# The backward Monte Carlo method for semiconductor device simulation

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## Abstract

A backward Monte Carlo method for the numerical solution of the semiconductor Boltzmann equation is presented. The method is particularly suited to simulate rare events. The general theory of the backward Monte Carlo method is described, and several estimators for the contact current are derived from that theory. The transition probabilities for the construction of the backward trajectories are chosen so as to satisfy the principle of detailed balance. This property guarantees stability of the numerical method and allows for a clear physical interpretation of the estimators. A symmetric sampling method which generates wave vectors always in pairs symmetric to the origin can be shown to yield zero current exactly as thermal equilibrium is approached. The properties of the different estimators are evaluated by simulation of an *n*-channel MOSFET. Quantities varying over many orders of magnitude can be resolved with ease. Such quantities are the drain current in the sub-threshold region, the high-energy tail of the carrier distribution function, and the so-called acceleration integral which varies over 30 orders in the example shown.

## Keywords

Backward Monte Carlo method Semiconductor Boltzmann equation Device simulation Electron distribution function MOSFET Rare events## 1 Introduction

The Boltzmann Equation (BE) describes the motion of charge carriers in a semiconductor as semi-classical and can be used to model the electrical properties of semiconductor devices. The BE can be solved by means of the Monte Carlo method [1, 2] for realistic device structures and for realistic scattering and band structure models. The physically transparent and commonly used forward Monte Carlo (FMC) method, however, shows severe drawbacks in terms of computation time and statistical error when statistically rare events are to be simulated. A solution method that overcomes this drawback is the backward Monte Carlo (BMC) method. This method was introduced in the field of semi-classical transport at the end of the 1980s [3, 4]. These early algorithms turned out to be numerically unstable, as the carrier energy tends to grow indefinitely on a trajectory that is followed back in time. A numerically stable algorithm was proposed in 2003 [5]. Since the backward transition rates are chosen to obey the principle of detailed balance, a runaway of the carrier energy along a backward trajectory is avoided. From a practical point of view, this means that the scattering rates of the forward method can be used in the backward method as well [5].

The principle of the BMC method for the solution of a boundary value problem is to choose a set of states in phase space and trace trajectories from these states back in time until a contact is reached. The value of the given distribution function (DF) at the contact determines the statistical weight of the backward trajectory and consequently its contribution to the estimator of interest. To calculate a current which is controlled by an energy barrier, one would typically choose states at the top of the barrier and estimate the current from these states, see Fig. 1. If the barrier is high, a forward trajectory is very unlikely to reach the top of the barrier, whereas in the backward method only these unlikely states are considered, and no computation time is wasted with the vast majority of trajectories that do not overcome the barrier.

## 2 Theory of the backward Monte Carlo method

*q*is the charge of the carrier. The carrier’s group velocity \(\mathbf {v}\) is related to the band energy \(\mathcal {E}(\mathbf {k})\) by \(\mathbf {v}=\hbar ^{-1}\nabla \mathcal {E}(\mathbf {k})\). A phase space trajectory satisfying the initial conditions \(\mathbf {K}_0(t_0)=\mathbf {k}_0\) and \(\mathbf {R}_0(t_0)=\mathbf {r}_0\) can be obtained by formal integration of the equations of motion.

*D*and has to be supplemented by boundary and initial conditions. The distribution function is commonly normalized as \( \int _{D}\,\mathrm {d}^3 r\int \,\mathrm {d}^3 k\, f(\mathbf {k},\mathbf {r},t) = {4\pi ^3} N_D(t), \) with \(N_D\) denoting the number of carriers contained in the semiconductor domain

*D*. The scattering operator \(Q=Q_\mathrm{g}-Q_\mathrm{l}\) consists of a gain and a loss term, respectively.

If many-body effects such as carrier–carrier scattering and degeneracy are to be considered, one has to resort to the established approximations [7].

### 2.1 Integral form of the Boltzmann equation

### 2.2 Probability density functions

*S*, one can define a PDF of the after-scattering states \(\mathbf {k}_a\).

### 2.3 The backward MC method

The integral equation can be solved by means of the Markov-chain Monte Carlo method [3, 4]. Starting from (7) will result in a backward Monte Carlo method, which means that the time instants \(t_i\), when in the simulation scattering events occur, form a descending sequence: \(t_0> t_1>t_2 >\cdots \).

Conversely, the more familiar forward Monte Carlo method is based on the adjoint equation of (7), see [8]. In the course of the simulation, an ascending sequence of time instants, \(t_0< t_1<t_2 < \cdots \), will be generated.

*f*in a given phase space point \(\mathbf {k}_0,\mathbf {r}_0\) is estimated by the following sample mean:

*N*denotes the number of trajectories, and

*n*(

*s*) is the order of the

*s*-th numerical trajectory, which is the number of scattering events occurring in the interval \([0, t_0]\). In this work, we discuss two specific choices of the estimator \(\mu ^{(n)}\).

### 2.4 Transition rate derived from mathematical considerations

Although the MC algorithm based on the estimator (18) is consistently derived from the integral form of the BE, computer experiments reveal a stability problem. The particle energy becomes very high when the trajectory is followed backward in time. The initial distribution takes on very small values at high energies, so that many realizations of the estimator will be very small. With small probability, the particle energy will stay low, where the initial distribution is large. These rare events give large contributions to the estimator, resulting in a large variance. The computer experiments show that the variance increases rapidly with time. However, for a given time *t* the variance of the estimator is finite.

### 2.5 Transition rate derived from physical considerations

The time evolution of the particle energy can be understood from a property of the scattering rate known as the principle of detailed balance. This property ensures that in any system particles scatter preferably to lower energies. If the backward transition rate (10) is employed for trajectory construction, in the simulation the principle of detailed balance is inverted, and scattering to higher energies is preferred.

*l*-th scattering event is denoted by \(\varDelta \mathcal {E}_l\).

*M*is the number of backward trajectories started from the point \((\mathbf {k}_0, \mathbf {r}_0)\). Note that the backward trajectory is constructed in the very same manner as a forward trajectory. Using the forward PDF (14) to generate the free-flight time means that we have inverted the time axis and are progressing along the negative time axis. The selection of the scattering mechanism and the calculation of the after-scattering state are also identical to the forward algorithm.

## 3 Current estimators

*yz*-plane located at \(x=x_0\) is obtained by integration.

*z*-direction, the current becomes

*W*is the device width. In principle, the integral in (27) could be evaluated by numerical integration, whereby the values of the distribution function at the discrete points \((\mathbf {k}_0,\mathbf {r}_0)\) are estimated by (24). However, it is more convenient to employ Monte Carlo integration instead. For this purpose, the current has to be expressed as an expectation value. This is accomplished by introducing a PDF \(p_0(\mathbf {k}_0,y_0)\) which can be chosen freely, and reformulating (27) as:

*N*is the number of sampling points. In the next step, the unknown distribution function

*f*in (28) is expressed through the estimator (24). This would in general convert (29) into a double sum of \(N\times M\) elements. However, it is sufficient to start only one backward trajectory from each sampling point \((\mathbf {k}_{0,i}, y_{0,i})\), so that one can set \(M=1\) in (24). With this assumption the estimator \(\mu \) in (28) becomes:

### 3.1 The boundary distribution

*Z*(

*T*) defined as [12]:

*C*in (31).

### 3.2 Injection from an equilibrium Maxwellian

### 3.3 Injection from a velocity-weighted Maxwellian

### 3.4 Injection from a non-equilibrium Maxwellian

*w*is defined as

### 3.5 Injection from the equilibrium concentration

*n*in (38) represents the actual carrier concentration as obtained from a device simulation. With the normalization integral in (54) defined as

*A*in (52) becomes

*Z*by \(N_C = Z/(4\pi ^3)\). Also, the normalization factor

*B*will be expressed through an energy \(\bar{E}_C\) defined as

*I*would be scaled by the factor \(\mathrm {e}^{-\beta _D \varDelta \mathcal {E}}\). In other words, increasing the barrier height by some energy increment will result in an exponential decrease in current. This means that the exponential dependence of the thermionic current on the barrier height can be directly deduced from the current estimator (61).

### 3.6 Symmetric sampling

In thermodynamic equilibrium, the current will vanish due to the symmetry of the distribution function. Estimating the current in equilibrium by the BMC method will result in positive values of the estimator for nearly half of the backward trajectories, and in negative values for the other half. In the sample mean, these positive and negative values will cancel to a large extent, and the current will be nearly zero. However, the current will not be exactly zero because we have to work with finite sample sizes. This type of statistical error, however, can be easily eliminated by always generating positive and negative values of the estimator in pairs. When starting a backward trajectory from a state \((\mathbf {k}_0, y_0)\), we also start another one from the opposite momentum state \((-\,\mathbf {k}_0, y_0)\). As we will show below, this procedure will give \(I=0\) exactly in thermal equilibrium without statistical error. One can also expect that this procedure will reduce the statistical error in situations close to thermal equilibrium.

In thermal equilibrium, the Fermi level \(E_\mathrm{F}\) is constant throughout the device. Regardless of at which contact a backward trajectory ends, the Fermi level encountered will always be \(F_n = E_\mathrm{F}\). The estimator (63) will identically vanish, and so will the current.

### 3.7 Estimation of the statistical error

Since all backward trajectories are statistically independent, one can easily find expressions for the statistical error of the simulation result.

*w*defined by (47) and the random variable \(\xi \) defined by

## 4 Multi-band semiconductors

*n*.

*C*remains unchanged.

The method of sampling an equilibrium trajectory yields random injection states of the form \((n_0, \mathbf {k}_0)\), where \(n_0\) is the initial band index.

## 5 The combined backward-forward MC method

In semiconductor devices, there are various processes which are caused by carriers with energies above a certain energy threshold. Such processes are impact ionization, carrier injection into the oxide, and the generation of interface traps due to hot carriers. To model such processes, only carriers with energies above the threshold need to be considered, whereas carriers with lower energies have no effect. Therefore, a good approximation in the modeling of such processes is to consider only those high energetic carriers that are able to surmount the energy barrier, and neglect the vast majority of carriers close to thermal equilibrium that get reflected on either side of the barrier.

## 6 Results and discussion

The BMC method has been implemented in the full-band Monte Carlo simulator VMC [13]. Backward trajectories are constructed in the same manner as forward trajectories. Routines for the computation of the free flight and the after-scattering states can be used without modification.

As a test structure for the proposed simulation method, we use a planar n-channel MOSFET with a gate length of \(L_\mathrm{G} = {65}~\hbox {nm}\), an effective oxide thickness of \(t_\mathrm {ox} ={2.5}~\hbox {nm}\), and a channel width of \(W=1\,\upmu \hbox {m}\). Device geometry and doping profiles have been obtained by process simulation [14]. A sketch of the device structure is shown in Fig. 3. Room temperature is assumed for all simulations (\(T_\mathrm{D}={300}\hbox { K}\)).

### 6.1 Transfer characteristics

Further, the statistical error of the BMC method is depicted in Fig. 5. In a MOSFET the current component due to carriers injected at the source contact is nearly independent of the drain voltage, whereas the current component of carries originating from the drain contact depends strongly on the drain voltage. At \(V_\mathrm{DS}={2.2}\hbox { V}\) the back diffusion current from the drain is extremely small, and the total current is dominated by forward diffusion, which will result in a low variance. At \(V_\mathrm{DS}={50}\hbox { mV}\), on the other hand, the back diffusion current is significant, and a stronger compensation of the two current components takes place, which will result in a higher variance. This explanation, using the forward time picture also holds true in the backward time picture. There a vast difference in the two current components is reflected by a significant difference in the statistical weights of the forward and backward diffusing carriers.

In Fig. 6 we compare the computation times for a given error tolerance of \(10^{-2}\). In the on-state (\(V_\mathrm{GS}= {2.2}\hbox { V}\)), BMC is about five times faster than FMC. Although in this operating point, the energy barrier in the channel is almost completely suppressed, many electrons injected at the source contact get reflected by the geometrical constriction at the source-channel junction. Since the BMC method need not simulate these reflected carriers, it shows a clear gain also in the on-state. The last point that could be simulated with FMC within a reasonable time was \(V_\mathrm{GS}= {0.8}\hbox { V}\). In this operating point, BMC is about 2300 times faster than FMC as shown in Fig. 6.

### 6.2 Output characteristics

Figure 7 compares the output characteristics computed by three different methods. As shown in Fig. 8, the statistical error decreases with increasing \(V_\mathrm{DS}\), a trend already discussed in the previous section. The figure also shows that the variance of the symmetric estimator (63) is lower in the entire range of drain voltages. Especially at low \(V_\mathrm{DS}\), where the device is approaching thermal equilibrium, the variance of the non-symmetric estimator tends to explode, whereas the variance of the symmetric estimator shows only a slight increase. In this regime, variance reduction by the symmetric estimator is particularly effective.

### 6.3 Injection from a non-equilibrium distribution

The injection distribution \(f_0\) can be freely chosen and does not have any influence on the expectation value, but it does affect the estimator’s variance. We demonstrate this fact by generating the random states \(\mathbf {k}_0\) from a non-equilibrium Maxwellian. The operating point is \(V_\mathrm{GS}={0.6}~\hbox {V}\) and \(V_\mathrm{DS}={2.2}\hbox { V}\). The current is calculated using (46) in conjunction with the estimators (40) and (42). Figure 9 shows the independence of the estimated current from the injection temperature \(T_0\).

### 6.4 Energy distribution function

### 6.5 Hot carrier degradation

*g*(

*E*) the density of states, and

*v*(

*E*) the group velocity. For the purpose of MC estimation, we convert (78) into a \(\mathbf {k}\)-space integral.

We used the combined backward/forward MC method to evaluate the acceleration integral. The statistical average is calculated from the forward trajectories using the before-scattering method [18]. In this simulation, \(10^{10}\) scattering events have been computed. To enhance the number of numerical trajectories at high energies the injection temperature \(T_0\) has been raised significantly (5000 and 10,000 \(\hbox {K}\)).

## 7 Conclusion

A stable backward method has been implemented in a full-band Monte Carlo device simulator and used to study the electrical characteristics of an *n*-channel MOSFET. The method allows one to calculate the current in the entire sub-threshold region including the leakage current in the off-state. Symmetric current estimators are proposed which produce less statistical error than the non-symmetric ones. This improvement is achieved for all operating conditions and is particularly large when thermal equilibrium is approached.

The current through a plane is calculated by Monte Carlo integration of the current density. For this integration, one has to assume a distribution of the sampling points which in the present case are the initial wave vectors of the backward trajectories. By assuming a Maxwellian distribution at elevated temperature the method will generate more sampling points at higher energies. This method of statistical enhancement reduces the statistical error of quantities that depend on the high-energy tail of the distribution function. It is shown that the estimated current is independent of the injection temperature, whereas the statistical error shows a clear minimum where the injection distribution most closely resembles the actual distribution. The proposed backward Monte Carlo method is able to estimate the energy distribution function in a chosen point in the (*r*, *k*) phase space with desired accuracy. The high-energy tail of the distribution can be calculated point-wise. As an illustrative application we have estimated the so-called acceleration integral in the channel of a MOSFET and compared to values from the literature.

## Notes

### Acknowledgements

Open access funding is provided by TU Wien (TUW). The financial support of the the Austrian Research Promotion Agency (FFG), project MORAFLASH (contract No. 850660) is thankfully acknowledged. The computational results presented have been achieved in part using the Vienna Scientific Cluster (VSC).

## Supplementary material

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