With increasing complexity of semiconductor devices the need for accurate simulations increases as well.
Iso-Thermal device simulations in one, two, and three dimensions are performed routinely and the physical models entering such simulations are constantly improved and also extended to new materials. The interest in non-isothermal simulations, however, has been considerably lower,despite the fact that heat generation and dissipation in for instance power devices are certainly not negligible effects. As both the power handling capability of power electronic devices and the number of very large scale integration (VLSI) components on a single chip increase, the interest in device heating and in how to reduce it is expected to become progressively larger.
Modeling Heat Generation in a Semi-Conductor Device
It is well known that for an electrolyte solution
Gibbs equation is given by
$\ d U^o = TdS - PdV+ \sum_{i} \mu^{(0)}dN_i$
where $\ U^o$ is the internal energy in the absence of an externally applied potential, T is the absolute temperature, S is the
entropy, P the pressure, $\ N_i$ the number of ions of the ith species in volume V, and , $\mu_{0} is the electrochemical potential of the ith species, containing a contribution from the electrostatic potential of the ions. Of course, this relation can equally well be applied to semiconductors with “ions” replaced by electrons (i=e and holes (i=h).
Since $\ U^o$ is a homogeneous function of S, V, and $\ N_i$ Euler’s theorem on homogeneous functions implies
$\ U^o = TS - PV + \sum_{i} \mu^{(0)}dN_i$
leading to the Gibbs-Duhem relation
$\ SdT - Vdp + \sum_{i} d\mu^{(0)}N_i$ = 0
Using lower-case letters to denote densities of extensive variables ($\ u^0$= U^0/V$, etc.), we have
$\ u^o = Ts - P + \sum_{i} \mu^{(0)}dN_i$
$\ sdT - dp + \sum_{i} \mu^{(0)}n_i$ = 0
whereby
$\ du^o = Tds - P + \sum_{i} \mu^{(0)}dn_i$
We are, however, concerned with the transfer of particles between phases, and in this case we must include in the change in energy the work done by the (possibly time varying) external potential as the particles are transferred. The internal energy at the phase located at r is
$\ u= u^0 + \phi\sum_{i}q_in_i$,
where $\ q_i$ is the charge of the ith species. Here $\phi$=$\phi(r,t)$ represents the extra potential felt by a charged particle at phase r due to the presence of a possibly time-varying external potential.
Thus we deduce
$\frac{\partial u}{\partial t} = T\frac{\partial s}{\partial t} + \sum_{i} \mu_{i}\frac{\partial n_i}{\partial t} + \frac{\partial \phi}{\partial t}\sum_{i}q_in_i$
where $\mu_i = \mu^{(0)}_i + q_i\phi_i$
is the total electrochemical potential including the effects of an external field. Since time variations in both the potential contribution \phi in the above equation and in the potential \psi are exclusively due to the application of a time varying
external field, we infer that
$\frac{\partial \phi}{\partial t} = \frac{\partial \psi}{\partial t}$
in each phase. With a notation more appropriate for semiconductors,
$\frac{\partial u_n}{\partial t} + \frac{\partial u_p}{\partial t}= T\frac{\partial s}{\partial t} + \mu^{(n)}\frac{\partial n}{\partial t} + \mu^{(p)}\frac{\partial p}{\partial t} + q(n - p)\frac{\partial \psi}{\partial t}$ --(1)
To evaluate the time derivative of the entropy density s we choose to express s as a function of T, n, and p:
$\ ds = \left(\frac{\partial s}{\partial T} \right)_{n,p}dT + \left(\frac{\partial s}{\partial n} \right)_{T,p}dn + \left(\frac{\partial s}{\partial p} \right)_{T,n}dp$
Furthermore, the specific heat of the electron-hole subsystem at constant volume, $\ c^{(e,h)}_v$, is given by
$\ c^{(e,h)}_v = T\left(\frac{\partial s}{\partial T} \right)_{n,p}$
Using the
Maxwell"s_equations
$\left(\frac{\partial s}{\partial T} \right)_{T,p} = - \left(\frac{\partial \mu^{(n)}}{\partial T} \right)_{n,p} and \left(\frac{\partial s}{\partial p} \right)_{T,n} = - \left(\frac{\partial \mu^{(p)}}{\partial T} \right)_{n,p}$
adding this to (1) and also considering the contribution of
phonon energy,
$\left(\frac{\partial u_L}{\partial t} \right)= c^{(L)}_v\left(\frac{\partial T}{\partial t} \right)$
we get
$\frac{\partial u}{\partial t} = c_v\frac{\partial T}{\partial t} - q\left(\phi_n - T\left(\frac{\partial \phi_n}{\partial T} \right)_{n,p} \right)\frac{\partial n}{\partial t} + q\left(\phi_p - T\left(\frac{\partial \phi_p}{\partial T} \right)_{n,p} \right)\frac{\partial p}{\partial t} + q(p - n)\frac{\partial \psi}{\partial t}$
where $\ c = c^{(eh)}_v + c^{(L)}_v$
When applied to the electron-hole subsystem, the phenomenological equation of
Irreversible Thermodynamics state that
$\ J^n = L^{nn}\bigtriangledown\left( - \frac{ \mu^{(n)}}{T} \right) +L^{np}\bigtriangledown\left( - \frac{ \mu^{(p)}}{T} \right) + L^{nu}\bigtriangledown\left(\frac{ 1 }{T} \right)$
$\ J^p = L^{pn}\bigtriangledown\left( - \frac{ \mu^{(n)}}{T} \right) +L^{pp}\bigtriangledown\left( - \frac{ \mu^{(p)}}{T} \right) + L^{pu}\bigtriangledown\left(\frac{ 1 }{T} \right)$
$\ J^u_{eh} = L^{un}\bigtriangledown\left( - \frac{ \mu^{(n)}}{T} \right) +L^{up}\bigtriangledown\left( - \frac{ \mu^{(p)}}{T} \right) + L^{uu}\bigtriangledown\left(\frac{ 1 }{T} \right)$
In absence of a magnetic field, the
Onsager relations state that
$\ L^{\alpha\beta} = (L^{\beta\alpha})^T, \alpha, \beta = n,p,u$
If we restrict attention to cubic crystals, in which case the transport tensors are diagonal with identical diagonal elements, the Onsager relations become particularly simple, introducing introduce transport coefficients.
$\ J_n = -qn\mu_n\bigtriangledown\phi_n + \sigma_{np}\bigtriangledown\phi_p + qnD^T_n\bigtriangledown T$
$\ J_p = \sigma_{np}\bigtriangledown\phi_n - qp\mu_p\bigtriangledown\phi_p + qnD^T_p\bigtriangledown T$
$\ J^Q_{(eh)} = qnD^T_nT\bigtriangledown\phi_n - qpD^T_pT\bigtriangledown\phi_p - \lambda^Q\bigtriangledown T$
where
$\mu_n = electron mobility$
$\mu_h = hole mobility$
$\sigma_np = transport coefficient$
Also the electrical current densities
$\ J_n = -qJ^n and J_p = qJ^p$
and also changed and also changed from $\ J^u_{eh}$ to the heat current density $\ J^Q_{eh}$ in order to preserve some of the symmetry in the matrix of transport coefficients. The two current densities are related according to
$\ J^Q_{eh} = J^u_{eh} - \phi_nJ_n - \phi_pJ_p$
In conventional simulation of semiconductor devices,based on the drift-diffusion model, the transport coefficient $\sigma_np $ is zero.
It is possible to express $\ J^Q_{eh}$, in terms of $\ J_n, J_p, and \bigtriangledownT $. It can now be show that
$\ J^Q_{eh} = - \Pi_n J_n + \Pi_p J_p - \kappa_{eh}\bigtriangledownT$
where
$\Pi_n = \frac{1}{1 - \theta}\left(\Pi^o_n - \theta_n\Pi^o_p)$,
$\Pi_p = \frac{1}{1 - \theta}\left(\Pi^o_p - \theta_p\Pi^o_n)$,
$\theta_n = \frac{\sigma_{np}}{qn\mu_n}$, $\theta_p = \frac{\sigma_{np}}{qn\mu_p}$
$\theta = \theta_n\theta_p$
$\Pi^0_n = T\frac{D^T_n}{\mu_n}, \Pi^0_p = T\frac{D^T_p}{\mu_p}$
$\kappa_{eh} = \frac{1}{1-\theta}\left(\kappa^0_{eh} + \frac{2}{T}\sigma_{np}\Pi^0_n\Pi^0_p -\theta\lambda^Q )$
The quantities with a zero superscript are the values of the corresponding quantities in the approximation $\sigma_np=O$, i.e., when we neglect electron-hole scattering in the non-diagonal transport coefficients.
Combining the expressions for $\ J^Q_{eh}$ and adding contribution from lattice we obtain
$\ J^u = - \left(\Pi_n - \phi_n)J_n + \left(\Pi_p + \phi_p)J_p -\kappa\bigtriangledown T $
where
$\kappa = \kappa_{eh} = \kappa_L $
is the thermal conductivity. The transport coefficients $\Pi_n$ and $\Pi_p$ are the Peltier coefficients and quantify the idea that the particles in a particle current also carry energy.
Substituting (1) in the adove equation we get
$\ c_v \frac{\partial T}{\partial t} + \bigtriangledown T \left( - \kappa \bigtriangledown T) = H$
where the heat generation term is
$\ H = \bigtriangledown \left( \left(\Pi_n - \phi_n \right)J_n - \left(\Pi_p + \phi_p \right)J_p \right)+ q\left( \phi_n - T \left(\frac{\partial \phi_n}{\partial T}\right)_{n,p} \right)\times \frac{\partial n}{ \partial t} - q\left( \phi_p - T \left(\frac{\partial \phi_p}{\partial T}\right)_{n,p} \right)\frac{\partial p}{\partial t}$
This is valid for cubic crystals with a position-dependent band structure of any complexity in a generalized drift-diffusion approximation and with Fermi-Dirac statistics. By using the continuity equations, other forms of H, not containing explicit time derivatives of n and p, are possible.
