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5.1 Semi-Classical Model of Charge Transport

5.1.1 Poisson’s Equation

The Poisson equation correlates the electrostatic potential V to a given charge distribution ρˆ . If the permittivity tensor is expressed as ˜ϵ , one can write the Poisson equation as

∇ ⋅ ˜ϵ ⋅ ∇V = -ρˆ.
(5.1)

The charge distribution can be expressed as ˆρ = q(p - n + C), where n (p) represents the electron (hole) concentration per unit volume, q is the fundamental charge unit, and the concentrations of ionized impurities and the dopant are added up as the fixed charge concentration C. These fixed charges can originate from charged impurities of donor (ND) and acceptor (NA) type and from trapped electrons (C1) and holes (C2)

C = ND - NA - C1 + C2.
(5.2)

The electric field (Ẽ) is related to V by

˜E = - ∇V .
(5.3)

5.1.2 Continuity Equations

The charge transport model is derived from the continuity equation which takes care of mass conservation (time is represented by ˜t )

∂ˆρ ∇ ⋅ J + q---= 0. ∂˜t
(5.4)

The total current density is J. One can decompose the contributions of the current J = Jn + Jp, where Jn (Jp) denotes the electron (hole) current density. Assuming all immobile charges as fixed with respect to time, ∂∂ˆρ˜t = q∂(p-∂˜tn) as ∂C- ∂˜t = 0. Then, the continuity equation (Equation 5.4) can be separated into an electron and a hole related parts,

∂(p - n) ∇ ⋅ (Jn + Jp) + q---------= 0. ∂˜t
(5.5)

This step enables to write the electron and hole related contributions as two independent equations,

∂n ∇ ⋅ Jn - q-- = qR, ∂˜t
(5.6a)

∂p ∇ ⋅ Jp + q---= - qR. ∂˜t
(5.6b)

where the net generation-recombination rate is represented as R.

5.1.3 Drift-Diffusion Equations

As the Poisson equation (Equation 5.1) and the two continuity equations (Equation 5.6a and Equation 5.6b) involve five unknown quantities (viz. V , n, p, Jn, and Jp), one needs two more conditions to make the equation system complete. Now, there are two major effects which lead to current flow in semiconductors (e.g. silicon). First, the drift of charged carriers due to the influence of an electric field, and second, the diffusion current due to a concentration gradient of the carriers. It is hereby mentioned that the drift-diffusion model considers the temperature to be constant throughout the device [179], both for the charge carriers and the lattice.

The charged carriers in a semiconductor subjected to an electric field are accelerated and acquire a certain drift velocity. The orientation depends on the charge state, holes are accelerated in direction of the electric field and electrons in the opposite direction. The magnitude of the drift velocity depends on the probability of scattering events. At low impurity concentration, the carriers mainly collide with the crystal lattice. When the impurity concentration is increased the collision probability with the charged dopants through Coulomb interaction becomes more and more likely, thus reducing the drift velocity with increasing doping concentration [180].

The drift component is expressed using the concept of carrier mobility, which is the proportionality factor between the electric field strength (Ẽ) and the average carrier velocity. If one denotes μn (μp) as the electron (hole) mobility (assumed isotropic in the channel under investigation), in the low electric field range one can write the corresponding average carrier velocities as vn = -μnẼ and vp = μpẼ. Then the drift current density for the electrons and the holes can be expressed as JDrift n = -qnvn = qnμnẼ and JDrift p = qpvp = qpμpẼ. The signs are justified as electrons (holes) move against (with) the field direction. One can express the conductivities as σn = qnμn for the electrons and σp = qpμp for the holes, then JDrift n = σnẼ and JDrift p = σpẼ.

A concentration gradient of carriers leads to carrier diffusion. This is because of their random thermal motion which is more probable in the direction of the lower concentration. The electron current contribution due to the concentration gradient is written as JDiffusion n = qDn∇n and the hole current as JDiffusion p = -qDp∇p, where Dn and Dp are the diffusion coefficients for electrons and holes. For the non-degenerate semiconductors and in thermal equilibrium, one can relate the diffusion coefficient with the mobility by using Einstein’s relation

( KBT ) Dn = ----- μn, q
(5.7a)

( KBT ) Dp = ----- μp, q
(5.7b)

where KB is the Boltzmann’s constant, and T is the temperature. The thermal voltage (VT ) is described as

V = KBT--. T q
(5.8)

Thus, the diffusion coefficients can be written as Dn = μnVT and Dp = μpVT . Here, the non-degenerate semiconductors are defined as semiconductors for which the Fermi energy is at least 3KBT away from both the conduction and the valence band edges [117, 179]. One can assume the considered semiconductor to be lightly doped and thus non-degenerate, as it is the case in spintronic applications [173]. However, when the doping becomes very high, the Fermi level rises more and more towards the conduction band and the transport becomes degenerate. In such a case, the transport equations must be expressed in the language of chemical potentials of the carriers instead of their concentrations [173, 181].

Combining the current contributions of the drift and the diffusion effect one gets

˜ Jn = qn μnE + qDn ∇n,
(5.9a)

Jp = qpμp˜E - qDp ∇p.
(5.9b)

By inserting Equation 5.9a into Equation 5.6a and Equation 5.9b into Equation 5.6b one obtains

∂n ∇ ⋅ (μnn ∇V - μnVT ∇n ) +--˜ = - R, ∂t
(5.10a)

∂p ∇ ⋅ (μpp ∇V + μpVT ∇p ) - --˜= R. ∂t
(5.10b)

From the equations Equation 5.1, Equation 5.10a, and Equation 5.10b, one can obtain the unknown parameters n, p, and V . Despite the clear limitations for the description of state-of-the-art devices, these set of equations are still widely used in TCAD applications due to their least qualitative results and their computational inexpensiveness.

5.1.4 Quasi-Fermi Levels

The thermal equilibrium does not demand a position-independent potential. For instance

EC (r ) = EC,0 (r) - qV (r),
(5.11a)

E (r) = E (r) - qV (r), V V,0
(5.11b)

Ei (r) = Ei,0(r) - qV (r),
(5.11c)

denoting the conduction band edge energy EC, the valence band edge energy EV , and the intrinsic Fermi level energy Ei, respectively. Treating the situation away from thermal equilibrium complicates the matter. One can reformulate Equation 5.9a with Equation 5.3 as

Jn = qμnVT ∇n - qnμn ∇V ( (n ) (n ) ) = qμnn VT -i ∇ -- - ∇V ( n ni ) = qμ n ∇ V ln (n-) - V , n T ni ◟-------◝◜------◞ -ϕn
(5.12)

where ni is the intrinsic carrier concentration. This shows that the drift and the diffusive contribution can be merged into one quantity. This quantity can be related to the quasi-Fermi level as follows [182] -qϕn = EF,n - Ei,0. Therefore, the current depends on the gradient of the quasi-Fermi levels

Jn = n μn∇EF,n,
(5.13a)

Jp = nμp∇EF,p,
(5.13b)

The drift-diffusion current relations consider position dependent band edge energies, EC(r) for the conduction band and EV (r) for the valence band, and position dependent effective masses, which are included in the effective density of states NC for the electrons and NV for the holes, can be expressed as [183, 184]

( ( E ) (N ) ( n )) Jn = qn μn ∇ --C-- V + VT --C- ∇ ---- , q n NC
(5.14a)

( ( EV ) ( NV ) ( p )) Jp = qp μp ∇ ----- V - VT ---- ∇ ---- . q p NV
(5.14b)

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