The transport equation for the minority electrons in the neutral p region now includes the generation term given by equation (5.66a). The generation rate is the number of electrons and holes produced per unit wavelength. Therefore, the minority electron density appearing in the transport equation will be the electron density per unit
wavelength, which is denoted as np. The total density is therefore np = JnpdX. And
the transport equation is the following,
where Rp(X) is the reflectivity at the surface of the p region, a^X) is the absorption
coefficient in the p region, and uM = 1 – — where no = 1 is the index of refraction
V I np)
of vacuum and np is the index of refraction of the p region.
Rewrite equation (5.140a) as follows,
where Np = np – npo, Ln, and Fp(X) is the photon flux that enters the p
f w-0-Rp (5141)
Equation (5.140b) is an inhomogenous, second order, linear differential equation with constant coefficients. The solution consists of the sum of the solution to the homogenous equation [equation (5.102)] and a particular solution, f(x).
Np = ApeLn + BpY1^ + f (x ) (5.142)
If the right hand side of equation (5.140b) (the inhomogenous part) is the sum of two exponential functions of x, then the particular solution  is the following.
f (x )= qeqx + C3eC“x (5.143)
It turns out that E2(x) in equation (5.140b) can be approximated by a simple exponential function (Appendix A).
E2 (x)* boe-b‘x (5.144)
In equation (5.140b) the incident radiation is isotropic. If the incident radiation is normal, then bo=1/2 and, b1=1 in equation (5.144) and um=0
Using equation (5.144) for E2(apx) and, E2
N’ = + C2e UM
in equation (5.140b) yields the following equations for Ci and C:
Thus, the solution is the following.
The contribution to the solution of the incident radiation is uncoupled from the in-the – dark solution. Hence, for the perfect p-n junction, the current density is just the sum of in-the-dark current density and the radiation caused current density. This becomes evident in the following development.
The boundary conditions that must be applied to determine Ap and Bp are the following.
Substituting equation (5.146) in (5.148) yields the following,
С1 – Фп )Ap ~(1 + Фп )Bp = -0p1 [ap OOb1Ln +Фп ] + 6p2
в„ < C-R. ЬМ, t(,) (5.150a)
SnLn = <
D v Vx
n vn i n
+ip [(1+ Ф;) eu +(1 – Ф;К ] [- 0p1e-apb1x + 0p2e
Jf (X)= 2eboFp (X)= 2ebo (1 – Rp (5.161)
The parameter, pn, is the optical depth based on the minority carrier electron diffusion length, Ln. The current density per unit wavelength, Jf, is the current density that would result if every absorbed photon of wavelength, X, produced current. In other words, it is the maximum possible current density that can be attained under illumination. Thus,
Jn ^ 1.
In the following sections, expressions for the dimensionless photon generated currents in the n region, jq,, and depletions region, Jd, are developed. The sum,
Jdn + J^ + Jd is called the internal quantum efficiency and is discussed in detail in Section 5.10.
As mentioned in Section 5.7, for an efficient PV cell, u =—^ < 1. Therefore, if
p L„ ’
< 1, then the exponential terms and DENp in equation
(5.159b) can be approximated by power series. If only the linear term in up is retained, then equation (5.159b) reduces to the following result.
up = — <<1 p L
Hence, the photon produced current is a linear function of a^A,) so that a large absorption coefficient is desirable in this case. Note also that Jqn is independent of the surface recombination parameter, фш and the parameter, b1. Thus, for up ^ 1 the
photon generated current is the same for both isotropic and normal incidence radiation[see discussion following equation (5.144)].
Ratio of p neutral region length to minority electron diffusion length
u =x /L
p p n
several values of В =b La (X) and ф =S L /D =.1.
In general, there is an optimum value of up for producing a maximum value of Jqn. This is illustrated in Figure 5.10, where bJnq is shown as a function of up for several values of Pn when uM = 0. In part a), ф„ = 0.1, and in part b), фи = 1. Increasing pn
results in larger values for jqn, as well as, smaller values for up where the maximum
occurs. For P ^ да, equation (5.159b) yields the following.
Ratio of p neutral region length to minority electron diffusion length
u =x /L
p p n
Figure 5.10b) – Effect of dimensionless neutral p region length, x /L on the photon produced minority electron diffusion current for several values of P =b La (X) and ф =S L /D =1.
decreases (problem 5.8). Thus, large surface recombination velocity, Sn, is detrimental. However, the value of up for maximum Jqn is not greatly affected.
Equation (5.159) applies for isotropic incident radiation, such as in a TPV system. The dimensionless current when the incident radiation is at zero angle of incidence, such as in a solar PV application, is obtained by setting bo = U, b1 = 1 [see equation (5.144)], and uM = 0 in equation (5.159).
РП = Mp (x) (5.164b)
and for РП ^ ® ,
lim Jqn =————- 1————- (5.164c)
РП coshup +фП sinhup
Now compare jqn for isotropic incident radiation to the zero angle of incidence case for
Under these conditions, equations (5.163) and (5.154c) yield the following result(bo= U for zero angle of incidence).
For efficient PV cells, Rip and Rp are small compared to 1. Therefore,
Values of the constants bo and bi depend upon the exponential approximation used for
E2(ax). The approximations in Appendix A [equations (A-13) and (A-14)] give bo
It is possible, therefore, that a PV cell in a TPV application, where the incident radiation is isotropic, produces a significantly lower photon current density than when it is used in a solar application, where the incident photon radiation is at zero angle of incidence. In both cases, incident photon flux is assumed to be the same. However, as mentioned in the discussion of equation (5.162), if up ^ 1, then the photon generated
current is the same for both isotropic and normal incidence radiation when the incident photon flux is the same for both cases. Thus, depending upon the optical depth, pn, and the ratio of the neutral region length, xp, to the minority carrier diffusion length, Ln, the photon produced current for a PV array in a TPV application may or may not be affected by the isotropic nature of the incident radiation.
Before proceeding to calculate the photon generated minority hole current density in the n region of the p-n junction, the solution to equation (5.140) when Pn = b1ap(X)Ln = 1, or pn = uM, is considered. As equations (5.146) and (5.159) show, there are singularities in the solutions for Np and Jnq when pn = 1, or pn = uM. Thus
the particular solution given by equation (5.145) does not apply for pn = 1 or pn = uM.
If pn = 1 or pn = uM, the particular solution is of the form Coxe b1“px or C^xe uM . Using those particular solutions for Np, the results for jpn and jqn can be determined (problem 5.7). Thus for pn = 1 and uM = 0, the solution for jqn is the following.
Now consider the minority hole current density, jp, in the n region. Again, if the
assumptions of section 5.7.1 are applied, the transport equation for minority holes in the
n region is similar to equation (5.140). Also, рП is the hole density per unit of
wavelength and pn = JpndA, . The hole transport equation is,
where derivation of the generation term, G(A, x) is left as an exercise (problem 5.9). In that derivation it is assumed that the p, n, and depletion regions all have the same index of refraction, nc. Therefore, the generation rate is the following.
i(X, x) = 2Fp (X)an (X)| E2 [(ap – ad )xp +(ad – an )xp + anx ]
where ad(A) is the absorption coefficient in the depletion region, a^A,) is the absorption
coefficient in the n region, uM = . 1 – — and Fp(A) is given by equation (5.141).
Again, approximate the exponential integral, E2(x) by equation (5.144). As a result, the hole transport equation is the following,
where p = pn – p2o and Lp is the diffusion length of holes in the n region.
Similar to the solution for Np [equation (5.146)], the solution for Pn’ is the following,
0n1 = 2b^pFo2^^ ^ ^ EXP [“Vp (ap – ad )- b1Xn (ap – an )] F p
Pp = b1an (^)Lp
The constants An and Bn are determined by the boundary conditions.
At x = xn,
and at x = f,
Applying equation (5.173) to equation (5.170) yields the following.
DENn = 2[(l + фр)eu – +(l-фр)e-un ]
Now calculate the hole diffusion current at the edge of the depletion region (x = xn),
– j; (5.183)
where J’p is the hole saturation current density without illumination and is given by
^ — x
equation (5.122) where un =———— -, and pno = pno. The photon produced hole
current density is Jqp and is given in dimensionless form by the following expressions.
coshu – e
The photon produced hole current density i – the ■ region, J^p, is significantly less than the photon produced electron current density, J’n, in the p region. This occurs because most of the photon flux is absorbed in the p and depletion regions. The terms
exp[-b1xp (ap – ad)]e~VdX – and exp -^Ljl(ap – ad)
and (5.184b) account for the photon absorption in the p and depletion regions.
Just as in the case for Np, there are singularities in the solutions for P – at Pp = 1,
and Pp = uM. For Pp = 1, the particular solution to equation (5.169) is of the form Coxe~b‘a-x. Using that result and assuming uM = 0, P – can be determined. Thus, the following result for the dimensionless photon saturation current density (problem 5.10) is obtained.
Exp [-Vp (ap – an )-VdX – ]f ^ ,A. t П Л , U – u
•L =—————– :——————— ^jcos^n t^smbu– [^u – фp U e –
Pp =1, um = 0
For a thick – region (u ^ да), equation (5.184b) has the following asymptotic value.
Obviously, if the dimensionless width of the n region, u^ becomes large the photon produced hole current density is independent of the surface recombination parameter, фр, as equation (5.186) indicates.
Under certain conditions, similar to the photon produced electron current density, Jnq in the p region, there is an optimum value of the neutral region length, un, for
producing maximum jpq. This is illustrated in Figure 5.11 where
jqpb1 exp|^b1xp (ap -ade~biaAl is shown as a function of un for uM = 0 and several
values of Pp with фр = 0.1 in part a) and фр = 1 in part b).
Ratio of n neutral region length to minority hole diffusion length
un = (I – Xn )/Lp
Figure 5.11a) – Effect of dimensionless neutral n region length on the photon generated minority hole diffusion current for u =0, several values of the optical depth, P =b L a (X)
m p 1 p n
and surface recombination parameter, ф =S L /D =.1.
p p p p
As can be seen, for large Pp and фр = 1, the photon produced hole current density has a maximum value greater than the asymptotic value given by equation (5.186). However, for Pp < 1 and фp = 0.1 and also for all Pp when фp = 1, the maximum value of Jqp is the asymptotic value. It would, therefore, appear that a large (> 1) value of the neutral p region length, un, is desirable. However, un > 1 for фp < 1 means the saturation current density, Jsp, is near its maximum value (Figure 5.8). Whereas, the desired result is that Jsp should be as small as possible.
and surface recombination parameter^ =S L /D =1.
p p p p
The results for jqn and J^ have been derived for the case where the photons are incident on the p side of the junction. Results for J^ and J^n when photons are incident on the n side of the junction are obtained as follows. To obtain J^, replace pn by Pp, фи by фр Rp by Rn, and up by u,, in equation (5.159). To obtain J^n, replace Pp by pn, фp by фп, Rp by Rn, and u by up, in equation (5.184).