Derivation Of Ideal Diode Equation 362p4u
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Derivation of the Ideal Diode Equation for Photovoltaics Christiana Honsberg Basic Equations: 1.
Poisson's equation:
v
∂ E dx 2.
=
ρ q = ( p ( x) − n( x ) − N A− + N D+ ) ε ε
Transport equations:
v dn( x) J n = qµ n n( x) E + qDn dx
(first term is drift, second is diffusion)
v dp ( x) J p = qµ p p ( x) E − qD p dx 3.
Continuity equations: General conditions
Under thermal equilibrium and steady state conditions
dn 1 ∂J n = − (U − G ) dt q ∂x
1 ∂J n = (U − G ) q ∂x
dp 1 ∂J p = + (U − G ) dt q ∂x
1 ∂J p = −(U − G ) q ∂x
where U and G are the recombination and generation rates in the particular material and depend on the details of the device and may also depend on distance.
Applying above equations to a pn junction: Depletion approximation The above equations can be difficult to solve in closed form unless several simplifying assumptions are made. However, the equations are readily solved using numerical approaches, and there are many device simulators that perform this task. In addition to the assumption of a one-dimensional device, the most central simplifying assumption in determining a closed form solution to the above equations is the depletion approximation. The depletion approximation assumes that the electric field in the device is confined to some region of the device. According to this assumption, the device can then be broken up into regions that have an electric field and those that do not. This is shown below for a pn junction, where Regions I and III do not have an electric field (called quasi-neutral regions or QNR) and Region II has an electric field (which is called space-charge or depletion region).
Region I
No electric field
Region II
Region III
No electric field
Figure 1: Schematic showing the regions with and without the electric field according to the depletion approximation. The depletion approximation greatly simplifies the solution to the basic semiconductor equations since in the regions with no electric field, Poisson’s Equation and the drift transport term in the transport equations “disappear”. In this case in the regions I and III, the equations reduce from 3 to 2 equations which have n(x) or p(x) as the only variable, and hence can be more readily solved in closed form. Christiana Honsberg
Derivation of the ideal diode equation for solar cells
General Procedure using the depletion approximation: Divide the device into regions with an electric field and without an electric field. 1. Solve for electrostatic properties in the depletion region (Region II on the diagram). This solution depends on the doping profile assumed. Here we will restrict the calculations to constant doping. 2. Solve for the carrier concentration and current in the quasi-neutral regions (Regions I and III on the diagram) under steady-state conditions. The steps in this are: a. Determine the general solution for the particular device. The general solution will depend only on the types of recombination and generation in the device. b. Find the particular solution, which depends on the surfaces and the conditions at the edges of the depletion region. 3. Find the relationship between the currents on one side of the depletion region and the currents on the other side. This depends on the recombination/generation mechanisms in the depletion region.
Part 1: Solving for static properties in region with electric field Assumptions are: (1) Depletion approximation: the electric field is confined to a particular region. (2) No free carriers (n(x), p(x) = 0 ) in depletion region. We can assume no free carriers since the electric field sweeps them out of the depletion region quickly. No free carriers means (1) transport equations drop out and (2) no recombination or generation, so the continuity equation becomes
1 ∂J n = (U − G ) = 0 . This means that Jn is constant q ∂x
across the depletion region. Similarly, Jp is also constant across the depletion region. (3) Abrupt or step doping profile (NA+, ND+ are constant). (4) All dopants are ionised (NA+= NA, ND+ = ND). (5) One-dimensional device. The only equation left to solve is Poisson’s Equation, with n(x) and p(x) =0, abrupt doping profile and ionized dopant atoms. Poisson’s equation then becomes:
v ∂E ρ q = = (− N A + N D ) dx ε ε
⎧ − qN A ⎪⎪ ε ε when - x p ≤ x < 0 ˆ dE ρ = where ρ = ⎨ 0 s qN D dx ε 0ε s ⎪ when 0 ≤ x < xn ⎪⎩ ε 0ε s
or
ε0 is the permittivity in free space, and εs is the permittivity in the semiconductor and xp and xn are the edges of the depletion region in the p- and n-type side respectively, measured from the physical junction between the two materials.
− qN A ⎧ − qN A ⎪⎪ ∫ ε dx = ε x + C1 s The electric field then becomes E = ⎨ qN D qN D ⎪∫ dx = x + C2 ⎪⎩ ε s ε
for - x p ≤ x < 0 for 0 ≤ x < xn
The integration constants C1 and C2 can be determined by using the depletion approximation, which states that the electric field must go to zero at the boundary of the depletion regions. This gives:
E(x = −xp ) = 0
Christiana Honsberg
⇒ C1 =
− qN A
ε
xp
and
E ( x = xn ) = 0
⇒ C2 =
− qN D
εS
Derivation of the ideal diode equation for solar cells
⎧ − qN A ⎪⎪ ε (x + x p ) E=⎨ s − qN D ⎪ (xn − x ) ⎪⎩ ε s
when - x p ≤ x < 0 when 0 ≤ x < xn
The maximum electric field occurs at the junction between the p- and n-type material. Further, we know that the electric field lines must be continuous across the interface, such the electric field in the p-type side and the n-type side must equal each other at the interface or when x = 0. Putting x = 0 in the above equation for electric field and setting the two values of E equal to each other gives: N A x p = N D xn . This equation makes physical sense since it states that the total charge on one side of the junction must be the same as the total charge on the other. In other words, if the electric field is confined to the depletion region, then the net charge in Region II must be zero, and hence the negative charge and the positive charge must be equal. NAxpA is the total negative charge, since NA is the charge density and xpA is the volume of the depletion region (A is the cross-sectional area and xp is the depth). Similarly, NDxnA is the positive charge. The cross sectional area (A) is the same and cancels out.
ρ
qND Ê
xp xn
x
dÊ dx
=
x
ρ ε
qNA Êmax
(a)
(b)
Figure 2: (a) Doping concentration in a pn junction. The dotted lines are the actual net charge density (the tails are exaggerated) and the solid line represents the assumed charge density in the depletion approximation. (b) The electric field in a pn junction. To find the voltage as a function of distance, we integrate the equation for the electric field.
⎧ qN A qN A ⎛ x ⎞ ⎪⎪ ∫ − E ( x)dx = ∫ ε (x + x p )dx = ε ⎜ 2 + x p ⎟ x + C3 ⎝ ⎠ V ( x) = ⎨ ⎪ − E ( x)dx = qN D ( xn − x )dx = qN D ⎛⎜ xn − x ⎞⎟ x + C4 ∫ ε ⎪⎩∫ ε ⎝ 2⎠
for - x p ≤ x < 0 for 0 ≤ x < xn
We are usually interested in the potential difference across the junction and can arbitrarily set one side to zero. Here we define the voltage on the p-type side as zero, such that at x= −xp, V=0. This gives the constant C3 as:
C3 =
qN A 2 qN A (x + x p )2 for - x p ≤ x < 0 . x p which gives V ( x) = 2ε 2ε
We can find C4 by using the fact that the potential on the n-type side and p-type side are identical at the interface, such that:
V p ( x = 0) =
qN A 2 qN A ⎛ x⎞ qN A 2 x p = Vn ( x = 0) = x p . Overall, V(x) is: ⎜ xn − ⎟ x + C4 or C4 = 2ε 2ε ⎝ 2⎠ 2ε
Christiana Honsberg
Derivation of the ideal diode equation for solar cells
⎧ qN A (x + x p )2 ⎪⎪ 2ε V ( x) = ⎨ qN ⎛ x⎞ qN A 2 ⎪ D ⎜ xn − ⎟ x + xp ⎪⎩ ε ⎝ 2⎠ 2ε
for - x p ≤ x < 0 for 0 ≤ x < xn
The total voltage is plotted below. ψ dψ dx
ψ 0 – VA
= –Ê
x
Figure 3: Plot of the voltage across a pn junction, assuming that the voltage on the p-type side is zero. The maximum voltage across the junction is a x= xn, which is:
V ( x = xn ) =
q (N D xn2 + N A x 2p ) . 2ε
This voltage is also equal to the built-in voltage across the pn junction, V0, (which we can find from the difference in Fermi-levels between the n and p-type material), giving
V0 =
q (N D xn2 + N A x 2p ) 2ε
Using N A x p = N D xn in the above equation and rearranging allows xp and xn to be determined. They are:
⎡ 2ε (V0 ) ⎛ ⎞⎤ NA ⎜⎜ ⎟⎟⎥ xn = ⎢ ⎣ q ⎝ N D (N A + N D ) ⎠⎦
12
and
⎡ 2ε (V0 ) ⎛ ⎞⎤ ND ⎜⎜ ⎟⎟⎥ xp = ⎢ ⎣ q ⎝ N A ( N A + N D ) ⎠⎦
12
From these equations, the maximum electric field and total depletion width can be determined.
2q Eˆ max = −
ε
(
W = x p + xn = xp = W
Vo + N1D
)
2ε Vo q
(
1 NA
1 NA
+
)
1 −1 ND
ND NA and xn = W N A + ND N A + ND
where Vo =
kT ⎛ N A N D ⎞ ⎟ is the built-in voltage and is calculated separately. ln⎜ q ⎜⎝ ni2 ⎟⎠
Christiana Honsberg
Derivation of the ideal diode equation for solar cells
Part 2: Solving for Regions I and III: no electric field In Regions I and III, the depletion approximation states that there is no electric field. Therefore, Poisson’s Equation becomes zero, and the drift term in the transport equation (qµnnE or qµppE) also becomes zero. The continuity equations remain:
The transport equations then become:
J n = qDn
dn( x) dx
J p = − qD p
dp ( x) dx
(1a)
1 ∂J n = (U − G ) q ∂x
(2a)
(1b)
1 ∂J p = −(U − G ) q ∂x
(2b)
For electrons, differentiating (1a) above and substituting into (2a) above gives:
For holes, differentiating (1b) substituting into (2b) above gives:
dJ n d 2 n( x ) = qDn = q(U ( x) − G ( x) ) , or: dx dx 2
dJ p
d n( x) U ( x) − G ( x ) = dx 2 Dn
dx
= − qD p
(3a)
and
d 2 p ( x) = − q(U ( x) − G ( x) ) or: dx 2
d 2 p ( x) U ( x) − G ( x) = dx 2 Dp
2
above
(3b)
Equations (3a) and (3b) are general under steady state conditions as long as the depletion approximation holds (which assumes low injection) and as long as drift and diffusion are the only transport mechanisms. For steady state and low injection, these are the equations that must be solved to determine the electrical characteristics, of the IV equation, of a device. The solution to these second order differential equations involves firstly finding equations for U and G to determine the general solution to the differential equation, and secondly determining boundary conditions to find the particular solution. Dominance of the minority carrier currents In either Region I or Region III, the total current consists of two current components, Jn (the current composed of electrons) and Jp (the current composed of holes). However, we will solve only for the minority carrier current in each material, since the recombination of minority carrier control the current flow. Later, we will also calculate the majority carrier currents Jp (in p-type material) based on the fact that the total current is constant. 2a: (i) Finding The Recombination Rate: Assumption of Low Injection The general form of the recombination rate for SRH recombination is:
U=
(np − n ) 2 i
τ p 0 (n + n1 ) + τ n 0 ( p + p1 )
where n1, p1 are the number of carriers in recombination sites, no, po are the electron and hole concentration in equilibrium and τp0 and τn0 are the minority carrier lifetimes for holes and electrons. In p-type material and under low injection (low bias) conditions, p >> n also p ≈ po. Further, we will assume that n >> n1 and p >> p1 and that the lifetimes do not vary dramatically in n- and p-type material (i.e., that τn0 p >>τp0 n). The above equation then reduces to the recombination rate for electrons in p-type material as:
U=
(np − n ) = (np − n p ) = (np τ n0 p
2 i
0
τ n0 p
0
− n0 p0 ) (n − n0 ) ∆n = = τ n 0 p0 τ n0 τn 0
This is the low injection form of the recombination rate and it is commonly used in closed form solutions of pn junctions as it greatly simplifies the mathematics. For holes in n-type material, U =
Christiana Honsberg
∆p
τp
.
Derivation of the ideal diode equation for solar cells
2a: (ii) Finding the Generation Rate: In general G will be given by the equation:
G = αN S exp(− αx ) Note that NS may vary with time and may be change with wavelength, such that in general NS will be a function of time and wavelength denoted by NS(λ,t), which also makes the generation rate G a function of wavelength and time. However, here we are solving only for steady state and ignore any time dependence. Wavelength dependence is often included by summing the result obtained at one wavelength over all wavelengths of interest. The exponential form of the generation rate generally makes the differential equation to be solved a nonhomogeneous second order differential equation. Therefore, approximation to the generation rate are often that the generation is constant (valid when the dimensions of interest are small compared to α-1), that there is an impulse generation at the surface, or that the generation is zero. 2b: Finding the general solution The general solution to the differential equation in (3a) and (3b) will depend on the equation for U and G. There are several common general solutions. These are shown below, where ζ (x) is any function dependent only on x, and for the pn junction equations ζ (x) will usually correspond to one of n(x), p(x), ∆n(x) or ∆p(x). A and B are constants that need to be determined by the boundary conditions. C and K are semiconductor or device constants. C for pn junction equations is usually Ln or Lp (the minority carrier diffusions length) and K is often a constant generation term, G. Differential equation of the form
General Solution
When Used
d 2ζ ( x) ζ ( x) = 2 dx 2 C
ζ ( x) = Ae − x C + Be x C
Bulk recombination, no generation
d 2ζ ( x ) ζ ( x ) K = 2 + 2 dx 2 C C
ζ ( x) = Ae − x C + Be x C + K
Bulk recombination, constant gen.
d 2ζ ( x) =0 dx 2
ζ ( x) = Ax + B
Zero recombination and generation
d 2ζ ( x) =K dx 2
ζ ( x) =
K 2 x + Ax + B 2
Zero recombination, constant gen.
2c: Finding the particular solution (all equations will be for electrons in p-type material) The particular solution depends on the conditions of each region at the edges or the regions. For many semiconductor devices, at least one edge will be a pn junction, and hence Boundary Condition #1 below applies to many semiconductor devices. 1. Boundary condition at the edge of a depletion region of a pn junction:
( (e
In p-type material: n p ( x = 0 ) = n p 0 e qV
kT
or ∆n p ( x = 0 ) = n p 0 e qV
In n-type material pn ( x = 0 ) = pn 0 e qV
kT
or ∆pn ( x = 0 ) = pn 0
kT
qV kT
− 1)
− 1)
This boundary condition holds as long as the device is in low injection. 2. Possible boundary conditions at a semiconductor surface The other boundary condition may depend on the surface of the device. The surface recombination velocity, Sr, determines the conditions at the surface. Some common boundary conditions for surfaces are listed below for p-type material. Location of surface Surface is far away from the junction (W>>Ln) Surface is within a few Christiana Honsberg
Boundary Description The minority carrier concentration must be finite as x → ∞ Surface recombination is “infinitely”
Equation n(x → ∞) = finite ∆n(x=W)=0
Derivation of the ideal diode equation for solar cells
diffusion lengths of junction Surface is within a few diffusion lengths of junction Surface next to a light generation source
fast (Sr = ∞) All carriers that reach the surface recombine. Surface recombination is finite.
Impulse light generation at surface with no surface recombination
qDn
dn = qS r ∆n dx
J ( x = 0) = qG
2d: Finding diffusion currents in Regions I and III. Once we have a form of n(x) and p(x) from the procedures described in 2(b) and 2(c), we can readily find the minority carrier currents by using the general equations:
J p = − qD p
dp ( x) dx
and
J n = qDn
dn( x) . dx
Part 3: Finding total current To find the total current, we note that the TOTAL current in the device must be constant, independent of distance as long as there is not a that can extract or inject carriers and as long as the device is under steady state conditions. This can be shown by:
dJ T d (J p + J n ) dJ p dJ n = = + = − q (U p + G p ) + q(U n + Gn ) = q (Gn − G p ) − q (U n − U p ) dx dx dx dx
Since each electron generates a holes and each recombining electron also uses up a hole, Un = Up and Gn= Gp so that the derivative of JT is 0 and JT is a constant. Physically, the continuity equation is stating that the total number of electrons and holes cannot change in the semiconductor (in steady state), and hence the total current also cannot change. Therefore if we find JT anywhere in the device, we have found it everywhere in the device. It is most convenient to find the total current at the edges of the depletion regions. Since we know the currents in Regions I and III, to calculate the total current, we need to do two things. (1) for the fact that the distance variable x is not the same in Region I and Region III and (2) find the current at the depletion region edges. 1. Making the distance variable the same. In our solutions, the distance variable, x, in the above equations is usually not the same for the different regions of the device. Typically, we define x' in Region I (here p-type region with an electron minority carrier current) as the distance from depletion region edge and increasing further into Region I. The other distance variable x, is defined as zero at the other depletion region edge and is increasing into Region III (here n-type material with a hole minority carrier current). Using these definitions, the transport equations are:
dn( x′) . dx′ dn( x′) dx′ dn( x′) dn(− x + W ) Since x = − x' + W, =− and then the current becomes J n ( x) = − qDn or dx′ dx dx dx J n ( x) = − J n ( x′) = − J n (− x + W ) . J p = −qD p
dp ( x) dx
and
J n = qDn
2. Current across the depletion region. Previously, we stated that the generation was zero and the number of free carriers was small, so the recombination could also be neglected the depletion region. As previously stated, under these conditions, the change in current across the depletion region is zero and we can find the total current just as the sum of the currents at the edges of Region I and III, as shown below:
J total = J n ( x = 0) + J p (− x = 0) A more accurate solution includes the change in Jn and Jp across the depletion region, and we find the total current by: J total = ( x = 0) + J p ( x = 0) + ∆J p −dep where ∆Jp-dep is the change in Jp across the depletion region. We could solve for ∆Jn-dep via the continuity equation. The continuity equations, repeated below, give Christiana Honsberg
Derivation of the ideal diode equation for solar cells
current dependence on recombination and generation. xp
becomes J p −depletion = ∆J p = −
∫ (U
x = xn
depletion
1 ∂J p = −(U − G ) and in the depletion region this q ∂x
− Gdepletion ) . Often, the recombination term is ignored and G is
(
)
assumed to be a constant, such that ∆J p = G xn + x p = GW .
Christiana Honsberg
Derivation of the ideal diode equation for solar cells
Example 1: General solution for low injection recombination and constant generation, wide base pn junction. A wide base junction is one in which the surface are far away enough from the junction edges such that they to not impact the recombination properties of carriers injected into the QNR under forward bias. Step 1: Solve for properties in depletion region. As in most devices, the solution for the electrostatic properties in the depletion region does not change, and so is not repeated. Step 2: Solve for carrier concentrations and currents in quasi-neutral regions The solution below is shown in detail only for the n-type material (in which there is a hole current). Step 2a: Find U and G. We will set G equal to a constant and in the n-type material U =
∆p
(in p-type material, U =
τp
∆n
τn
).
Step 2b: Find general solution Using low injection recombination and constant generation gives the equation:
d 2 p (U − G ) ∆p G = = − 2 dx Dp D pτ p D p
d 2 p d 2 ∆p since ∆p = pn − pn 0 (where pn0 is a constant), so the derivative (and second = dx 2 dx 2 derivative) of ∆p(x) is the same as the derivative of p(x). In addition for simplicity, we introduce a variable change using: L p = D pτ p . Note that
The overall differential equation now becomes:
d 2 ∆p ∆p Gτ p = 2 − 2 , which has a general solution: dx 2 Lp Lp
∆p d 2 p d 2 ∆p G ∆p Gτ p = = − = − 2 or 2 2 dx dx D pτ p D p L2p Lp ∆p ( x) = Ae
−x
Lp
+ Be
x
Lp
+ Gτ p
For electrons (p-type material), the differential equations and solutions are:
∆n d 2 ∆n G ∆n Gτ n = − = − 2 2 dx' Dnτ n Dn L2n Ln
and ∆n( x ' ) = Ae
− x′
Ln
+ Be
x′
Ln
+ Gτ n
Step 2c: Particular solution for wide base diode We need two boundary conditions these are: (1) At the edge of the depletion region, pn ( x = 0 ) = pn 0 e qV
kT
(2) The minority carrier concentration must be finite even as x tends to infinity. This can only be achieved if B = 0. Since B = 0, the general solution for holes then becomes
∆p ( x) = p ( x) − pn 0 = Ae
−x
Lp
+ Gτ p
at x=0, ∆p( x=0 ) = pn ( x = 0 ) − pn 0 = pn 0 e qV
Christiana Honsberg
kT
− pn 0 = A + Gτ p
Derivation of the ideal diode equation for solar cells
(
Rearranging gives A = pn 0 e qV
kT
− 1) − Gτ p
Plugging A back in gives:
[
∆p = pn 0 (e qV
kT
]
− 1) − Gτ p e
−x
Lp
[
p ( x) = pn 0 (e qV
+ Gτ p or
kT
]
− 1) − Gτ p e
− x Lp
+ Gτ p + pn 0
The equation for electrons in p-type material, ∆n(x′), can be similarly derived as:
[
∆n( x′) = n p 0 (e qV
kT
]
− 1) − Gτ n e
− x′
Ln
+ Gτ n
This is plotted below for G=0. Carrier concentration on linear scale
Carrier concentration on log scale small change in majority carrier concentration n
p
p n
ln(n) , ln(p)
pp0
n
p
p
n
pn0
np0
x'
x a
nn0
n, p
pn0
np0
≈≈ ≈
≈
calculated minority carrier concentration
a
b
Distance, x
b
Differentiating and plugging into equation for current gives:
J p ( x) = − qD p J n ( x′) = qDn
[
dp ( x) qD p = pn 0 (e qV dx Lp
dn( x′) qD = − n n p 0 (e qV dx′ Ln
[
]
kT
− 1) − Gτ p e
kT
− 1) − Gτ n e − x′ Ln
− x Lp
]
Making the change from x to x′ gives
J n ( x) = − J n ( x′) =
[
qDn n p 0 (e qV Ln
kT
]
− 1) − Gτ n e ( − x+ w) Ln
Step 3: Find total current The change in the current across the depletion region is:
∂J n = ∆J n = q (U n − depletion − Gn − depletionn ) ∂x Assuming that there is no generation and recombination, then ∆Jn = 0 and
J total = J n ( x = 0) + J p ( x = −W ) This case is shown in the graph below. Christiana Honsberg
Derivation of the ideal diode equation for solar cells
Current Density, J
J total Jn
Jp
p-type doped
n-type doped Jp
Jn
x'
distance
x a
b
If there is a constant generation across the depletion region, then ∆J n − dep = qGxn , where xn is the depletion width in the p-type material and xn +xp = W. Jn at the edge of the depletion region in the p-type material is:
J n ( x = 0) = −
qDn n p 0 Ln
(e
qV kT
− 1) +
qDn n p 0 qV qDn (e Gτ n = − Ln Ln
kT
− 1) + qGLn
Jn at the edge of the depletion region in the n-type material is:
J n ( x' = 0) = J n ( x = 0) + ∆J n − dep = −
qDn n p 0 Ln
(e
qV kT
)
− 1 + qG ( Ln + xn + x p )
An analogous equations exists for Jp, and the total current is:
⎡ D ⎤ Dp J total = ⎢q n n p 0 + q p n 0 ⎥ e qV Lp ⎢⎣ Ln ⎥⎦
[
⎡ D n2 D p ni2 ⎤ qV = ⎢q n i + q ⎥e L p N D ⎥⎦ ⎢⎣ Ln N A
[
kT
kT
]
− 1 − qG (Ln + L p + W )
]
− 1 − qG (Lnτ n + L pτ p )
Typically, we write the equation in the form:
[
J = J 0 e qV
kT
]
[
− 1 − J SC or I = I 0 e qV
kT
]
− 1 − I SC
where
⎡ Dn ni2 D p ni2 ⎤ +q J 0 = ⎢q ⎥ L p N D ⎥⎦ ⎢⎣ Ln N A
Christiana Honsberg
Derivation of the ideal diode equation for solar cells
Example 2: General solution for narrow base diode, assumption no recombination in base. Step 1: Solve for properties in depletion region. As in most devices, the solution for the electrostatic properties in the depletion region does not change, and so is given here. Step 2: Solve for carrier concentration and current in quasi-neutral regions Step 2a: Find U and G. We will set G equal to a constant and U = 0 . Step 2b: Find general solution We still start out with the same equation derived from the continuity equations. However, in this case the recombination is zero, so the equation becomes:
d 2 ∆n (U n − Gn ) G G = =− =− 2 Dn Dn Dn dx The general solution is:
∆n( x) = −
G 2 x + Ax + B 2 Dn
Step 2c: Particular solution for narrow base diode with high recombination at edges. We need boundary conditions and these are:
(
(1) At the edge of the depletion region, ∆n p ( x = 0 ) = n p 0 e qV
kT
− 1)
(2) The excess minority carrier concentration ∆n must be zero at x = W, or ∆n(x = W)=0.
(
The first boundary condition gives B = n p 0 e qV
kT
− 1)
The second boundary conditions gives
0=− A=
G W 2 + AW + n p 0 (e qV 2 Dn
n p 0 qV G (e W− 2 Dn W
kT
kT
− 1) , which simplifies to A =
n p 0 qV G (e W− 2 Dn W
kT
− 1) or
− 1)
Substituting these equations into the general solutions gives the equation for the carrier concentration:
∆n ( x ) =
n p 0 qV G 2 ⎡ G (e x +⎢ W− 2 Dn W ⎣ 2 Dn
kT
⎤ − 1)⎥ x + n p 0 (e qV ⎦
kT
− 1)
The current is found by differentiating the carrier concentration:
J n ( x) = qDn
n p 0 qV ⎡G d∆n ( x ) G (e = qDn ⎢ x+ W− 2 Dn dx W ⎣ Dn
kT
⎤ − 1)⎥ ⎦
Simplifying this gives:
J n ( x) = qGx + q
Dn G W − q n p 0 (e qV 2 W
kT
− 1)
Step 3: Find total current The change in the current across the depletion region is given by the general equation:
Christiana Honsberg
Derivation of the ideal diode equation for solar cells
∂J n = ∆J n = q (U n − depletion − G n − depletionn ) ∂x If there is a constant generation across the depletion region and no recombination, then ∆J n − dep = qGx n , where xn is the depletion width in the p-type material. Jn at the edge of the depletion region in the p-type material is:
J n ( x = 0) = q
Dn n p 0 qV G (e W −q 2 W
kT
− 1)
Jn at the 0edge of the depletion region in the n-type material is:
J n ( x' = 0) = J n ( x = 0) + ∆J n − dep = q
Dn n p 0 qV G (e W −q 2 W
kT
− 1) + qGx n
An analogous equations exists for Jp, and the total current is:
Dp ⎡ Dn ⎤ W ⎞ ⎛W J total = ⎢q n p0 + q p n 0 ⎥ e qV kT − 1 − qG⎜ emitter + base + Wdep ⎟ Wbase 2 ⎠ ⎝ 2 ⎣ Wemitter ⎦ 2 ⎡ D p ni ⎤ qV kT Dn ni2 W ⎞ ⎛W = ⎢q +q − 1 − qG⎜ emitter + base + Wdep ⎟ ⎥e Wbase N D ⎥⎦ 2 ⎝ 2 ⎠ ⎢⎣ Wemitter N A
[
[
Christiana Honsberg
]
]
Derivation of the ideal diode equation for solar cells
Summary: Surface Condition
U (for n-type J0 material)
Widebase diode
Far away
U=
Narrow Base diode
Infinite
τp
U=0
∆p = 0 Infinite ∆p = 0
General case
∆p
qDn
dn = qS r ∆n dx
Christiana Honsberg
U=
U=
∆p
τp ∆p
τp
⎡ D n2 D p ni2 ⎤ J 0 = ⎢q n i + q ⎥ L p N D ⎥⎦ ⎢⎣ Ln N A
⎡ D p ni2 ⎤ Dn ni2 J 0 = ⎢q +q ⎥ Wbase N D ⎥⎦ ⎢⎣ Wemitter N A 2 ⎛ qD n 2 ⎛ WN ⎞ ⎞ ⎛ WP ⎞ qD p ni n i ⎜ ⎜ ⎟⎟ ⎜ ⎟ ⋅ coth ⎜ + J0 = coth ⎟ L N ⎜ L ⎟⎟ ⎜ Ln N A L n p D p ⎝ ⎠ ⎝ ⎠⎠ ⎝
⎛ ⎛W ⎞ D ⎛W ⎛W ⎞ ⎛W ⎞ D ⎜ cosh⎜ N ⎟ + p sinh ⎜⎜ N cosh⎜⎜ P ⎟⎟ + n sinh ⎜⎜ P ⎟⎟ 2 ⎟ ⎜ 2 ⎜ qD n ⎝ Ln ⎝ Lp ⎠ S p Lp ⎝ Ln ⎠ + qD p ni ⎝ Ln ⎠ S n Ln J0 = ⎜ n i ⋅ ⎛W ⎞ ⎛W ⎜ Ln N A Dn cosh ⎛⎜ WP ⎞⎟ + sinh ⎛⎜ WP ⎞⎟ L p N D D p cosh⎜ N ⎟ + sinh ⎜ N ⎜L ⎟ ⎜L ⎟ ⎜ ⎜ ⎟ ⎜L S n Ln S L L ⎝ n ⎠ ⎝ n ⎠ p p ⎝ p ⎠ ⎝ p ⎝
Derivation of the ideal diode equation for solar cells
⎞ ⎞⎟ ⎟⎟ ⎠ ⎟ ⋅⎟ ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ ⎠
Poisson's equation:
v
∂ E dx 2.
=
ρ q = ( p ( x) − n( x ) − N A− + N D+ ) ε ε
Transport equations:
v dn( x) J n = qµ n n( x) E + qDn dx
(first term is drift, second is diffusion)
v dp ( x) J p = qµ p p ( x) E − qD p dx 3.
Continuity equations: General conditions
Under thermal equilibrium and steady state conditions
dn 1 ∂J n = − (U − G ) dt q ∂x
1 ∂J n = (U − G ) q ∂x
dp 1 ∂J p = + (U − G ) dt q ∂x
1 ∂J p = −(U − G ) q ∂x
where U and G are the recombination and generation rates in the particular material and depend on the details of the device and may also depend on distance.
Applying above equations to a pn junction: Depletion approximation The above equations can be difficult to solve in closed form unless several simplifying assumptions are made. However, the equations are readily solved using numerical approaches, and there are many device simulators that perform this task. In addition to the assumption of a one-dimensional device, the most central simplifying assumption in determining a closed form solution to the above equations is the depletion approximation. The depletion approximation assumes that the electric field in the device is confined to some region of the device. According to this assumption, the device can then be broken up into regions that have an electric field and those that do not. This is shown below for a pn junction, where Regions I and III do not have an electric field (called quasi-neutral regions or QNR) and Region II has an electric field (which is called space-charge or depletion region).
Region I
No electric field
Region II
Region III
No electric field
Figure 1: Schematic showing the regions with and without the electric field according to the depletion approximation. The depletion approximation greatly simplifies the solution to the basic semiconductor equations since in the regions with no electric field, Poisson’s Equation and the drift transport term in the transport equations “disappear”. In this case in the regions I and III, the equations reduce from 3 to 2 equations which have n(x) or p(x) as the only variable, and hence can be more readily solved in closed form. Christiana Honsberg
Derivation of the ideal diode equation for solar cells
General Procedure using the depletion approximation: Divide the device into regions with an electric field and without an electric field. 1. Solve for electrostatic properties in the depletion region (Region II on the diagram). This solution depends on the doping profile assumed. Here we will restrict the calculations to constant doping. 2. Solve for the carrier concentration and current in the quasi-neutral regions (Regions I and III on the diagram) under steady-state conditions. The steps in this are: a. Determine the general solution for the particular device. The general solution will depend only on the types of recombination and generation in the device. b. Find the particular solution, which depends on the surfaces and the conditions at the edges of the depletion region. 3. Find the relationship between the currents on one side of the depletion region and the currents on the other side. This depends on the recombination/generation mechanisms in the depletion region.
Part 1: Solving for static properties in region with electric field Assumptions are: (1) Depletion approximation: the electric field is confined to a particular region. (2) No free carriers (n(x), p(x) = 0 ) in depletion region. We can assume no free carriers since the electric field sweeps them out of the depletion region quickly. No free carriers means (1) transport equations drop out and (2) no recombination or generation, so the continuity equation becomes
1 ∂J n = (U − G ) = 0 . This means that Jn is constant q ∂x
across the depletion region. Similarly, Jp is also constant across the depletion region. (3) Abrupt or step doping profile (NA+, ND+ are constant). (4) All dopants are ionised (NA+= NA, ND+ = ND). (5) One-dimensional device. The only equation left to solve is Poisson’s Equation, with n(x) and p(x) =0, abrupt doping profile and ionized dopant atoms. Poisson’s equation then becomes:
v ∂E ρ q = = (− N A + N D ) dx ε ε
⎧ − qN A ⎪⎪ ε ε when - x p ≤ x < 0 ˆ dE ρ = where ρ = ⎨ 0 s qN D dx ε 0ε s ⎪ when 0 ≤ x < xn ⎪⎩ ε 0ε s
or
ε0 is the permittivity in free space, and εs is the permittivity in the semiconductor and xp and xn are the edges of the depletion region in the p- and n-type side respectively, measured from the physical junction between the two materials.
− qN A ⎧ − qN A ⎪⎪ ∫ ε dx = ε x + C1 s The electric field then becomes E = ⎨ qN D qN D ⎪∫ dx = x + C2 ⎪⎩ ε s ε
for - x p ≤ x < 0 for 0 ≤ x < xn
The integration constants C1 and C2 can be determined by using the depletion approximation, which states that the electric field must go to zero at the boundary of the depletion regions. This gives:
E(x = −xp ) = 0
Christiana Honsberg
⇒ C1 =
− qN A
ε
xp
and
E ( x = xn ) = 0
⇒ C2 =
− qN D
εS
Derivation of the ideal diode equation for solar cells
⎧ − qN A ⎪⎪ ε (x + x p ) E=⎨ s − qN D ⎪ (xn − x ) ⎪⎩ ε s
when - x p ≤ x < 0 when 0 ≤ x < xn
The maximum electric field occurs at the junction between the p- and n-type material. Further, we know that the electric field lines must be continuous across the interface, such the electric field in the p-type side and the n-type side must equal each other at the interface or when x = 0. Putting x = 0 in the above equation for electric field and setting the two values of E equal to each other gives: N A x p = N D xn . This equation makes physical sense since it states that the total charge on one side of the junction must be the same as the total charge on the other. In other words, if the electric field is confined to the depletion region, then the net charge in Region II must be zero, and hence the negative charge and the positive charge must be equal. NAxpA is the total negative charge, since NA is the charge density and xpA is the volume of the depletion region (A is the cross-sectional area and xp is the depth). Similarly, NDxnA is the positive charge. The cross sectional area (A) is the same and cancels out.
ρ
qND Ê
xp xn
x
dÊ dx
=
x
ρ ε
qNA Êmax
(a)
(b)
Figure 2: (a) Doping concentration in a pn junction. The dotted lines are the actual net charge density (the tails are exaggerated) and the solid line represents the assumed charge density in the depletion approximation. (b) The electric field in a pn junction. To find the voltage as a function of distance, we integrate the equation for the electric field.
⎧ qN A qN A ⎛ x ⎞ ⎪⎪ ∫ − E ( x)dx = ∫ ε (x + x p )dx = ε ⎜ 2 + x p ⎟ x + C3 ⎝ ⎠ V ( x) = ⎨ ⎪ − E ( x)dx = qN D ( xn − x )dx = qN D ⎛⎜ xn − x ⎞⎟ x + C4 ∫ ε ⎪⎩∫ ε ⎝ 2⎠
for - x p ≤ x < 0 for 0 ≤ x < xn
We are usually interested in the potential difference across the junction and can arbitrarily set one side to zero. Here we define the voltage on the p-type side as zero, such that at x= −xp, V=0. This gives the constant C3 as:
C3 =
qN A 2 qN A (x + x p )2 for - x p ≤ x < 0 . x p which gives V ( x) = 2ε 2ε
We can find C4 by using the fact that the potential on the n-type side and p-type side are identical at the interface, such that:
V p ( x = 0) =
qN A 2 qN A ⎛ x⎞ qN A 2 x p = Vn ( x = 0) = x p . Overall, V(x) is: ⎜ xn − ⎟ x + C4 or C4 = 2ε 2ε ⎝ 2⎠ 2ε
Christiana Honsberg
Derivation of the ideal diode equation for solar cells
⎧ qN A (x + x p )2 ⎪⎪ 2ε V ( x) = ⎨ qN ⎛ x⎞ qN A 2 ⎪ D ⎜ xn − ⎟ x + xp ⎪⎩ ε ⎝ 2⎠ 2ε
for - x p ≤ x < 0 for 0 ≤ x < xn
The total voltage is plotted below. ψ dψ dx
ψ 0 – VA
= –Ê
x
Figure 3: Plot of the voltage across a pn junction, assuming that the voltage on the p-type side is zero. The maximum voltage across the junction is a x= xn, which is:
V ( x = xn ) =
q (N D xn2 + N A x 2p ) . 2ε
This voltage is also equal to the built-in voltage across the pn junction, V0, (which we can find from the difference in Fermi-levels between the n and p-type material), giving
V0 =
q (N D xn2 + N A x 2p ) 2ε
Using N A x p = N D xn in the above equation and rearranging allows xp and xn to be determined. They are:
⎡ 2ε (V0 ) ⎛ ⎞⎤ NA ⎜⎜ ⎟⎟⎥ xn = ⎢ ⎣ q ⎝ N D (N A + N D ) ⎠⎦
12
and
⎡ 2ε (V0 ) ⎛ ⎞⎤ ND ⎜⎜ ⎟⎟⎥ xp = ⎢ ⎣ q ⎝ N A ( N A + N D ) ⎠⎦
12
From these equations, the maximum electric field and total depletion width can be determined.
2q Eˆ max = −
ε
(
W = x p + xn = xp = W
Vo + N1D
)
2ε Vo q
(
1 NA
1 NA
+
)
1 −1 ND
ND NA and xn = W N A + ND N A + ND
where Vo =
kT ⎛ N A N D ⎞ ⎟ is the built-in voltage and is calculated separately. ln⎜ q ⎜⎝ ni2 ⎟⎠
Christiana Honsberg
Derivation of the ideal diode equation for solar cells
Part 2: Solving for Regions I and III: no electric field In Regions I and III, the depletion approximation states that there is no electric field. Therefore, Poisson’s Equation becomes zero, and the drift term in the transport equation (qµnnE or qµppE) also becomes zero. The continuity equations remain:
The transport equations then become:
J n = qDn
dn( x) dx
J p = − qD p
dp ( x) dx
(1a)
1 ∂J n = (U − G ) q ∂x
(2a)
(1b)
1 ∂J p = −(U − G ) q ∂x
(2b)
For electrons, differentiating (1a) above and substituting into (2a) above gives:
For holes, differentiating (1b) substituting into (2b) above gives:
dJ n d 2 n( x ) = qDn = q(U ( x) − G ( x) ) , or: dx dx 2
dJ p
d n( x) U ( x) − G ( x ) = dx 2 Dn
dx
= − qD p
(3a)
and
d 2 p ( x) = − q(U ( x) − G ( x) ) or: dx 2
d 2 p ( x) U ( x) − G ( x) = dx 2 Dp
2
above
(3b)
Equations (3a) and (3b) are general under steady state conditions as long as the depletion approximation holds (which assumes low injection) and as long as drift and diffusion are the only transport mechanisms. For steady state and low injection, these are the equations that must be solved to determine the electrical characteristics, of the IV equation, of a device. The solution to these second order differential equations involves firstly finding equations for U and G to determine the general solution to the differential equation, and secondly determining boundary conditions to find the particular solution. Dominance of the minority carrier currents In either Region I or Region III, the total current consists of two current components, Jn (the current composed of electrons) and Jp (the current composed of holes). However, we will solve only for the minority carrier current in each material, since the recombination of minority carrier control the current flow. Later, we will also calculate the majority carrier currents Jp (in p-type material) based on the fact that the total current is constant. 2a: (i) Finding The Recombination Rate: Assumption of Low Injection The general form of the recombination rate for SRH recombination is:
U=
(np − n ) 2 i
τ p 0 (n + n1 ) + τ n 0 ( p + p1 )
where n1, p1 are the number of carriers in recombination sites, no, po are the electron and hole concentration in equilibrium and τp0 and τn0 are the minority carrier lifetimes for holes and electrons. In p-type material and under low injection (low bias) conditions, p >> n also p ≈ po. Further, we will assume that n >> n1 and p >> p1 and that the lifetimes do not vary dramatically in n- and p-type material (i.e., that τn0 p >>τp0 n). The above equation then reduces to the recombination rate for electrons in p-type material as:
U=
(np − n ) = (np − n p ) = (np τ n0 p
2 i
0
τ n0 p
0
− n0 p0 ) (n − n0 ) ∆n = = τ n 0 p0 τ n0 τn 0
This is the low injection form of the recombination rate and it is commonly used in closed form solutions of pn junctions as it greatly simplifies the mathematics. For holes in n-type material, U =
Christiana Honsberg
∆p
τp
.
Derivation of the ideal diode equation for solar cells
2a: (ii) Finding the Generation Rate: In general G will be given by the equation:
G = αN S exp(− αx ) Note that NS may vary with time and may be change with wavelength, such that in general NS will be a function of time and wavelength denoted by NS(λ,t), which also makes the generation rate G a function of wavelength and time. However, here we are solving only for steady state and ignore any time dependence. Wavelength dependence is often included by summing the result obtained at one wavelength over all wavelengths of interest. The exponential form of the generation rate generally makes the differential equation to be solved a nonhomogeneous second order differential equation. Therefore, approximation to the generation rate are often that the generation is constant (valid when the dimensions of interest are small compared to α-1), that there is an impulse generation at the surface, or that the generation is zero. 2b: Finding the general solution The general solution to the differential equation in (3a) and (3b) will depend on the equation for U and G. There are several common general solutions. These are shown below, where ζ (x) is any function dependent only on x, and for the pn junction equations ζ (x) will usually correspond to one of n(x), p(x), ∆n(x) or ∆p(x). A and B are constants that need to be determined by the boundary conditions. C and K are semiconductor or device constants. C for pn junction equations is usually Ln or Lp (the minority carrier diffusions length) and K is often a constant generation term, G. Differential equation of the form
General Solution
When Used
d 2ζ ( x) ζ ( x) = 2 dx 2 C
ζ ( x) = Ae − x C + Be x C
Bulk recombination, no generation
d 2ζ ( x ) ζ ( x ) K = 2 + 2 dx 2 C C
ζ ( x) = Ae − x C + Be x C + K
Bulk recombination, constant gen.
d 2ζ ( x) =0 dx 2
ζ ( x) = Ax + B
Zero recombination and generation
d 2ζ ( x) =K dx 2
ζ ( x) =
K 2 x + Ax + B 2
Zero recombination, constant gen.
2c: Finding the particular solution (all equations will be for electrons in p-type material) The particular solution depends on the conditions of each region at the edges or the regions. For many semiconductor devices, at least one edge will be a pn junction, and hence Boundary Condition #1 below applies to many semiconductor devices. 1. Boundary condition at the edge of a depletion region of a pn junction:
( (e
In p-type material: n p ( x = 0 ) = n p 0 e qV
kT
or ∆n p ( x = 0 ) = n p 0 e qV
In n-type material pn ( x = 0 ) = pn 0 e qV
kT
or ∆pn ( x = 0 ) = pn 0
kT
qV kT
− 1)
− 1)
This boundary condition holds as long as the device is in low injection. 2. Possible boundary conditions at a semiconductor surface The other boundary condition may depend on the surface of the device. The surface recombination velocity, Sr, determines the conditions at the surface. Some common boundary conditions for surfaces are listed below for p-type material. Location of surface Surface is far away from the junction (W>>Ln) Surface is within a few Christiana Honsberg
Boundary Description The minority carrier concentration must be finite as x → ∞ Surface recombination is “infinitely”
Equation n(x → ∞) = finite ∆n(x=W)=0
Derivation of the ideal diode equation for solar cells
diffusion lengths of junction Surface is within a few diffusion lengths of junction Surface next to a light generation source
fast (Sr = ∞) All carriers that reach the surface recombine. Surface recombination is finite.
Impulse light generation at surface with no surface recombination
qDn
dn = qS r ∆n dx
J ( x = 0) = qG
2d: Finding diffusion currents in Regions I and III. Once we have a form of n(x) and p(x) from the procedures described in 2(b) and 2(c), we can readily find the minority carrier currents by using the general equations:
J p = − qD p
dp ( x) dx
and
J n = qDn
dn( x) . dx
Part 3: Finding total current To find the total current, we note that the TOTAL current in the device must be constant, independent of distance as long as there is not a that can extract or inject carriers and as long as the device is under steady state conditions. This can be shown by:
dJ T d (J p + J n ) dJ p dJ n = = + = − q (U p + G p ) + q(U n + Gn ) = q (Gn − G p ) − q (U n − U p ) dx dx dx dx
Since each electron generates a holes and each recombining electron also uses up a hole, Un = Up and Gn= Gp so that the derivative of JT is 0 and JT is a constant. Physically, the continuity equation is stating that the total number of electrons and holes cannot change in the semiconductor (in steady state), and hence the total current also cannot change. Therefore if we find JT anywhere in the device, we have found it everywhere in the device. It is most convenient to find the total current at the edges of the depletion regions. Since we know the currents in Regions I and III, to calculate the total current, we need to do two things. (1) for the fact that the distance variable x is not the same in Region I and Region III and (2) find the current at the depletion region edges. 1. Making the distance variable the same. In our solutions, the distance variable, x, in the above equations is usually not the same for the different regions of the device. Typically, we define x' in Region I (here p-type region with an electron minority carrier current) as the distance from depletion region edge and increasing further into Region I. The other distance variable x, is defined as zero at the other depletion region edge and is increasing into Region III (here n-type material with a hole minority carrier current). Using these definitions, the transport equations are:
dn( x′) . dx′ dn( x′) dx′ dn( x′) dn(− x + W ) Since x = − x' + W, =− and then the current becomes J n ( x) = − qDn or dx′ dx dx dx J n ( x) = − J n ( x′) = − J n (− x + W ) . J p = −qD p
dp ( x) dx
and
J n = qDn
2. Current across the depletion region. Previously, we stated that the generation was zero and the number of free carriers was small, so the recombination could also be neglected the depletion region. As previously stated, under these conditions, the change in current across the depletion region is zero and we can find the total current just as the sum of the currents at the edges of Region I and III, as shown below:
J total = J n ( x = 0) + J p (− x = 0) A more accurate solution includes the change in Jn and Jp across the depletion region, and we find the total current by: J total = ( x = 0) + J p ( x = 0) + ∆J p −dep where ∆Jp-dep is the change in Jp across the depletion region. We could solve for ∆Jn-dep via the continuity equation. The continuity equations, repeated below, give Christiana Honsberg
Derivation of the ideal diode equation for solar cells
current dependence on recombination and generation. xp
becomes J p −depletion = ∆J p = −
∫ (U
x = xn
depletion
1 ∂J p = −(U − G ) and in the depletion region this q ∂x
− Gdepletion ) . Often, the recombination term is ignored and G is
(
)
assumed to be a constant, such that ∆J p = G xn + x p = GW .
Christiana Honsberg
Derivation of the ideal diode equation for solar cells
Example 1: General solution for low injection recombination and constant generation, wide base pn junction. A wide base junction is one in which the surface are far away enough from the junction edges such that they to not impact the recombination properties of carriers injected into the QNR under forward bias. Step 1: Solve for properties in depletion region. As in most devices, the solution for the electrostatic properties in the depletion region does not change, and so is not repeated. Step 2: Solve for carrier concentrations and currents in quasi-neutral regions The solution below is shown in detail only for the n-type material (in which there is a hole current). Step 2a: Find U and G. We will set G equal to a constant and in the n-type material U =
∆p
(in p-type material, U =
τp
∆n
τn
).
Step 2b: Find general solution Using low injection recombination and constant generation gives the equation:
d 2 p (U − G ) ∆p G = = − 2 dx Dp D pτ p D p
d 2 p d 2 ∆p since ∆p = pn − pn 0 (where pn0 is a constant), so the derivative (and second = dx 2 dx 2 derivative) of ∆p(x) is the same as the derivative of p(x). In addition for simplicity, we introduce a variable change using: L p = D pτ p . Note that
The overall differential equation now becomes:
d 2 ∆p ∆p Gτ p = 2 − 2 , which has a general solution: dx 2 Lp Lp
∆p d 2 p d 2 ∆p G ∆p Gτ p = = − = − 2 or 2 2 dx dx D pτ p D p L2p Lp ∆p ( x) = Ae
−x
Lp
+ Be
x
Lp
+ Gτ p
For electrons (p-type material), the differential equations and solutions are:
∆n d 2 ∆n G ∆n Gτ n = − = − 2 2 dx' Dnτ n Dn L2n Ln
and ∆n( x ' ) = Ae
− x′
Ln
+ Be
x′
Ln
+ Gτ n
Step 2c: Particular solution for wide base diode We need two boundary conditions these are: (1) At the edge of the depletion region, pn ( x = 0 ) = pn 0 e qV
kT
(2) The minority carrier concentration must be finite even as x tends to infinity. This can only be achieved if B = 0. Since B = 0, the general solution for holes then becomes
∆p ( x) = p ( x) − pn 0 = Ae
−x
Lp
+ Gτ p
at x=0, ∆p( x=0 ) = pn ( x = 0 ) − pn 0 = pn 0 e qV
Christiana Honsberg
kT
− pn 0 = A + Gτ p
Derivation of the ideal diode equation for solar cells
(
Rearranging gives A = pn 0 e qV
kT
− 1) − Gτ p
Plugging A back in gives:
[
∆p = pn 0 (e qV
kT
]
− 1) − Gτ p e
−x
Lp
[
p ( x) = pn 0 (e qV
+ Gτ p or
kT
]
− 1) − Gτ p e
− x Lp
+ Gτ p + pn 0
The equation for electrons in p-type material, ∆n(x′), can be similarly derived as:
[
∆n( x′) = n p 0 (e qV
kT
]
− 1) − Gτ n e
− x′
Ln
+ Gτ n
This is plotted below for G=0. Carrier concentration on linear scale
Carrier concentration on log scale small change in majority carrier concentration n
p
p n
ln(n) , ln(p)
pp0
n
p
p
n
pn0
np0
x'
x a
nn0
n, p
pn0
np0
≈≈ ≈
≈
calculated minority carrier concentration
a
b
Distance, x
b
Differentiating and plugging into equation for current gives:
J p ( x) = − qD p J n ( x′) = qDn
[
dp ( x) qD p = pn 0 (e qV dx Lp
dn( x′) qD = − n n p 0 (e qV dx′ Ln
[
]
kT
− 1) − Gτ p e
kT
− 1) − Gτ n e − x′ Ln
− x Lp
]
Making the change from x to x′ gives
J n ( x) = − J n ( x′) =
[
qDn n p 0 (e qV Ln
kT
]
− 1) − Gτ n e ( − x+ w) Ln
Step 3: Find total current The change in the current across the depletion region is:
∂J n = ∆J n = q (U n − depletion − Gn − depletionn ) ∂x Assuming that there is no generation and recombination, then ∆Jn = 0 and
J total = J n ( x = 0) + J p ( x = −W ) This case is shown in the graph below. Christiana Honsberg
Derivation of the ideal diode equation for solar cells
Current Density, J
J total Jn
Jp
p-type doped
n-type doped Jp
Jn
x'
distance
x a
b
If there is a constant generation across the depletion region, then ∆J n − dep = qGxn , where xn is the depletion width in the p-type material and xn +xp = W. Jn at the edge of the depletion region in the p-type material is:
J n ( x = 0) = −
qDn n p 0 Ln
(e
qV kT
− 1) +
qDn n p 0 qV qDn (e Gτ n = − Ln Ln
kT
− 1) + qGLn
Jn at the edge of the depletion region in the n-type material is:
J n ( x' = 0) = J n ( x = 0) + ∆J n − dep = −
qDn n p 0 Ln
(e
qV kT
)
− 1 + qG ( Ln + xn + x p )
An analogous equations exists for Jp, and the total current is:
⎡ D ⎤ Dp J total = ⎢q n n p 0 + q p n 0 ⎥ e qV Lp ⎢⎣ Ln ⎥⎦
[
⎡ D n2 D p ni2 ⎤ qV = ⎢q n i + q ⎥e L p N D ⎥⎦ ⎢⎣ Ln N A
[
kT
kT
]
− 1 − qG (Ln + L p + W )
]
− 1 − qG (Lnτ n + L pτ p )
Typically, we write the equation in the form:
[
J = J 0 e qV
kT
]
[
− 1 − J SC or I = I 0 e qV
kT
]
− 1 − I SC
where
⎡ Dn ni2 D p ni2 ⎤ +q J 0 = ⎢q ⎥ L p N D ⎥⎦ ⎢⎣ Ln N A
Christiana Honsberg
Derivation of the ideal diode equation for solar cells
Example 2: General solution for narrow base diode, assumption no recombination in base. Step 1: Solve for properties in depletion region. As in most devices, the solution for the electrostatic properties in the depletion region does not change, and so is given here. Step 2: Solve for carrier concentration and current in quasi-neutral regions Step 2a: Find U and G. We will set G equal to a constant and U = 0 . Step 2b: Find general solution We still start out with the same equation derived from the continuity equations. However, in this case the recombination is zero, so the equation becomes:
d 2 ∆n (U n − Gn ) G G = =− =− 2 Dn Dn Dn dx The general solution is:
∆n( x) = −
G 2 x + Ax + B 2 Dn
Step 2c: Particular solution for narrow base diode with high recombination at edges. We need boundary conditions and these are:
(
(1) At the edge of the depletion region, ∆n p ( x = 0 ) = n p 0 e qV
kT
− 1)
(2) The excess minority carrier concentration ∆n must be zero at x = W, or ∆n(x = W)=0.
(
The first boundary condition gives B = n p 0 e qV
kT
− 1)
The second boundary conditions gives
0=− A=
G W 2 + AW + n p 0 (e qV 2 Dn
n p 0 qV G (e W− 2 Dn W
kT
kT
− 1) , which simplifies to A =
n p 0 qV G (e W− 2 Dn W
kT
− 1) or
− 1)
Substituting these equations into the general solutions gives the equation for the carrier concentration:
∆n ( x ) =
n p 0 qV G 2 ⎡ G (e x +⎢ W− 2 Dn W ⎣ 2 Dn
kT
⎤ − 1)⎥ x + n p 0 (e qV ⎦
kT
− 1)
The current is found by differentiating the carrier concentration:
J n ( x) = qDn
n p 0 qV ⎡G d∆n ( x ) G (e = qDn ⎢ x+ W− 2 Dn dx W ⎣ Dn
kT
⎤ − 1)⎥ ⎦
Simplifying this gives:
J n ( x) = qGx + q
Dn G W − q n p 0 (e qV 2 W
kT
− 1)
Step 3: Find total current The change in the current across the depletion region is given by the general equation:
Christiana Honsberg
Derivation of the ideal diode equation for solar cells
∂J n = ∆J n = q (U n − depletion − G n − depletionn ) ∂x If there is a constant generation across the depletion region and no recombination, then ∆J n − dep = qGx n , where xn is the depletion width in the p-type material. Jn at the edge of the depletion region in the p-type material is:
J n ( x = 0) = q
Dn n p 0 qV G (e W −q 2 W
kT
− 1)
Jn at the 0edge of the depletion region in the n-type material is:
J n ( x' = 0) = J n ( x = 0) + ∆J n − dep = q
Dn n p 0 qV G (e W −q 2 W
kT
− 1) + qGx n
An analogous equations exists for Jp, and the total current is:
Dp ⎡ Dn ⎤ W ⎞ ⎛W J total = ⎢q n p0 + q p n 0 ⎥ e qV kT − 1 − qG⎜ emitter + base + Wdep ⎟ Wbase 2 ⎠ ⎝ 2 ⎣ Wemitter ⎦ 2 ⎡ D p ni ⎤ qV kT Dn ni2 W ⎞ ⎛W = ⎢q +q − 1 − qG⎜ emitter + base + Wdep ⎟ ⎥e Wbase N D ⎥⎦ 2 ⎝ 2 ⎠ ⎢⎣ Wemitter N A
[
[
Christiana Honsberg
]
]
Derivation of the ideal diode equation for solar cells
Summary: Surface Condition
U (for n-type J0 material)
Widebase diode
Far away
U=
Narrow Base diode
Infinite
τp
U=0
∆p = 0 Infinite ∆p = 0
General case
∆p
qDn
dn = qS r ∆n dx
Christiana Honsberg
U=
U=
∆p
τp ∆p
τp
⎡ D n2 D p ni2 ⎤ J 0 = ⎢q n i + q ⎥ L p N D ⎥⎦ ⎢⎣ Ln N A
⎡ D p ni2 ⎤ Dn ni2 J 0 = ⎢q +q ⎥ Wbase N D ⎥⎦ ⎢⎣ Wemitter N A 2 ⎛ qD n 2 ⎛ WN ⎞ ⎞ ⎛ WP ⎞ qD p ni n i ⎜ ⎜ ⎟⎟ ⎜ ⎟ ⋅ coth ⎜ + J0 = coth ⎟ L N ⎜ L ⎟⎟ ⎜ Ln N A L n p D p ⎝ ⎠ ⎝ ⎠⎠ ⎝
⎛ ⎛W ⎞ D ⎛W ⎛W ⎞ ⎛W ⎞ D ⎜ cosh⎜ N ⎟ + p sinh ⎜⎜ N cosh⎜⎜ P ⎟⎟ + n sinh ⎜⎜ P ⎟⎟ 2 ⎟ ⎜ 2 ⎜ qD n ⎝ Ln ⎝ Lp ⎠ S p Lp ⎝ Ln ⎠ + qD p ni ⎝ Ln ⎠ S n Ln J0 = ⎜ n i ⋅ ⎛W ⎞ ⎛W ⎜ Ln N A Dn cosh ⎛⎜ WP ⎞⎟ + sinh ⎛⎜ WP ⎞⎟ L p N D D p cosh⎜ N ⎟ + sinh ⎜ N ⎜L ⎟ ⎜L ⎟ ⎜ ⎜ ⎟ ⎜L S n Ln S L L ⎝ n ⎠ ⎝ n ⎠ p p ⎝ p ⎠ ⎝ p ⎝
Derivation of the ideal diode equation for solar cells
⎞ ⎞⎟ ⎟⎟ ⎠ ⎟ ⋅⎟ ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ ⎠
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