Lecture 6 Thermionic Engines
IT
M
,
n o
e
t
h
l
• Review
C
a
g rm
n
e
• Richardson formula
a
h
G /T ion
© lar rs
• Thermionict engines
e
h
o
v
g
i
S
n
r
• Schottky
barrier
and
diode
t
o
y
c
p
C
e
o
r
i
y
• pn
and
diode
C junction
g
D
r
7• discussion
7 ne
9
9
9
9
E
.
.
l
2 r 2 ca
i
o
r
F ct
e
l
E
–WARREN M. ROHS
ENOW HEAT AND MASS TRANSFER LABORATORY, MIT
ROHSENOW
MIT
Review: Fermi-Dirac Distribution
T
I
M
• Average number of electrons in the state
,
n o
e
t
h
1
l
Fermi-Dirac
=
f = ∑ nAe
C
a
Distribution
⎛ E −g
μ⎞
m
exp⎜⎜an ⎟⎟ + 1er
G⎝ k T/T⎠h ion
© lar rs
t
e
h
o
v
g
At T=0K, μ is called
i
S
n
r
t
y ec Co Fermi level, E
p
o
C Dir rgy f=1 for E<μ
7 97 ne
9
f=0 for E>μ
9
9
E
.
.
2 r 2 ca l
i
o
r
F ct
e
l
E
− ( E − μ ) /( k BT )
n = 0 ,1
B
FERMI-DIRAC DISTRIBUTION
1
0.8
f
1000 K
0.6
0.4
100 K 300 K
0.2
0
-0.1
-0.05
0
E-μ (eV)
0.05
0.1
–WARREN M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY, MIT
Review: Electron Density
T
I
2
2
+
k
h 2 (k x2n+, kM
y
z )
E − Ec = he
o
t
l
2am
C
g rm
n
e
a
h
G /T ion
© lar rs
t
⎡ E−μ⎤
2V o
e
h
N = 2 ∑ ∑ ∑ f (E
g, T ) =t8πS ∫ ∫nv∫ dk dk dk exp⎢− k T ⎥
i
r
y ec Co
⎣
⎦
p
o
∞
C Dir rgy
e
n= ∫D
(7E ) f ( E
,7μ , T )dE
n
9
9
9
E
.
.
E9
l
2 r 2 ca
⎛ E −μ ⎞
⎛ E −μ ⎞
⎛o
2 πm κtriT ⎞
F
⎟⎟ = N exp⎜⎜ −
⎟⎟
⎟⎟ exp⎜⎜ −
n = 2⎜⎜
c
⎝ Eleh
⎠
⎝ k T ⎠
⎝ k T ⎠
π /a π /a π /a
Nx / 2 Ny / 2 Nz / 2
x
3
−Nx / 2 −N y / 2 −Nz / 2
y
z
−π / a −π / a −π / a
B
c
3/ 2
*
B
c
c
c
2
B
B
–WARREN M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY, MIT
Different Solids
T
I
M
,
Energy
n o
e
t
h
l
C
a
g rm
n
e
a
h
G /T ion
© lar rs
t
e
h
o
v
g
i
S
n
r
t
o
y
c
p
C
e
o
r
C D i rg y
71 k/(π/a)907 ne1
0
1
0
9
0
9
9
E
.
.
l
2Metal r 2 ca
i
o
r
F ct Semiconductor
e
l
E
Energy
Energy
Energy
Energy
Energy
Gap Eg
Energy
Donor
Energy Level
Acceptor
Acce
ptor
Energy
Ene
rgy Level
Fermi
Ferm
Fermi
i
Level
Level
k/(π/a)
k/(π/a)
Insulator
n-type
Semiconductor
Se
miconductor
1
k/(π/a)
/a)
p-type
p-type
Semiconductor
Semiconductor
–WARREN M. ROHS
ENOW HEAT AND MASS TRANSFER LABORATORY, MIT
ROHSENOW
MIT
Work Function and Affinity
Light
T
I
Electron
M
,
n
e
Metal to
h
l
Electrodes
C
a
g rm
n
e
a
h
n
G /T ioCurrent
© lar rs
Vacuum
t
e
h
o
v
g
i
S
n
r
t
o
y
c
p
Affinity
C
e
Work Function
W
o
r
C D i rg y
7 97 ne
E
9
9
9
E
Fermi.Level
E
.
l
2 r 2 ca
Bandgap E
i
o
r
F ct
e
l
E
E
c
f
Ef
g
v
–WARREN M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY, MIT
Vacuum
Electrons Just Outside the Metal
T
I
h (k + k, M
+k )
n o
E − μ −W =
Work
e
Function W
2m l t
h
C
a
g
m
r
n
Fermi Level μ
e
Electron a
particlehFlux
Surface
n
G /T iLeaving
o
r
©
s
a
r
t
l
e
h
o
v
g
i
S
n
r
t
o
=
J
v f
y
∑
∑
∑
c
p
C
e
o
r
C D i rg y
hk
2V
e
7
7
n
=
dk dk dk
f
9
9
∫
∫
∫
9
9
E
m
8π
2. r 2. cal
i
o
r
F ct
e
l
E
2
2
x
2
y
2
z
k y =π / a k z =π / a
FERMI-DIRAC DISTRIBUTION
1
0.8
p
k x > 0 k y = −π / a k z = −π / a
x
π /a π /a π /a
1000 K
0.6
x
x
3
0.4
0.2
0
-0.1
y
z
0 −π / a −π / a
100 K 300 K
-0.05
0
E-μ(e V)
0.05
0.1
–WARREN M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY, MIT
Electron Flux Out of Surface
T
I
M
,
n o
e
t
h
l
C
hk
2
1
a
J =
dk dk dk
g
m
∫
∫
∫
r
n
8π
m a ⎛ E −e
μ⎞
⎜
G exp⎜⎝/TkhT ⎟⎟⎠ +io1n
© lar rs
t
h
o
⎛ ⎡ veh (k + k + k )⎤
1
⎞
k
h
2
g
i
S
⎜ − ⎢n
⎟
r
W
≈
+
exp
dk
dk
dk
⎥
t
o
⎜
⎟
c
m
k
T
8π ∫ ∫ ∫ py
2
m
⎥
⎢
C
⎦
e y⎝ ⎣
⎠
o
r
i
C
g
D
r
⎞
⎛
W
m 7
7 ⎜⎜ − ne⎟⎟
(k T )9exp
=
9
9
h
2.π
9
⎝l Ek T ⎠
.
2 r 2 ca
i
o
r
F ct
e
l
E
π /a π /a π /a
x
p
x
3
y
z
0 −π / a −π / a
B
π /a π /a π /a
2
x
x
3
y
2
x
2
y
2
z
z
0 −π / a −π / a
B
2
2
3
B
B
–WARREN M. ROHS
ENOW HEAT AND MASS TRANSFER LABORATORY, MIT
ROHSENOW
MIT
Electrical Current Density
T
I
M
,
n
⎛ W ⎞
⎛ eW
⎞
em
⎜
⎟
⎜
(k T ) exp⎜ − ⎟ = AT exp⎜h− ⎟⎟l to
J =
2π h
C
⎝ k T⎠
⎝ k Ta⎠
O.W. Richardson
g
m
1928 Nobel Prize
n er
a
Richardson Formula
or h
n
G
T
o
/
i
Richardson-Dushman
Equation
r
©
s
t oEmission
la ver
For Thermionic
h
g tS n
i
r
y ec Co
p
emk
A ir
o
y
=C
120 D
A=
Richardson
Constant
g
r
2π 7
h
cm K e
7
n
9
9
9
9
E
2. r 2. cal
⎛ ϕ ⎞
i
o
r
t exp⎜⎜ − k T ⎟⎟ More General
J =F(1 − r ) AT
c
e
⎝
⎠
l
E
2
e
2
2
B
3
B
2
B
2 3
2
B
2
2
e
B
–WARREN M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY, MIT
Contact Potential
T
I
M
,
n o
e
t
h
l
C
a
g rm
n
e
a
h
G /T ion
© lar rs
t
e
h
o
v
g
i
S
n
r
t
o
y
c
p
C
e
o
r
C D i rg y
7 97 ne
9
9
9
E
.
.
l
2 r 2 ca
i
o
r
F ct − e(ϕ − ϕ ') = W − W ' = −eVc
e
l
E
WS = χ + qϕn
Eo
Eo
χ
WS
WM
Ec
EF
qϕn
EF
Ev
a) metal and semiconductor far apart
M S
Eo
qφbi
WM
EF
qϕBn
WS
Eo
χ
Ec
EF
qϕn
Ev
b) metal and semiconductor in intimate contact
Courtesy of Jesús del Alamo. Used with permission.
–WARREN M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY, MIT
Vacuum Diode
T
I
M
,
n o
Cathode
e
Anode
t
h
l
eV
C
a
W
g
m
r
n
J
e
a
h
G /T ion
© lar W rs
t
e
h
o
J
v
g
i
S
n
r
t
o
y
c
p
μ
C
e
o
r gy
C ⎛ WD+ieV
r
⎞
-eV =(W -W )
7 exp⎜⎜9−7 ne⎟⎟
J =9
AT
9
9
kE
T ⎠
.
.
⎝
l
2 r 2 ca
At equilibrium,
i
o
r
⎛
⎞
W
t
F
⎜
⎟⎟
c
J = ATeexp⎜ −
J=J -J =0
El ⎝ k T ⎠
c
a
a
c
c
2
c
c
c
c
a
c
B
2
a
a
B
c
a
–WARREN M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY, MIT
Thermionic Generator
T
I
M
T >T
,
n
e
Anode T
o
Cathode T
t
h
lW
C
a
g
m
r
n
J
e
a
h
G /T ion
© lar W rs
t
e
h
o
J
v
g
i
S
μn eV
r
t
y ec Co
p
μ
o
y
C ⎛ WD+ireV +reV
g
⎞
-eVo=μa-μc
e
7
7
⎜
⎟
−
J =9AT exp9
n
⎜
⎟
k
T
9
9
E
⎠
2. r 2. ⎝ cal
At equilibrium,
i
o tr⎛ W ⎞
F
⎟⎟
J = ATec
exp⎜⎜ −
J=J -J =0
l
k
T
⎝
⎠
E
c
a
a
c
a
a
c
c
c
o
a
2
c
c
c
o
c
B c
2
a
a
a
B a
c
a
–WARREN M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY, MIT
Thermionic Generator
T
I
M
,
⎛ W + eV + eV ⎞
⎛ Wen
⎞
o
t
h
⎜
⎟
⎜
⎟
exp
0
−
−
AT exp⎜ −
=
AT
⎟
⎜ Ck T ⎟ al
k
T
⎝
⎠
⎝
⎠
g
m
r
n
e
a
⎛ T ⎞ W + eV + eV
W
h
n
2 ln⎜⎜ ⎟⎟ =
− G
T
o
/
i
k T
k T
r
©
s
⎝T ⎠
t ola er
h
g t S nv
i
r
y ec Co
Open Circuit Voltage:
p
o
C Dir rgy
⎞7 k T e⎛ T ⎞
W 7⎛ T
−9
V =99 ⎜⎜ 9
1⎟⎟ + 2 nln⎜⎜ ⎟⎟
E
.
.
e
T
l
⎝
⎠
2 r 2 ca e ⎝ T ⎠
i
o
r
F ct
e
l
E
2
c
c
2
o
c
a
a
B c
c
c
a
c
B a
o
B c
a
c
a
B a
B c
c
o
a
a
–WARREN M. ROHS
ENOW HEAT AND MASS TRANSFER LABORATORY, MIT
ROHSENOW
MIT
Under Operation
T
I
M
,
Anode T
n o
Cathode T
e
t
h
l
C
a
⎛
⎞
⎛ W ⎞
+
+
W
eV
eV
J
m
⎟
⎜⎜ −
⎟⎟
−
J = AT exp⎜⎜ − ng
AT
exp
r
⎟
k T e ⎠
⎝a
⎝ k T ⎠
h
G⎛ W +/TeV ⎞ ion ⎛ W ⎞
©
s
⎟⎟ −rAT
⎟⎟
= AT
exp⎜⎜ −lar
exp⎜⎜ −
t
J
k T e
h
⎝
⎠
⎝ k T ⎠
o
v
g
i
S
n
r
t
o
y
c
p
C
e
o
r
y
P = JV
C DiPower Output
g
r
7 97 ne
9
9
9
E
R
.
.
l
2 r 2 ca
i
o
r
F ct
e
l
E
a
c
a
2
c
2
c
a
a
c
B a
B c
2
c
2
a
c
a
a
B c
B a
load
–WARREN M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY, MIT
Heat Transfer: Electron Heat Flux
T
I
Kinetic Energy:
M
,
n
e
o
⎛h k ⎞ h 1
2
t
⎟⎟ C
J =
dk dk dk ⎜⎜
l
∫
∫
∫
a
8π
−μ ⎞
⎝ 2mg⎠ exp⎛⎜ Em
⎟⎟ + 1
r
n
⎜
e
a
k T ⎠
⎝
h
G /T ion
r
2k T
©
s
a
r
t
=
J
l
e
h
o
e
v
g
i
S
n
r
t
o
y
c
Total Heat frompCathode
to Anode:
C
e
o
r
i
y
C
g
D
r
k
T
2
⎛7W
⎞
7 n⎟ Je
J 9
=⎜
− V9+
W
e
e
9
⎝ .9
⎠
E
.
l
2 Heat
2fromcCathode
a
Total
to Anode:
μ
r
eV
i
o
r
F ⎛ Wct 2k T ⎞
J = ⎜le +
⎟J
E e e
π /a π /a π /a
k ,c
2
x
3
y
2
x
z
0 −π / a −π / a
B
B c
c
a
Wa
B c
q ,c
c
c
c
a
q ,a
⎝
B a
⎠
μa
a
–WARREN M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY, MIT
Net Heat Input
T
I
M
,
n
Anode T
Cathode T
Q = J −hJ e + Q to+ Q
l
C
a
g rm
J
n
e
a
Q
--Radiation
heat
h
rad
n
Gtransfer
T
o
/
i
between
cathode
r
©
s
a
r
t and
l
J
anode
e
h
o
v
g
i
S
n
r
t
o
y
c
p
C
e
Q
o
r
cond---Conduction heat
i
y
C D rg loss through leads, and
7 97 ne gas in between
9
9
E
.
.
R9
l
2 r 2 ca
i
o
r
F ct
e
l
E
a
c
h
q ,c
q ,a
rad
cond
a
c
load
–WARREN M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY, MIT
Efficiency
IT
Power
η=
e to
h
l
Qh
C
a
g rm
n
e
a
When radiation is neglected
h
G /T ion
© lar rs
t
e
h
o
v
g
i
n
r
o
y
c
p
C
e
o
r
CeV =W Di r y
e
7
7
n
9
9
9
9
E
2. r 2. cal
i
o
r
F ct
e
l
E
Thermal efficiency - ηt percent
βa = βc
θ=0
80
θ = 0.2
60
θ = 0.4
40
20
a
θ = 0.6
θ = 0.8
a
12
S.W. Angrist,
Direct Energy Conversion,
Allyn and Bacon, Boston, 1965
15
18
βa =
21
24
eVa
kTa
Figure by MIT OpenCourseWare.
–WARREN M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY, MIT
Experimental Demonstration
T
I
M
,
n o
e
t
h
l
C
a
g rm
n
e
a
h
G /T ion
© lar rs
t
e
h
o
v
g
i
S
n
r
t
o
y
c
p
C
e
o
r
C D i rg y
e
7
7
n
9
9
9
9
E
.
.
l
2
Hatsopoulos2
and Kaye,
JAP,
1958.
a
r
c
i
o
r
F ct
e
l
E
Image removed due to copyright restrictions.
Please see Fig. 3 in Hatsopoulos, George N., and
Joseph Kaye. "Measured Thermal Efficiencies of
a Diode Configuration of a Thermo Electron Engine."
Journal of Applied Physics 29 (1958): 1124-1125.
–WARREN M. ROHS
ENOW HEAT AND MASS TRANSFER LABORATORY, MIT
ROHSENOW
MIT
Real Device Issues
T
I
M
,
Due to Spacen
Charge
e
T >T
o
t
h
l
Additional δ
C
a
Anode T
Cathode T
g rm
n
W
e
a
h
G /T ion
J
© lar rs
t
e
h
o
Wv
g
i
S
n
r
t
o
y
c
J
p
C
e
μ
eV
o
r
i
y
C D rg
μ
7 97 ne
9
9
9
E
.
.
l
• Large
work
function,
2 r 2 ca high temperature
i
• Cesium
to reduce
space charge, but reliability problem
o
r
t
F
c
• Vacuumeoperation or plasma operation (filled with gas)
El
c
a
a
c
a
a
c
c
c
o
a
–WARREN M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY, MIT
Recent Trends
T
I
M
,
n o
e
t
h
l
C
a
g rm
n
e
a
h
G /T ion
© lar rs
t
e
h
o
v
g
i
S
n
r
t
o
y
c
p
C
e
o
r
C Di rgy • Negative electron
7 97 ne
affinity materials
9
9
9
E
• Small gap devices
.
.
l
Electron
Affinity
Materials
2Negative
2
a
r
• Solid-state
(Diamond)
c
i
o
r
t
F
thermionics
c
Smith et al., Diamond
and related
e
materials, 15, l2082 (2006)
E
Courtesy of Elsevier, Inc., http://www.sciencedirect.com. Used with permission.
–WARREN M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY, MIT
Schottky Barrier
T
I
M
,
Vacuum
Vacuum
nSchottkyo Barrier
e
t
h
l
C
a
g
Affinity
m
r
n
Work Function W
e
a
h
G /T ion
© lEar rs
E
t
Δ
Δ
e
h
E
Fermi Level E
o
v
g
i
S
n
r
Bandgap
E
E
t
o
y
c
p
C
e
o
r
Metal
Semiconductor
C D i rg y E
7 97 ne
9
9
9
E
Interface
Metal
.
.
N-type
Semiconductor
l
2 r 2 ca
i
o
r
F ct
e
l
E
c
c
f
f
g
f
v
–WARREN M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY, MIT
P-type Schottky Barrier
T
I
M
,
n o
e
t
h
l
C
a
g rm
n
e
a
h
G /T ion
© lar rs
t
e
h
o
v
g
i
S
n
r
t
o
y
c
p
C
e
o
r
C D i rg y
7 97 ne
9
9
9
E
.
.
l
2 r 2 ca
i
o
r
http://w3.ualg.pt/~pjotr/Images/Schottky.png
F ct
e
l
E
Courtesy of Peter Stalllinga. Used with permission.
–WARREN M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY, MIT
Schottky Diode
Richardson Formula
T
I
M
⎞ ⎡ ⎛ eV ⎞ ⎤
⎛ Δ
,
⎟ − 1⎥
J = AT exp⎜⎜ −n ⎟⎟ ⎢exp⎜
e⎝ k T ⎠⎢⎣to ⎜⎝ k T ⎟⎠ ⎥⎦
E
Δ
h
l
C
a
E
g rm
n
e
a
h
n
G
T
o
E
/
i
r
©
s
Δ
t ola er
eV h
g t S nv
E
i
r
y ec Co
Forward Bias
p
o
C Dir rgy
7 97E ne
9
Δ .9
9
E
.
l
2 E r 2 ca
eV
i
o
r
F ct
e
l
Reverse
E Bias
2
c
B
B
f
c
f
c
f
–WARREN M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY, MIT
pn Junction
T
I
M Negative
Electron Energy Increasingly
,
n o
e
t
Build-In Potential
h
l
C
a
g rm
n
p-Type
a
V e
G /Th ion
© lar rs
t
e
h
o
v
g
i
S
n
r
t
o
y
c
p
C
e
n-TYPE
o
- - ir
y
+ + +C + g
-D
r
- e
7
7
+
+
+
+
Space Charge
n
9
9
9
9
E
Region
.
.
l
2 + r+2+ + c-aHole Energy
i
Increasingly Positive
o
r
t Region
F Space Charge
c
e
l
E
Ec
Ec
Ef
Ef
Ev
n-TYPE
n-Type
Ev
bi
p-Type
p-Type
–WARREN M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY, MIT
pn Junction Basics
T
I
M
,
n o
• For nondegenerate semiconductor
e
t
h
l
C
a
g rm
n
⎛ G
Ea− μ ⎞ he
n
⎟
n = N exp⎜⎜ −
T
o
/
i
⎟
r
©
k
T
s
t ⎝ ola ⎠ er
h
g t⎛ Sμ − E n⎞v
i
r
⎜⎜ − Co ⎟⎟
py= N exp
c
p
e ⎝ yk T ⎠
o
r
i
C D rg
7 97 ne ⎛ E ⎞
9
9
9
E
pn
=
.
.
l
2 r 2 ca N N exp⎜⎜⎝ − k T ⎟⎟⎠ = n
i
o
r
F ct
e
l
E
c
c
B
v
v
B
g
c
v
2
i
B
–WARREN M. ROHS
ENOW HEAT AND MASS TRANSFER LABORATORY, MIT
ROHSENOW
MIT
pn Junction Basics: Built-in Potential
T
I
Electron Energy Increasingly Negative
M
⎛ E −μ ⎞
,
n
⎟⎟
N = Ne exp⎜⎜ −o
Build-In Potential
t
h
k T ⎠
⎝
l
C
a
g
m
p-Type
r
n
V
e
a
h
G E/T − μ =ioknT ln⎛⎜⎜ N ⎞⎟⎟
© lar rs
N ⎠
t
⎝
e
h
o
v
g
i
S
n
r
t
o
y
⎛N
c
p
C
n-TYPE
e y E − μ = E − k T ln⎜⎜
o
r
i
C D rg
⎝N
7 Space9Charge
7 ne
9
Region
9
9
E
.
.
l
Hole
Energy
2
2 a
Increasingly r
Positive ic
FeVo = cEtr − E = E − k T ln⎛⎜ N N ⎞⎟ = k T ln⎛⎜ N N
e
l
⎜N N ⎟
⎜ n
E
c ,n
D
c
B
bi
c
c ,n
B
D
⎞
⎟
⎟
A ⎠
v
c, p
g
v
bi
c, p
c ,n
g
B
⎝
A
B
c
D
D
A
⎠
B
⎝
2
i
⎞
⎟
⎟
⎠
–WARREN M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY, MIT
pn Junction Basics: Space Charge Region
T
I
M
,
Electron Energy Increasingly Negative
2eεn⎛ N +
N
o
t
⎜
=
w
h
Build-In Potential
C
e ⎜⎝alN N
g rm
n
e
a
p-Type
V
h
One
Only
n
G /T Side
o
i
r
©
s
a
r
t
l
e
h
o
2ε
v
g
i
S
nw = eN V
r ct
o
y
p ire
C
o
y
n-TYPE C
g
D
r
7 Space97Charge ne
Debye Length
9
9
9
E
.
.
l
Region
2
2
a
r
Hole Energy
c
2ε
i
o
r
w=
k T
Increasingly
t
F Positive
c
e N
e
l
E
s
A
A
D
D
⎞
⎟⎟Vbi
⎠
bi
s
bi
B
s
B
2
B
–WARREN M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY, MIT
pn Junction I-V
T
I
M
,
eV / k T
n o
J = J s (e
− 1)
e
t
h
l
C
a
g
m
r
n
Saturation Current
e
a
h
G /T ion
r
©
s
a
r
t
l
⎛ 1 ha
⎞
⎛ E ⎞
a
1
e
o
v⎟⎟ exp⎜⎜ − ⎟⎟
J = eN N ⎜⎜rig
+
S
n
N c
τ t N oτ ⎠ ⎝ κ T ⎠
y
⎝
p ire
C
o
y
C
g
D
r
(m
/s)
a7---hole diffusivity
e
7
n
9
9
a ---electron
diffusivity
9
9
E
.
l
time
2. τ r---hole
2 recombination
a
c
τ ---electron
recombination time
i
o
r
F ct
e
l
E
B
h
s
c
G
v
A
h
e
h
D
e
B
2
e
h
e
–WARREN M. ROHS
ENOW HEAT AND MASS TRANSFER LABORATORY, MIT
ROHSENOW
MIT
Compare Schottky diode and pn diode
T
I
M
,
n o
e
t
h
l
C
a
g rm
Δ
n
e
a
E
h
G /T ion
© lar rs
t
E
e
h
o
v
g
i
S
n
r
t
o
y
c
p
C
e
⎡ ⎛ eV ⎞ ⎤ o
r
i
y
eV / k T
⎟ −C
1⎥
J = J ⎢exp⎜
g
D
(
J
=
J
e
− 1)
r
T ⎟⎠ ⎥⎦
s
⎢⎣ ⎜⎝ k 7
7 ne
9
9
9
9
E
.
.
l
2 r⎛ 2Δ ⎞ ca
⎞
⎛ 1 a
⎛ E ⎞
a
1
⎟
⎜
⎜⎜ −
⎟⎟
exp
J
=
eN
N
+
⎜⎜ −
⎟⎟ i
J = AT exp
o
⎟
⎜
r
N
τ ⎠ ⎝ κ T⎠
F ⎝ k cT t⎠
⎝N τ
e
l
E
p-type
electron
diffusion
ψe
C
V
hole
diffusion A
ψh
Ec
B
D
Ev
c
n-type
xC
f
xA
x
xD
xB
B
s
B
h
2
s
s
B
c
e
G
v
A
h
D
e
B
–WARREN M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY, MIT
p-type
ptype
ψe
C
V
hole
diffusion A
ψh
B
Current and
Energy
T
I
Distribution
,M
Ec
electron
el
ectron
diffusion
n o
e
t
h
l
C
a
g rm
n
e
a
h
G /T ion
© lar rs
t
e
h
o
v
g
i
S
n
r
t
o
y
c
p
C
e
o
r
C D i rg y
7 97 ne
9
9
9
E
.
.
l
2 r 2 ca
i
o
r
F ct
e
l
E
D
Ev
n-type
xC
xA
xB
x
xD
Energy Source
Energy
Distribution
Possiitive
tive
J
xc
xA
0
Positive
xB
xD
Jn
Negative
Jp
xc
x
xA
xB
xD
–WARREN M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY, MIT
x
MIT OpenCourseWare
http://ocw.mit.edu
2.997 Direct Solar/Thermal to Electrical Energy Conversion Technologies
Fall 2009
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