Simulation of traveling-wave resonant tunneling diode oscillator waveguides

Zoltán Jéhn

doi:10.1364/OE.466405

1. Introduction

The resonant tunneling diodes (RTDs) are exciting devices that possess negative differential conductance (NDC) at room temperature even at high frequencies [1–3]. It makes them a viable choice for fabricating communication devices in the terahertz frequency range, applying them as a gain medium [4,5]. One can create lumped element oscillators rather easily from RTDs: integrate them into a resonant circuit, which will balance out the losses in the system, thus creating an oscillator. Various types of RTD oscillators are demonstrated that can operate at room temperature up to even 2 THz [1–8] or reach reasonable output power. The devices can differ in the type of resonator structure they are integrated into and the type of RTD they are using. One type of structure is a waveguide resonator with integrated RTD between the waveguide plates: the traveling-wave RTD oscillator (TW RTD oscillator) [9,10]. These devices are similar to quantum cascade laser (QCL) structures with the modification that the gain medium is the RTD, and the waveguide serves as the cavity. RTDs work at room temperature; however, there is no experimental demonstration of room temperature THz QCLs structures. With this realization of an RTD oscillator, the area of the RTD is larger than for a conventional lumped element oscillator; hence the output power of these devices can be larger.

In this work, different waveguide geometries were analyzed for TW RTD oscillators. The geometries investigated here can not be analyzed as decoupling them into active and passive parts as for most RTD oscillators. For the here presented geometries, the RTD is integrated into the waveguide, not coupled to it as for the structures investigated in [11]. Both parts should be included in the same calculation, which is challenging with the currently available software tools [12,13]. With the time-dependent methods, the simulation of the active media can not converge, and generally, with eigenmode solvers, the analysis of systems with high losses is problematic. Hence a new simulation method is needed.

The simulation of such waveguide geometries is somewhat complicated since they consist of thin metal layers, lossy dielectrics, and dielectrics with negative differential conductance. The previously existing tools could not deliver reliable results; hence a more sophisticated simulation method is needed. The calculation technique in this work employs the full-wave method of moments technique (MOM) to find the propagation modes in the waveguides with a simplification to exclude spurious solutions.

2. Technique

Different methods exist to simulate optical and metallic waveguides [14,15]. If the waveguide plates are not perfect conductors, or the dielectric constant difference is rather large between the subsequent regions, then full-wave solvers should be used. Various full-wave analysis techniques have been developed for waveguides. For one type of method, a volumetric mesh is needed (FEM, FD, etc. [16–18]). There are other methods where only the surface mesh is needed on the boundary of each homogeneous domain (point matching, MOM, Method of fundamental solutions). Each method has its underlying problems originating from the accuracy or the complexity and size of the underlying equation system.

For the structures to be analyzed in the current work, the methods that employ volumetric meshes should be used with great care. The size of some geometrical features is at the same length-scale as the skin penetration depth; thus, the impedance boundary condition can not be applied in a straightforward manner. A very fine meshing of the entire domain is needed, increasing the number of equations that must be solved. On the other hand, if one tries to find the propagation mode in the form of a simple eigenvalue problem, the solution is likely spurious [19].

The employed technique is a method of moments method applied for two-dimensional structures analogous to [19], but the base functions are pulse functions.

2.1 MOM in 2D

In Lorentz gauge, the Helmholtz-equation for a three-dimensional homogeneous media can be written as:

(1)$$\Delta \vec{A} +k^2 \vec{A} ={-} \mu \vec{J}(\vec{r}),$$

where $\vec {A}$ is the vector potential and $\vec {J}(\vec {r})$ is the free spatial currents in the media. If one assumes the periodicity of the solution in one direction ($\vec {z}$), the following ansatz can be used:

(2)$$\vec{A}(\vec{r}) = \vec{A}_{\bot} (\vec{r}_p) exp(i k_z z) \qquad \vec{J}(\vec{r}) = \vec{J}_{\bot} (\vec{r}_p) exp(i k_z z),$$

where $\vec {r}_p$ is the coordinate in the plane perpendicular to the propagation direction, whereas $k_z$ is the propagation constant. Substituting the ansatz into the Helmholtz equation Eq. (1), one ends up with a Helmholtz equation in the cross-section of the waveguide:

(3)$$\Delta_{\bot} \vec{A}_{\bot} +(k^2-k_z^2) \vec{A}_{\bot} ={-}\mu\vec{J}_{\bot} .$$

For the homogeneous media, the Green function of the differential equation is:

(4)$$G(|\vec{r}_{p1}-\vec{r}_{p2}|) ={-}\frac{i}{4} H_0^2 (k_p|\vec{r}_{p1}-\vec{r}_{p2}|),$$

where $H_n^2$ is the n-th order Hankel function of the second kind and $k_p^2 = k^2-k_z^2$.

In order to calculate the induced $\vec {H}$-field and $\vec {E}$-field on the surface induced by the surface currents, they should be separated into longitudinal and cross-sectional currents:

(5)$$\vec{J}_{\bot} = \vec{e_z} J_z+\vec{e_p} J_p,$$

the vector potential can also be separated:

(6)$$\vec{A}_{\bot} = \vec{e_z} A_z+\vec{e_p} A_p.$$

In order to find the electromagnetic field distribution in the structure, one should find the equivalence currents on the interfaces to satisfy the interface boundary conditions for the $\vec {E}$ and $\vec {H}$-field. The vector potential is calculated from the Green functions, whereas the $\vec {E}$ and $\vec {H}$-fields are calculated from the vector potential:

(7)$$\vec{H} = \frac{1}{\mu} \nabla \times \vec{A},$$

(8)$$\vec{E} ={-} i \omega \vec{A} - \frac{i}{\omega \mu \epsilon} \vec{\nabla} ( \nabla \cdot \vec{A}).$$

2.2 Discretization

The surfaces of the structure are discretized into segments. The current at segment $n$ induces field on segment $m$ which is depicted in Fig. 1. Let the $\vec {H}_{mn}$ magnetic field be created by the segment $n$ on the position of segment $m$ in centroid approximation.

(9)$$ \vec{H}_{mn} \cdot \vec{e}_m \equiv H_{p,nm} = \frac{ k_z}{4} w_n J_{p,n} \cdot \vec{e}_n \times \vec{e}_m \cdot H_0^2(k_p |\vec{r}_{nm}|) $$

(10)$$ + \frac{-i k_p}{4} w_n J_{z,n} \cdot \frac{\vec{e}_m \times \vec{r}_{nm} }{|\vec{r}_{nm}|} \cdot H_0^{2'}(k_p |\vec{r}_{nm}|) $$

(11)$$\vec{H}_{mn} \cdot \vec{e}_z \equiv H_{z,nm} = \frac{-i k_p}{4} w_n J_{p,n} \cdot \frac{\vec{r}_{nm} \times \vec{e}_n}{|\vec{r}_{nm}|} \cdot H_0^{2'}(k_p |\vec{r}_{nm}|)$$

Fig. 1. Current on segment $n$ induces field on segment $m$. The length of the two segments are $w_n$, and $w_m$, while the vector pointing from the middle of segment $n$ to segment $m$ is $\vec {r}_{nm}$.

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Similarly, the $\vec {E}_{mn}$ electric field is created by the segment $n$ at the position of segment $m$:

(12)$$ \vec{E}_{mn} \cdot \vec{e}_z \equiv E_{z,nm} ={-}\frac{\omega \mu}{4} \left(1- \frac{k_z^2}{k^2} \right) H_0^{2}(k_p |\vec{r}_{nm}|) w_n J_{z,n} $$

(13)$$ -\frac{\omega \mu}{4} H_0^{2'}(k_p |\vec{r}_{nm}|) \left(\vec{e}_n \cdot\frac{\vec{r}_{nm}}{|\vec{r}_{nm}|}\right) w_n J_{p,n} \frac{k_p i k_z}{k^2} $$

(14)$$ \vec{E}_{mn} \cdot \vec{e}_m \equiv E_{p,nm} ={-}\frac{\omega \mu}{4} \vec{e}_n \cdot \vec{e}_m H_0^{2}(k_p |\vec{r}_{nm}| ) w_n J_{p,n} $$

(15)$$ -\frac{\omega \mu}{4} \frac{(\vec{r}_{nm} \cdot \vec{e}_m)(\vec{r}_{nm} \cdot \vec{e}_n)}{|\vec{r}_{nm}|^2} H_0^{2''}(k_p |\vec{r}_{nm}| ) \frac{k_p^2}{k^2} w_n J_{p,n} $$

(16)$$ -\frac{\omega \mu}{4} \frac{(\vec{r}_{nm} \times \vec{e}_m)(\vec{r}_{nm} \times \vec{e}_n)}{|\vec{r}_{nm}|^3} H_0^{2'}(k_p |\vec{r}_{nm}| ) \frac{k_p}{k^2} w_n J_{p,n} $$

(17)$$ -\frac{\omega \mu}{4} (H_0^{2'}(k_p |\vec{r}_{nm}| ) \left(\vec{e}_m \cdot \frac{\vec{r}_{nm}}{|\vec{r}_{nm}|}\right) \frac{i k_z k_p}{k^2} )w_n J_{z,n} $$

The centroid approximation fails if $k_p |\vec {r}_{nm}| \ll 1$. In that case, the EM-field induced by a segment should be calculated analytically with the help of the small argument approximation of the Hankel functions. That is the proximity field calculation.

If the system is separated into $N$ number of segments, it creates $4\cdot N$ number of unknown currents: each segment has two sides, and on each side, there is an unknown longitudinal $J_z$ in-plane $J_p$ current. Also, the Eqs. (10, 11, 13, 17) create $4\cdot N$ equations. In order to find a possible propagation mode, one needs to search for the $k_z$ value where the created equation system has a non-trivial solution. To find the solution for the equation system is a nonlinear eigenvalue problem [20].

2.3 Excitation

Finding the non-trivial solution in a large structure can be challenging [19] due to the very small width of the resonances and the existence of spurious solutions. If the current distribution of the required mode is approximately known like in a TW oscillator waveguide—the field is confined in the RTD region—one could excite this mode easily by an external current source $J_p^*$ in the region of the RTD. Thus focusing on the targeted propagation mode in the system. These excitement ports are periodic in the $z$ direction with $k_z$ propagation constant. One could define a specific admittance for this additional excitement segment (for a port):

(18)$$Y^* = \frac{J_p^*}{\int_A^B \vec{E}_p d\vec{s} },$$

where the denominator corresponds to the induced voltage along the segment between its endpoints $A$ and $B$. It is important to point out that the induced voltage measured exactly along the segment diverges. One needs to integrate in the vicinity of the segment to find a limited numerical solution and not exactly along.

In that picture, the $Y^*$ admittance value corresponds to the specific admittance of a capacitor, which should be connected to the contacts of the periodically excited port. Thus the electromagnetic wave propagates with the predefined $k_z$ propagation constant in the waveguide. For a TW RTD waveguide, the task is to find the negative differential conductance, where $k_z$ propagation constant is real, i.e., the propagation is lossless.

2.4 Implementation

The previously defined equation system was discretized, assuming a constant current distribution at each segment. If the argument of the Hankel function $k_p r_{nm}$ is smaller than a threshold value (for example, $k_p r_{nm} < 0.1$), then the analytically calculated proximity values were used. With a Newton algorithm, the conductance in the RTD region and the value of $k_z$ was adjusted till $Y^*$ port admittance became 0. In other words, the goal was to find the solution of the equation: $Y^*(k_z, \sigma _{RTD})=0$. The method was compared with a commercial simulator for a simple geometry without active media (see Supplement 1), and the results agreed well.

3. Parallel plate waveguide (PP)

In a parallel plate waveguide, the RTD is integrated between metallic plates (See Fig. 2). In practice, the active medium, the RTD, does not fill the whole space between the waveguide plates; instead, an additional highly conductive n++ layer connects the RTD to other metal layers. One must calculate with an additional contact layer with significant contact resistance at the semiconductor metal interfaces.

Fig. 2. PP alignment with the two metal contacts and contact resistance layers. The semiconductor layers (RTD and n++) are plotted, respectively.

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In practice, the value of the contact resistance is around a couple of $\mathrm {\Omega \mu m^2}$ [21]. In these simulations, to compare the two waveguide structures, for one set of calculations, the contact resistance was chosen to be $\mathrm { 2.5 ~\Omega \mu m^2}$. For another set, it was neglected. The capacitance of the RTD layer can be estimated [22], in this work, a constant value of 10 fF/$\mathrm {\mu m^2}$ was used. The excitation ports in the RTD layers in Fig. 2 are depicted as arrows. The voltage is measured in the RTD region along the arrows to calculate the two-dimensional port impedance.

The dispersion relation of the PP waveguide for possible parameters is plotted in Fig. 3. For reference, the values gained from a simple transmission line method (TLM) [10] are also plotted, whereas the waveguide parameters are calculated as for a parallel plate capacitor. The propagation constant $k_z$ and the wave impedance $Z$ of the TW waveguide mode are:

(19)$$k_z^2 = XY \qquad Z^2 = \frac{X}{Y},$$

where:

(20)$$\begin{aligned} X & = R_{metal}+i \omega L \\ \frac{1}{Y} & = \frac{1}{ i \omega C_{RTD}+G_{RTD}} + \frac{1}{ i \omega C_{n+{+}}+G_{n+{+}}} \\ & + \frac{1}{i \omega C_{cont}+G_{cont}}. \end{aligned}$$

Fig. 3. Dispersion relation of a PP waveguide (for parameters, see Table 1). The left axis corresponds to the propagation constant (solid lines), whereas the right axis corresponds to the NDC of the RTD needed in order to support lossless propagation (dashed lines).

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Table 1. Parameters of the waveguide structures used in the simulations.

View Table

The $R_{metal}$ and $L$ parameters are the specific resistance and inductance of the waveguide plates that can be calculated from the distance of the waveguide plates and the skin effect. The $C$ and $G$ parameters are the specific capacitance and conductance of the RTD, n++ (InGaAs layer), and contact resistance layer. The specific contact resistance of the waveguide ($G_{cont}$) can be written with the areal contact resistance ($R_c$): $G_{cont} = w_{wg}/R_c$. Similarly the specific NDC of the RTD layer ($G_{RTD}$) with areal NDC of the RTD substrate ($g_{RTD}$): $G_{RTD} = g_{RTD} w_{wg} = \frac {\sigma _{RTD}}{d_{RTD}}w_{wg}$.

The analytically calculated result fits well the numerically simulated ones in the region of the operation frequency, hence validating that the analytical calculation is acceptable for PP waveguides. At low frequencies, the calculated areal NDC values are different for the two approaches because of the calculation method of the loss resistance in the metal waveguide plates. The assumption that resistance of the metal can be calculated by the skin penetration depth loses its validity at low frequencies since the penetration depth becomes larger than the thickness of the waveguide. At higher frequencies, the two calculation methods deviate, as similarly demonstrated for lossy semiconductor waveguides in [22]. From Eq. (20). as the NDC value of the RTD layer is decreasing, the real part of the admittance ($Re(Y)$) of the waveguide is increasing. After one point, $Re(Y)$ becomes positive, and a lossless propagation can not occur in the structure. After this point, the wave propagation in that mode is not possible; a cut-off frequency can be observed. On the other hand, in the more accurate MOM simulation, longitudinal currents start to flow in the n++ layer at higher frequencies, effectively decreasing the separation between the RTD and the region of the longitudinal currents. As the skin depth of the n++ layer becomes comparable to the thickness of the n++ layer, the conductance of the n++ layer effectively decreases. This cut-off can happen just at higher frequencies.

4. Co-planar waveguide (CPW)

The simple analytical method can not be used for a co-planar waveguide due to the more complicated current path. The current flows in a longer route in the n++ layer (InGaAs) for the CPW structures (see Fig. 4), making it lossier at first glance. On the other hand, the specific inductance is larger than for the PP waveguide due to the larger separation between the waveguide plates. The specific inductance and admittance for the CPW structure can be calculated from a separated static numerical simulation:

• Assuming PEC electrodes for the waveguide, the magnetic vector potential was set on the electrodes to 1 and 0 Tm. The induced longitudinal current in the waveguide plates was calculated from the transversal component of the $\vec {H}$ field. The ratio between the potential difference of the waveguide plates and the longitudinal current in the electrodes is the specific inductance of the waveguide.
• The specific admittance of the waveguide without the RTD layer was calculated in a similar way. The electric scalar potential was set to 1 and 0 V on the waveguide electrodes. The admittance was calculated back from the induced current density in the structure.

Fig. 4. Schematic of the CPW structure.

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In order to have lossless propagation, the NDC of the RTD layer in the waveguide was adjusted to have the imaginary part of the propagation constant at zero. In the function of the frequency, the real part of the propagation constant and the corresponding NDC of the RTD layer are plotted in Fig. 5. It is essential to point out that the contact resistance influences the results in two aspects: It changes the specific admittance $Y$ of the waveguide (see Eq. (20)), and additionally, it changes the current crowding at the connection of the n++ and metallization layer. Hence the current penetrates a broader area in the metallization layer, creating a higher inductance of the waveguide. A similar effect can be observed for CPW and PP waveguides. The TLM-based calculation neglects the longitudinal currents in the n++ layer, which is an appropriate assumption for the operation range. However, at higher frequencies, also a cut-off appears - at lower frequencies than for the PP waveguide. After this point, the full-wave calculation is needed.

Fig. 5. Dispersion relation of a CPW waveguide calculated with different models (for parameters, see Table 1) The left axis corresponds to the propagation constant (solid lines), whereas the right axis corresponds to the NDC of the RTD needed in order to support lossless propagation (dashed lines).

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5. Conclusion

In this work, the PP and CPW waveguide structures were compared with the help of MOM simulations. A new calculation method was developed to find the propagation parameters of the targeted mode in the structure. The possible spurious solutions and higher modes were successfully excluded by the introduction of the periodic excitement port. It was also successfully proven that the analytical treatment for the PP and CPW waveguide geometries has good agreement with the numerical simulations using full-wave calculation at low frequencies. Additionally, it was shown that these waveguide structures have a reasonable cut-off frequency which can not be correctly calculated with a simple analytic description. With the presented method, it is possible to correctly investigate TW RTD structure in the entire operational range. Because beyond frequencies where the penetration depth of the n++ layer is comparable to its thickness, a unique full-wave method is needed, like the one presented here.

Funding

Horizon 2020 Framework Programme (765426).

Acknowledgment

I would like to thank my supervisor Prof. Michael Feiginov, and Petr Ouředík, for the helpful comments and discussions. The author acknowledges TU Wien Bibliothek for financial support through its Open Access Funding Programme.

Disclosures

The author declares no conflicts of interest

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the author upon reasonable request.

Supplemental document

See Supplement 1 for supporting content.

References

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Symbol	Value	Description
$w_{w g}$	1 $μ m$	Width of the waveguide
$d_{n + +}$	500 $n m$	Thickness of the n++ layer
$σ_{n + +}$	$2 \cdot 10^{5} S / m$	Conductivity of the n++ layer
$d_{R T D}$	$100 n m$	Thickness of the RTD layer
$d_{s p}$	$200 n m$	Width of the electrode spacing (CPW)
$d_{A u}$	$300 n m$	Thickness of the metallization
$σ_{A u}$	$2 \cdot 10^{7} S / m$	Conductivity of the metallization layer
$R_{c}$	$2.5 Ω μ m^{2}$	Contact resistance
$ϵ_{n + +}$	$13.9$	Relative permittivity of InGaAs
$c_{R T D}$	$10 f F / μ m^{2}$	Areal capacitance of RTD

Simulation of traveling-wave resonant tunneling diode oscillator waveguides

Abstract

1. Introduction

2. Technique

2.1 MOM in 2D

2.2 Discretization

2.3 Excitation

2.4 Implementation

3. Parallel plate waveguide (PP)

4. Co-planar waveguide (CPW)

5. Conclusion

Funding

Acknowledgment

Disclosures

Data availability

Supplemental document

References

Supplementary Material (1)

Data availability

Cited By

Figures (5)

Tables (1)

Equations (20)

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