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Quantitative Model of Unilluminated Diode – part I
G.R. Tynan UC San Diego MAE 119
Lecture Notes
A Solar PV Cell is just a p-n junction (“diode”) illuminated by light….
P-type n-type
Photon Flux with E>Egap
Examine This p-n Junction…
Diode current-voltage characteristics
3 http://www.electronics-tutorials.ws/diode
Key elements: Quasineutral regions Quasi-neutral region
Net charge density ~ 0 Quasi-neutral region
Net charge density ~ 0
Key elements: Depletion region Depletion region With net charge,
finite E-field
What is needed for quantitative model of this diode?
• What are the electron and hole densities? • How do they move within the device?
– Diffusion due to concentration gradients – Motion under action of electric fields
• Given these items, what is the current in the device as a function of voltage across device?
What is needed for quantitative model of this diode?
• What are the electron and hole densities? • How do they move within the device?
– Diffusion due to concentration gradients – Motion under action of electric fields
• Given these items, what is the current in the device as a function of voltage across device?
Determining Charge Carrier Densities
• Find number of mobile electrons/volume in conduction band from – Density of allowed states, N(E) and – Energy distribution of e and h, f(E):
Charge Carrier Energy Distribution
• Charge carrier particle energies follow Fermi-Dirac probability distribution
EF = “Fermi Energy” characteristic upper limit for energy T = Temperature of particles f(E)dE = probability of finding particle between (E, E+dE)
Density of allowed charge carrier states
• Conduction Band Allowed State Density (states/vol-energy)
• Valence Band Allowed State Density (states/vol-energy:
NC E( ) = 8 2π me
*( )h3
3/2
E − EC( )1/2; E ≥ EC
NC E( ) = 8 2π me
*( )h3
3/2
E − EC( )1/2; E ≥ EC
Mobile Charge Carriers found from product of N(E)f(E):
• Find number of mobile electrons/volume in conduction band from N(E) and f(E):
Mobile Charge Carrier Density:
• Number of mobile electrons/volume in conduction band:
• Result:
CE E≥( ) ( )maxc
C
E
En f E N E dE= ∫
n = NC exp EF − EC( ) kT⎡⎣ ⎤⎦
NC = 2 2πme*kT
h2⎛⎝⎜
⎞⎠⎟
3/2
Mobile Charge Carrier Densities
• Number of mobile holes/volume in valence band:
• Result: p = NV exp EV − EF( ) / kT⎡⎣ ⎤⎦
p = NV (E) f (E)dE0
EV
∫
NV = 2 2πmh*kT
h2⎛⎝⎜
⎞⎠⎟
3/2
A few useful definitions:
• Useful to define “intrinsic concentration, ni :
• Since n=p in a pure semiconductor can write
• Which gives:
ni2 = np = NCNV exp EV − EC( ) / kBT⎡⎣ ⎤⎦
NV exp EV − EF( ) / kT⎡⎣ ⎤⎦ = NC exp EF − EC( ) / kT⎡⎣ ⎤⎦
ln2 2
c V VF
C
E E NkTEN
⎛ ⎞+= + ⎜ ⎟⎝ ⎠
Charge Carrier Densities – n-type doped semiconductors
• Charge neutrality tells us
• For n-type doped material
• And therefore in n-type material
p + n + ND+ − NA
− = 0
n = ND+ and ND
+ ≈ ND
p = n − ND+ << n
Charge Carrier Densities – p-type doped semiconductors
• Charge neutrality tells us
• For p-type doped material
• And therefore in p-type material
p + n + ND+ − NA
− = 0
p = NA− and NA
− ≈ NA
n = p − NA− << p
What is needed for quantitative model of this diode?
• What are the electron and hole densities? • How do they move within the device?
– Diffusion due to concentration gradients – Motion under action of electric fields
• Given these items, what is the current in the device as a function of voltage across device?
Resulting carrier density distribution
EXAMINE Depletion Region IN MORE DETAIL…
DEPLETION REGION”
Charge Distribution Across Unbiased p-n diode
Can Find Electric Potential in Depletion Region:
Charge Distribution From Semiconductor Material Theory E-field from Poisson’s Equation Potential Profile by Integrating E-field
Idealized Model of Charge Distribution Across Depletion Region:
Poisson’s eq’n relates charge density to E field:
Poisson’s Equation:
DD NN ≈+AA NN ≈−
Donor/Acceptor Ion Density
p, n distribution given by step functions seen in previous viewgraph
Integrate Poisson’s Equation to Find E(x); Integrate Again to find potential distribution
dEdx
= qε(p − n + ND
+ − NA− )
Idealized Model of Depletion Region:
( )2/1
0max112
⎥⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛+−−=
DAext NNVqE ψ
ε
RESULT:
( )2/1
0112
⎥⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛+−−=+=
DAextnp NNV
qllw ψε
DA
Dp NN
NWl
+=
DA
An NN
NWl+
=
Carrier concentration at edges of un-biased junction
With NO Ext. Voltage, Vext=0:
D
ipnnb N
nkToqeppp
2
)(00
≈−== ψ
A
inpb N
nkToqennn
a
2
00≈⎟
⎠⎞⎜
⎝⎛ −== ψ
What is needed for quantitative model of this diode?
• What are the electron and hole densities? • How do they move within the device?
– Diffusion due to concentration gradients – Motion under action of electric fields
• Given these items, what is the current in the device as a function of voltage across device?
Question: What happens when we apply an external voltage to this diode? In order to answer, need to first define current density, J (Amps/m2)…
Need to define current density, J: Consider a collection of n (p) electrons/unit volume (holes/unit volume) that are moving towards the right
Q: How many charges pass thru the surface per unit area/unit time? A: This is defined as the electron (hole) current density, Je (Jh )
Can be caused by E-field and/or by diffusion
Basic Equations of Semiconductor Device Physics
Poisson’s Equation: )( −+ −+−= AD NNnpqdxdE
DD NN ≈+AA NN ≈−
Donor/Acceptor Ion Density
dxdpqDpEqJ
dxdnqDnEqJ
eheh
eee
−=
+=
µ
µCurrent
Transport
S.S. Charge Conservation: )(1
1
GUdxdJ
q
GUdxdJ
q
h
e
−−=
−=U~Source G~Sink
Question: What happens when we apply an external voltage to this diode? A: Currents can begin to flow…
How does charge distribution respond to Vbias?
Jh = qpµhE − qDh
∂p∂x
In general current density depends on E and gradient, i.e.
We will simplify analysis by assuming that
qpµhE ≈ qDh
dpdx even if Jh ≠ 0
Now…
E ≈ kTq
1p
dpdx
And since we can integrate to find potential drop across depletion region…
E = −∂φ ∂x
e e
h h
kTDqkTDq
µ
µ
=
=
How does charge distribution respond to Vbias?
And since we can integrate to find potential drop across depletion region…
E = −∂φ ∂x
φ! −Va = + kT
qln
ppa
pnb
Which can be re-arranged to give:
pnb
= ppaexp
qφ!kT
⎡
⎣⎢
⎤
⎦⎥exp
qVa
kT⎡
⎣⎢
⎤
⎦⎥
How does charge distribution respond to Vbias?
And since we can integrate to find potential drop across depletion region…
E = −∂φ ∂x
φ! −Va = + kT
qln
ppa
pnb
Which can be re-arranged to give:
pnb
= ppaexp
qφ!kT
⎡
⎣⎢
⎤
⎦⎥exp
qVa
kT⎡
⎣⎢
⎤
⎦⎥
Get similar result for electrons…
How does charge distribution respond to Vbias?
And since we can integrate to find potential drop across depletion region…
E = −∂φ ∂x
φ! −Va = + kT
qln
ppa
pnb
Which can be re-arranged to give:
pnb
= ppaexp
qφ!kT
⎡
⎣⎢
⎤
⎦⎥exp
qVa
kT⎡
⎣⎢
⎤
⎦⎥
pn0
Minority carrier density w/o ext. bias
Get similar result for electrons… THIS IS A KEY RESULT!
Exponential Increase w/ Va>0 !
Forward Bias Increases Minority Charge Carrier Density at Edge
of Quasineutral region:
kTqV
D
ikTqV
nn
appapp
be
nn
epp2
0==
kTqV
A
ikTqV
pp
appapp
ae
Nn
enn2
0==
Concentration Of MINORITY CARRIERS at Edge of Depletion Region INCREASES EXPONENTIALLY W/ Ext. Bias è KNOWN AS MINORITY è CARRIER INJECTION
Minority Carrier Density at edge of quasineutral region increases EXPONENTIALLY forward bias
pnb= pn0
expqVa
kT⎡
⎣⎢
⎤
⎦⎥
npa= np0
expqVa
kT⎡
⎣⎢
⎤
⎦⎥