Space Charge Region beyond the Abrupt Approximation

The problem of the potential, electrical field and charge density in a space charge region is revisited. Within the Boltzmann approximation, the asymptotic solution is found analytically. The exact solution everywhere can be found from numerically integrating an analytical function. The solution is compared to the popular abrupt (or depletion) approximation and an analytical approximation is given.

The calculation of the distributions of charge density ρ, electrical field E, and potential ϕ within a space charge region is a long-standing problem of semiconductor physics.It occurs in Schottky diodes for the semiconductor side of the structure and for bipolar (p-n) diodes for both sides.Here, we formulate the 1D problem of the potential ϕ for one n-doped side (x ≥ 0) as where ρ denotes the charge density and ε s denotes the dielectric constant of the semiconductor.The boundary conditions (for a semi-infinite slab) are ϕ(0) = ϕ 0 and ϕ(x !∞) = 0. We neglect here image charge effects which occur for Schottky diodes and (less importantly) for heterostructure p-n diodes.
The potential drop across the space charge region is given by the built-in voltage (V bi > 0) and the applied bias (V ext > 0 for forward direction), ϕ 0 = À(V bi À V ext ) < 0. Within the Boltzmann approximation, the Poisson Equation (1) must be consistent with Here, β = e/(kT ) (e denotes the (positive) electron charge, k denotes the Boltzmann constant, and T denotes the temperature) and N D denotes the (in general locally varying) donor concentration.We assume here exhaustion, i.e., that all donors are ionized at the given temperature.If the finite ionization of the donors and partial compensation with an acceptor density N A is considered (p % 0), For ϕ = 0, charge neutrality demands that γ = À1 þ N D / (n 0 þ N A ).We note that for γ = 0, n 0 = N D and Equation ( 2) is recovered.With the donor binding energy E b D , the donor degeneracy ĝD (typically ĝD ¼ 2), the conduction band edge density of states N C , and α ¼ exp ÀE b D =kT À Á =ĝ D , the electron density n 0 is given by [1] The combined Equations ( 1) and (2) yield which is not integrable analytically.A common approximation is the abrupt (or depletion) approximation [1,2] where ρ = ρ 0 = eN D (for spatially constant N D ) for x ≤ 0 ≤ w 0 and ρ = 0 for x > w 0 , neglecting the exponential term for the thermal majority carriers in (5).Then, Equation (1) can be directly integrated; the potential φ is parabolic and (for 0 ≤ x ≤ w 0 ; φ x ≥ w 0 ð Þ¼0) given by with the width w 0 of the depletion layer with the Debye length The purpose of this article is to investigate the problem without neglecting the majority carriers in the space charge region.

Analytical Solutions 2.1. Asymptotic Solution
We consider first for constant N D the limit x !∞ (and ϕ !0).In this case, Equation (2) yields ρ % ÀeN D βϕ (note ϕ ≤ 0).Using the Poisson Equation ( 1), we find Thus, the asymptotic dependency of the potential is an exponential, ϕ(x !∞) = c exp(Àx/L D ).Also, the electrical field and the charge density have this exponential dependency.
For the case of Equation ( 3), thus also resulting in an exponential asymptotic.
For the charge distribution of Equation ( 2), This can be integrated for constant N D and E(ϕ !0) !0 to an analytical solution for E(ϕ): We note that for ϕ !0, the field in lowest order of ϕ is given by consistent with the result from Section 2.1.From (13), the field at the origin E(x = 0) = E(ϕ 0 ) = E m is found (the negative sign is for n-type material; in p-type material the positive root must be chosen): The exact solution yields E m = 0 for the flat-band case of ϕ 0 = 0; the approximation is valid for -βϕ 0 ≫ 1.The total charge Q (per area) is with Gauss' law: If we like to express the charge as If we use Equation ( 3) for the charge density, the integral yields [2] (also for E(ϕ !0) !0) In the approximation βϕ 0 ≫ 1,

Numerical Solution
Taking ϕ(0) = ϕ 0 as a starting value and using the analytically known derivative (choosing the negative root of Equation ( 18)), the potential can be constructed for constant N D via for a sufficiently small step size δx ( L D .We note that a similar scheme has been put forward in ref. [4].As example for a numerical calculation, we use GaAs material parameters, namely, ε s = 12.5ε 0 , N D = 10 16 cm À3 , T = 300 K, and ϕ(0) = À(V bi ÀV ext ) = À2 V (small reverse bias).The exact potential is depicted in Figure 1a together with the abrupt approximation φ.The position of w 0 is indicated.The charge density is shown in Figure 1b.The dash-dotted line represents the assumed (constant) charge distribution in the abrupt approximation.
For the given material parameters, the Debye length is L D = 42.2 nm which corresponds to the width of the transition regime of the charge density from ρ 0 to zero at the end of the depletion layer.

Approximate Analytical Solution
We investigate the Ansatz for the charge density (ρ 0 = eN D ): with a length L D % L 0 D that has approximately the correct asymptotic behavior, ρ ∝ exp(Àx/L D ).We note that this "Fermi" distribution is used in nuclear physics as radial charge distribution model.
The integral yields the electrical field given by E m which leads to the determination of w 0 : and with (15) yielding and thus the same result as in (17).We note that the condition -βϕ 0 ≫ 1 means that w=L D ð Þ 2 ≫ 1.The electrical field (22) can be analytically integrated to φ with the polylogarithm function [5,6] The integration constant of the potential was chosen such that φ x !∞ ð Þ¼ 0. The condition for the value of the potential at x = 0 yields the value of L 0 D (the approximations are for w 0 =L 0 D ≫ 1): Therefore, comparing to (8), [7] L 0 In Figure 2, we compare the exact solution with this approximate solution.The charge distribution from the ansatz (21), using w 0 from (24) and L 0 D from (29), is shown together with ρ φ À Á according to Equation (2); ideally they should be the same.For our approximate solution (21), the two distributions are fairly close.The smooth transition around x = w within about AEL D is reproduced quite well.For our numerical example, the charge error between the exact and approximate solution, Δρ ¼ ρ exact À ρ, is shown as blacked dashed line in Figure 2 and has a maximum of about 5% of ρ 0 .
For the charge distribution ρ, the value ρ 0 /2 is reached at x ¼ w 0 by design; as can be seen from Figure 2, the half-point of exact distribution is shifted a little from w.The error of the approximation can also be quantified by looking at the electrical field for ρ 1=2 ¼ ρ 0 =2.For the exact solution, from (2) ϕ 1=2 ¼ ln 2 ð Þ=β and thus using ( 14), while we find from (22) for Ẽ1=2 ¼ Ẽ w 0 ð Þ:

Conclusion
We have developed a good analytical approximation for the charge distribution in a semiconductor depletion layer with consideration of the thermal carriers (within the Boltzmann approximation).

Figure 1 .
Figure 1.a) Potential ϕ and b) charge density ρ across the depletion layer of a Schottky n-GaAs diode.Calculation parameters are ε s¼ 12.5ε 0 , V bi À V ext ¼ 2 V (small reverse bias), N D ¼ 10 16 cm À3 , T ¼ 300 K. Abrupt approximation is shown as dash-dotted lines, exact (numerical) calculation as solid lines.The dashed line in panel (a) indicates the depletion layer width w 0 in the abrupt approximation.

Figure 2 .
Figure 2. Charge distribution for the exact (numerical) solution (solid black line, same as in Figure1b) for the same material parameters as in Figure1.Charge distribution ρ from (21) (blue dashed line) and ρ ¼ ρ φ À Á from (2) (red dashed line).The black dashed line is the difference of the exact and approximate solution, Δρ ¼ ρ exact À ρ.