Carrier Thermal Conductivity: Analysis and Application to Submicron-Device Simulation

Within a correlation-function (CF) formalism, the kinetic coefficientsof charge carriers in semiconductors are studied under different conditions. For the case of linear response in equilibrium, thetransitions from the non-degenerate to the degenerate regimes as wellas from ballistic to diffusive conditions are discussed within ananalytical model. Generalizing the method to high-field transport innondegenerate semiconductors, the CFs are determined by Monte Carlo (MC) calculations for bulk silicon from which the appropriate thermalconductivity has been obtained and included into the hydrodynamic code HEIELDS. For an n+nn submicron structure the temperatureand velocity profiles of the carriers have been calculated with HFIELDS.


INTRODUCTION
Thermal conductivity of charge carriers is of fundamental interest in describing transport phenomena in bulk materials as well as electronic devices. To provide a microscopic theory for this coefficient, the CF approach represents a very effective method [1]. As a consequence of the fluctuation-dissipation theorem, the carrier transport coefficients may be determined by the spectrum of the fluctuations in the system. Results for the thermal conductivity available in literature are mainly based on relaxation-time approximations [2,3]. A weighting ofthe single relaxation time 59 with a power of the energy yields a generalization ofthe Wiedemann-Franz law (WFL) given by + c #nT, q (1) where c is the so called power law exponent, # is the mobility and T the electron temperature. This WFL together with the energy relaxation time is usually introduced within hydrodynamic approaches [4][5][6][7][8].
In this paper we present the appropriate set of CFs for the cases of linear response around thermodynamic equilibrium in the ballistic and diffusive regime under different degeneracy conditions [9], as well as the hot-carrier regime in the classical-diffusive condition [10,11] with its applications to the simulation of an n+nn+-structure.

LINEAR RESPONSE REGIME
In the linear regime, generalized fluxes are the response of the material to generalized externally applied forces mediated by the kinetic (or Kelvin-Onsager) coefficients L according to with #, u= 1, 2, jl and j2 denoting the electrical current and the energy-flux densities, respectively, while X1 and X2 are the generalized driving forces.

TLI (co)
For a finite one-dimensional conductor of length d limited by ideal (i.e. completely absorbing and thermalizing) contacts, if scattering is treated within the relaxation time approximation, the thermal conductivity n can be calculated analytically showing the transitions both from ballistic to diffusive (with increasing d) and from degenerate to non-degenerate (with increasing T) conditions. The static (i.e. co 0) limiting cases are given by [9] i,ndg,b nk2 T _ndg ndk3ff 2 T 1/2 77.
Here ag/g is the carrier transit time in the ballistic (bl) case under non-degenerate and degenerate conditions, respectively, while in the diffusive (df) cases denotes the relaxation time. Figure shows the behavior of (0) at fixed carrier density as function of and T. The maximum value is reached in the ballistic degenerate case.

FAR FROM EQUILIBRIUM (HOT-CARRIER) CASE
The CFs of microscopic fluxes are calculated with respect to stationary values at the given bias point and the thermodynamic temperature T is replaced by the noise-temperature spectrum Tn(E, co=0) [13] associated with velocity fluctuations at the given field E, where we restrict to the stationary case (co=0). The thermal conductivity parallel to the applied electric field, n(E, co=0) generalized to hot-carrier conditions is thus given by Eq. (6) with the following replacement: where cra is the differential conductivity. Under thermal-equilibrium conditions (i.e. E=0) T,(co) T, and standard linear-response formalism is recovered [14]. The CFs and conductivities entering the definition of n are calculated using MC simulations [15]. We have considered the case of extrinsic n-Si, doping concentration 10 v cm-3, at T= 300 K. Figure 2 from a) to d) reports the four longitudinal CFs C,(t) (fluctuations of flux variable along the electric field direction) at increasing values ofthe electric field normalized to their respective initial values. At increasing fields the correlations decay faster. C22(t) shows the fastest decay, which indicates that its decayrate is the sum of haomentum and energy rates. Thus we see that even at thermal equilibrium an approach based on a single time-scale is very poor.
We observe that the variances C1(0), C22(0) and C12(0)--C21(0) increase systematically with increasing electric field,which is due to hot-carrier effects. Figure 3  strong decrease at high fields may not be reproduced by a parameterized WFL.

DEVICE SIMULATIONS
To study the role of thermal conductivity in device csimulation, the transport in a Si submicron n + -n + n structure has been analyzed by means of the hydrodynamic device simulator HFIELDS [16].
The simulated structure is the same n+-n-n + diode analyzed in [17]: the high doped n+-regions have length 0.1 gm and doping concentration 51017 cm-3, the lower doped n-region has length 0.4gm and doping concentration 2 1015 cm-3.
The electron mobility #(e) and energy-relaxation time -(c) as a function of the mean electron energy c, required in the hydrodynamic simulator, are calculated from the velocity and energy versus field curves obtained from MC simulations.
For the thermal conductivity both the models obtained from the results of MC simulations and from the parameterized WFL are adopted. In Figure 5 the velocity profile along the device obtained with different models of the thermal conductivity are shown. When the WFL (1) is applied, the spike in the electron velocity curve can be remarkably reduced by setting the value of the power law exponent to c -2.1; at the same time the velocity profiles is smoothed with respect to the MC solution presented in [17] for the same structure. This can be ascribed to the slow decrease of the WFL thermal conductivity with increasing electric field. The CF approach yields to a stronger dependence of the thermal conductivity on the electric field, which allows to obtain a better comparison with MC data. The carrier temperature profile is shown in Figure 6: alsoin this case the WFL approach tends to smooth the profile with respectto the case when the results of the CF method are applied.

CONCLUSION
We have shown that the CF formalism provides an effective method forcalculating electronic transport parameters in different transportregimes. These parameters, calculated in the case of nonequilibriumand incorporated into the hydrodynamic simulatorhfields, allowfor a description of the electronic behavior of submicron structuresin agreement with the results of more sofisticated microscopic approaches.
of Stuttgart. Since 1996 is full professor at the University of Mfinster. His research interest is in the field of carrier dynamics and fluctuations in semiconductors far from equilibrium on ultrashort length or time scales.
Paola Golinelli, born 1969 in Rivoli (ITALY), received the Dr. in Physics degree in 1995 from Modena University. Her main research activity has been the generalization of charge carrier thermal conductivity to hot-carrier conditions.