for monolayer WSe₂ and 3.86 Å for monolayer SnSe₂. We created a common supercell of 6.66 Å from a 2 × 2 cell of WSe₂ and a √3 × √3 cell of SnSe₂, adding up to a 21 atom supercell and causing a stress of less than 1% for both materials. In Fig. 1 we plot the band structure of this system, exhibiting a broken gap configuration with the top of the valence band (VB) (made of WSe₂ states) being above the bottom of the conduction band (CB) (made of SnSe₂ states). The broken gap value is about 0.013 eV.

<figure><img src="image_2.png" /><figcaption>Fig. 1. Electronic band structure of the SnSe₂/WSe₂ heterostructure projected over the states of SnSe₂ (light blue) and WSe₂ (orange). In the inset, we show the stick-and-ball model of the SnSe₂/WSe₂ heterostructure, where green, magenta and orange balls are the Sn, W and Se atoms respectively.</figcaption></figure>

The model Hamiltonian we used for transport calculations is based on the envelope function method proposed by Ref. [7]. We simplified the problem by assuming that the only relevant coupling is due to the vertical tunneling between the minimum
of the CB of SnSe₂ and the maximum of the VB of WSe₂. This model can be naturally extended to any heterostructure where other bands become relevant to electron transport or other types of couplings become important.

In our simple case, the Schrödinger problem can then be described by a 2 × 2 Hamiltonian matrix:

$$ H = \begin{pmatrix} H_1(x, z_t) & H_T(x) \\ H_T^*(x) & H_2(x, z_b) \end{pmatrix} \left\{ \begin{aligned} H_1(x, z_t) &= -\frac{\hbar^2}{2m_{cx}} \nabla_x^2 + \frac{\hbar^2}{2m_{cy}} k_y^2 + E_0 + U(x, z_t) \\ H_2(x, z_b) &= \frac{\hbar^2}{2m_{vx}} \nabla_x^2 - \frac{\hbar^2}{2m_{vy}} (k_y - \Delta k_y)^2 + U(x, z_b) \end{aligned} \right. \quad (1) $$

where $m_{cx,cy,vx,vy} = (0.202, 0.427, 0.342, 0.342) \times m_0$ are effective masses, $\Delta k_y = 3.15 \text{ nm}^{-1}$ is the lateral momentum mismatch [8], $E_0$ the band offset between the two materials, $U(x, z)$ the potential energy, and we have assumed the system to be periodic along the lateral direction (y-axis) and that the direction of transport is on the x-axis. The band-to-band tunneling coefficient can be estimated as

$$ H_T(x) = -\frac{i\hbar}{m_0} \frac{U(x, z_t) - U(x, z_b)}{z_t - z_b} \frac{p_{c,v}}{E_c - E_v} \quad (2) $$

where $z_{t,b}$ are the vertical positions of the top layer SnSe₂ and bottom layer WSe₂, respectively, and the momentum matrix element

$$ p_{c,v} = -i\hbar \int_{\text{u.c.}} d\mathbf{r} u_c(\mathbf{r}) \nabla_z u_v(\mathbf{r}) \quad (3) $$

which we evaluated using DFT and resulted to be $3.98 \times 10^{-2} \text{ eV} \cdot \text{Å}$.

To validate the model, we computed the distance between the CB minimum and the VB maximum with the field-dependent expression derived from the effective Hamiltonian and plot the field-dependent curve in Fig. 2. Using Quantum ESPRESSO [9], we were able to add a gate inside our cell and simulate how the electronic properties evolve with the electric field created by the external gate, thus measuring the band evolution with the vertical electric field and assessing the bandgap opening. This is shown in Fig. 2, where we measured the gap of the heterostructure for different gate voltages. As can be seen, our simplified model reproduces very well the electric-field dependence resulting from DFT calculations.

### III. ROLE OF NON UNIFORM VERTICAL TUNNELING

Quantum transport calculations were performed by self-consistently solving the NEGF kinetic equations and the 2D non-linear Poisson equation. Importantly, phonon scattering was considered by adding specific self-energies computed within the self-consistent Born approximation by assuming the deformation potential approximation. The Hamiltonian matrix considered in the NEGF simulations was obtained by discretizing the one in Sec. II.

We are interested in an Esaki tunneling diode made with WSe₂ and SnSe₂ and an overlap region of length $L_{OV} =$

<figure><img src="image_3.png" /><figcaption>Fig. 2. Band gap of the heterostructure depending on the electric field measured at its centre. The field-dependent curve is computed using the two-band Hamiltonian while the points of the DFT curve are computed ab initio for each electric field applied on the gate</figcaption></figure>

6.66 nm, as shown in Fig. 3. WSe₂ and SnSe₂ constitute the p-doped source region and the n-doped drain region, respectively, while the vertical tunneling occurs in the overlap region made of the SnSe₂/WSe₂ heterostructure.

<figure><img src="image_4.png" /><figcaption>Fig. 3. Illustration of an Esaki diode based on the SnSe₂/WSe₂ heterostructure. The length of source and drain regions is 13 nm while the length of the overlap region is $L_{OV}$. The doping concentration is $-8 \times 10^{13} \text{ cm}^{-2}$ in the source and $8 \times 10^{13} \text{ cm}^{-2}$ in the drain.</figcaption></figure>

The I-V characteristics are reported in Fig. 4. To strengthen the importance of considering a non-uniform tunneling coupling in such systems, we compared the curves obtained with the field-dependent $H_T(x)$ in Eq. (2) and a constant $H_T$ as assumed in Ref. [10]. The simulations with constant coupling terms overestimate (underestimate) the current for reverse (forward) voltage bias. This is consistent with the bias dependence of $H_T(x)$ shown in Fig. 5, showing the absolute value of the tunneling coupling as a function of the position in the overlap region and the applied bias. Moreover, with the field-dependent $H_T$ model, we observe that the peak current is shifted to higher voltages compared to the case with constant $H_T$. The increase of the field-dependent $H_T$ with V, shown Fig. 5, is responsible of this shift as higher $H_T$ results in higher transmission. Hence we obtain a larger current for a higher voltage. Additional analysis comes from Fig. 6, which compares the current spectra obtained with the two models.
<figure><img src="image_5.png" /><figcaption>Fig. 4. Current-voltage characteristics of the Esaki diode with an overlap region of length $L_{OV} = 7$ nm made of the WSe$_2$/SnSe$_2$ heterostructure. The field-dependent curve was obtained using the expression of $H_T$ given in section II while the others were obtained by replacing this expression by a constant value.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 5. Colormap of $|H_T|$ as a function of the applied bias and of the x-axis in the overlap region of length $L_{OV} = 7$ nm.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 6. (Colormap) current spectra along the x-axis for an overlap length of $L_{OV} = 7$ nm. (Left) simulations employing field-dependent $H_T$; (Right) simulation with constant $H_T$. The color scale is the same for the two figures on the top and the two on the bottom.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 7. Current-voltage characteristic of the Esaki diode made with the WSe$_2$/SnSe$_2$ heterostructure for different lengths of overlap. These were obtained using the field-dependent expression of $H_T$.</figcaption></figure>

As can be seen, under reverse bias ($V = -0.1$ V), the current obtained with the field-dependent coupling is confined to the energy range, where $H_T(x)$ is nonzero (see Fig. 5), whereas in the case of a constant $H_T$, the current extends over a broader energy range.

Another interesting behaviour that we can notice in the I-V curve (Fig. 4) is that in reverse bias, the current corresponding to the field-dependent $H_T(x)$ tends to decrease for $V < -0.08$ V. This phenomenon is due to the low value of $H_T$ attained at negative voltages, as reported in Fig. 5, thus implying a decrease of the vertical tunneling current.

Finally, we studied the dependence of the current characteristics on the length of the overlap region. To this end, we considered not only $L_{OV} = 7$ nm, but also $L_{OV} = 13$ nm and $L_{OV} = 33$ nm. In Fig. 7 we report the I-V curves for these three overlap lengths. We notice that the forward bias parts

are identical, while in reverse bias the current increases with the overlap. This behaviour can be explained by considering that, in the reverse bias regime, the CB and VB in the middle of the heterostructure are practically inversed. As can be seen in Fig. 8 (top), the vertical tunneling current is therefore non null along all the overlap region, inducing a drain current that increases with $L_{OV}$. On the contrary, as illustrated in Fig. 8 (bottom), in the forward bias regime the vertical tunneling current between the two materials is mainly located on the edge of the overlap region due to the fact that the CB and VB are separated in the middle of the heterostructure. We have
<figure><img src="image_9.png" /><figcaption>Fig. 8. (Colormap) vertical tunneling current spectra along the x-axis for an overlap of $L_{OV}$ = 33 nm in (top) reverse bias and (bottom) forward bias.</figcaption></figure>

therefore a point-like tunneling device with a drain current almost independent of $L_{OV}$.

## IV. CONCLUSION

We proposed an efficient model to simulate tunnel devices based on large vdW heterostructures while keeping a complex dependence of tunneling coupling on the vertical electric field validated by DFT. The comparison with the case with constant values of $H_T$ highlights the effects of interlayer coupling on the current, which is highly sensitive to its variations. Our model allowed us to explain the non-trivial dependence of the I-V curves on the geometrical parameters of the Esaki diode based on the $WSe_2/SnSe_2$ vdW heterostructure.

## V. ACKNOWLEDGMENTS

This work was supported by the French "Agence Nationale de la Recherche" through the projects "Tunne2D" (ANR-21-CE24-0030). D.R. acknowledges support from the HPC resources of IDRIS, CINES, and TGCC under Allocation No. 2024-A0160914101 made by GENCI.

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