Semiconductor thermodynamics:
Peltier effect at a p–n junction
(corrected and extended version)
Jan-Martin Wagner, Hilmar Straube, Otwin Breitenstein
• Motivation: Peltier effect in lock-in thermography investigations of
photovoltaic devices
• General theory: basics of thermoelectricity and of the Peltier effect
• Microscopic interpretation of the local Peltier coefficient
• p–n junction: spatially varying, bias dependent Peltier coefficients
• Recent example: quantitative interpretation of Peltier contributions
in a LIT measurement
• Summary
1
Retreat 2009, Weimar
Motivation: Peltier effect in lock-in thermography (LIT)
investigations of photovoltaic devices
a) LIT image of a shunted mc-Si cell
1 mK
Simple quantitative interpretation
possible: local heating power directly
proportional to the current strength,
P = UbiasIlocal (energy conservation)
2a
Retreat 2009, Weimar
Motivation: Peltier effect in lock-in thermography (LIT)
investigations of photovoltaic devices
a) LIT image of a shunted mc-Si cell
b) LIT image of a CSG module showing
strong edge recombination and cooling
at the contacts
0
100
200
300
400
0
1 mK
250
375
500
Pixel
Simple quantitative interpretation
possible: local heating power directly
proportional to the current strength,
P = UbiasIlocal (energy conservation)
2b
125
cooling
-0.75
-0.50
-0.25
0
0.00
heating
0.25
0.50
0.75
_350mV_dat
Quantitative interpretation only possible
after correction for the Peltier effect:
heat transfer from contacts to edge
Retreat 2009, Weimar
General theory: thermoelectricity basics
Peltier effect: “heat current” (not: “flow”!) accompanying an electric current,
not directly observable (in contrast to temperature-gradient-driven heat flow)
3a
Retreat 2009, Weimar
General theory: thermoelectricity basics
Peltier effect: “heat current” (not: “flow”!) accompanying an electric current,
heat flow)
rnot directly observable (in contrast to temperature-gradient-driven
r
jQ – heat current density, ! – Peltier coefficient, j – electric current density,
" – heat conductivity (all local quantities, i.e., dependent on position)
r
r
Generalized Fourier law: jQ = ! j " #$T
3b
(!: heat energy per charge carr.)
Retreat 2009, Weimar
General theory: thermoelectricity basics
Peltier effect: “heat current” (not: “flow”!) accompanying an electric current,
heat flow)
rnot directly observable (in contrast to temperature-gradient-driven
r
jQ – heat current density, ! – Peltier coefficient, j – electric current density,
" – heat conductivity (all local quantities, i.e., dependent on position)
r
r
Generalized Fourier law: jQ = ! j " #$T
(!: heat energy per charge carr.)
Temperature change caused by local heating or a change in heat current
c – specific heat capacity, # – mass density, p – heating power density
r
!T
Heat conduction equation: c "
= p # $ % jQ
!t
3c
Retreat 2009, Weimar
“heat tone”
General theory: thermoelectricity basics
Peltier effect: “heat current” (not: “flow”!) accompanying an electric current,
heat flow)
rnot directly observable (in contrast to temperature-gradient-driven
r
jQ – heat current density, ! – Peltier coefficient, j – electric current density,
" – heat conductivity (all local quantities, i.e., dependent on position)
r
r
Generalized Fourier law: jQ = ! j " #$T
(!: heat energy per charge carr.)
Temperature change caused by local heating or a change in heat current
c – specific heat capacity, # – mass density, p – heating power density
r
!T
Heat conduction equation: c "
= p # $ % jQ
!t
“heat tone”
Seebeck effect (electric field caused by a temperature gradient):
$ – electric conductivity, E – (applied) electric field, % – Seebeck coefficient
Generalized Ohm’s law:
3d
r
r
j = ! E " #$T
(
)
Retreat 2009, Weimar
(%: voltage per kelvin)
General theory: thermoelectricity basics
Onsager relation (thermodynamics of irreversible processes)
for symmetrical coupling in linear description ! Kelvin relation:
4a
Retreat 2009, Weimar
! = "T
General theory: thermoelectricity basics
Onsager relation (thermodynamics of irreversible processes)
for symmetrical coupling in linear description ! Kelvin relation:
! = "T
“Pure” Peltier effect: isothermal transport of heat
(usually well approximated by LIT measurement)
r
r
Isothermal conditions (stationary): !T = 0 " j = # j $ %!T
Q
0
r
r
r
r
! “Heat tone” (observable effect): !" # jQ = !" # $ j = ! j # "$ ! $" # j
( )
4b
Retreat 2009, Weimar
General theory: thermoelectricity basics
Onsager relation (thermodynamics of irreversible processes)
for symmetrical coupling in linear description ! Kelvin relation:
! = "T
“Pure” Peltier effect: isothermal transport of heat
(usually well approximated by LIT measurement)
r
r
Isothermal conditions (stationary): !T = 0 " j = # j $ %!T
Q
0
r
r
r
r
! “Heat tone” (observable effect): !" # jQ = !" # $ j = ! j # "$ ! $" # j
( )
Effect: heat exchange at inhomogeneities of ! (e.g. jump at interfaces);
both signs (heating / cooling) are possible
4c
Retreat 2009, Weimar
General theory: thermoelectricity basics
Onsager relation (thermodynamics of irreversible processes)
for symmetrical coupling in linear description ! Kelvin relation:
! = "T
“Pure” Peltier effect: isothermal transport of heat
(usually well approximated by LIT measurement)
r
r
Isothermal conditions (stationary): !T = 0 " j = # j $ %!T
Q
0
r
r
r
r
! “Heat tone” (observable effect): !" # jQ = !" # $ j = ! j # "$ ! $" # j
( )
Effect: heat exchange at inhomogeneities of ! (e.g. jump at interfaces);
both signs (heating / cooling) are possible
Important: Redistribution of heat only, no global heat generation or
consumption
To obtain !" , we need to know !(x)! ! Microscopic view?
4d
Retreat 2009, Weimar
Microscopic interpretation of the local Peltier coefficient
Peltier heat transfer mechanisms:
– Thermal energy (excited states) of charge carriers ! !cc
– Stream of phonons being dragged along by the electric current ! !ph
5a
Retreat 2009, Weimar
Microscopic interpretation of the local Peltier coefficient
Peltier heat transfer mechanisms:
– Thermal energy (excited states) of charge carriers ! !cc
– Stream of phonons being dragged along by the electric current ! !ph
(n-type semicond.: “electron drag”, p-type semicond.: “hole drag”)
Roughly:
5b
! ph "
free
vLph
µcc
(long-wavelength phonons ! sample size!)
Retreat 2009, Weimar
Microscopic interpretation of the local Peltier coefficient
Peltier heat transfer mechanisms:
– Thermal energy (excited states) of charge carriers ! !cc
– Stream of phonons being dragged along by the electric current ! !ph
(n-type semicond.: “electron drag”, p-type semicond.: “hole drag”)
Roughly:
! ph "
free
vLph
µcc
(long-wavelength phonons ! sample size!)
Thermal energy of charge carriers:
Intuitively: 3kBT/2 above band edge (free electron/hole gas) – too simple!
Transport theory: conductivity-weighted average of band-structure energy
relative to the Fermi energy
!
# cc
5c
1 % (EF $ E) "(E) f (E) dE
=
e
% "(E) f !(E) dE
Retreat 2009, Weimar
Microscopic interpretation of the local Peltier coefficient
Non-degenerate semiconductor:
Boltzmann distribution, scattering time approximation (& ~ Er)
(r depends on scattering mechanism; r = –! for acoustic phonon scattering)
eff
! effective band-structure energy contribution: Eband = ( 52 + r)k B T above the
band edge,
6a
Retreat 2009, Weimar
Microscopic interpretation of the local Peltier coefficient
Non-degenerate semiconductor:
Boltzmann distribution, scattering time approximation (& ~ Er)
(r depends on scattering mechanism; r = –! for acoustic phonon scattering)
eff
! effective band-structure energy contribution: Eband = ( 52 + r)k B T above the
band edge, the latter relative to EF: !e / h = EC / V " EF , qe/h = ±e
!
(
eff
! e / h = "e / h + Eband
EC
EF
p-type
EF
EV
eff
!h
!h Eband
EV
6b
e/h
EC
eff
!e
!e Eband
n-type
)q
Retreat 2009, Weimar
Microscopic interpretation of the local Peltier coefficient
Non-degenerate semiconductor:
Boltzmann distribution, scattering time approximation (& ~ Er)
(r depends on scattering mechanism; r = –! for acoustic phonon scattering)
eff
! effective band-structure energy contribution: Eband = ( 52 + r)k B T above the
band edge, the latter relative to EF: !e / h = EC / V " EF , qe/h = ±e
!
(
eff
! e / h = "e / h + Eband
EC
e/h
EC
eff
!e
!e Eband
n-type
)q
EF
p-type
EF
EV
eff
!h
!h Eband
EV
Interpretation: EF is the free energy F; isothermal condition: Eint = F + TS
! The excess energy is heat; % = !/T: entropy per charge carrier
6c
Retreat 2009, Weimar
Microscopic interpretation of the local Peltier coefficient
Degenerate semiconductor: (e.g. solar cell emitter!)
– Large doping:
Fermi level inside the band (impurity deionization relevant)
7a
Retreat 2009, Weimar
Microscopic interpretation of the local Peltier coefficient
Degenerate semiconductor: (e.g. solar cell emitter!)
– Large doping:
Fermi level inside the band (impurity deionization relevant)
– Charge carrier contribution !cc small but not negligible (very roughly:
a few kBT/e); only band-structure energy relative to EF relevant
7b
Retreat 2009, Weimar
Microscopic interpretation of the local Peltier coefficient
Degenerate semiconductor: (e.g. solar cell emitter!)
– Large doping:
Fermi level inside the band (impurity deionization relevant)
– Charge carrier contribution !cc small but not negligible (very roughly:
a few kBT/e); only band-structure energy relative to EF relevant
– Phonon contribution !ph negligible:
(i) reduced free path (more dopants ! more scattering centers)
(ii) back-transfer of momentum from phonons to electrons (“phonon drag”,
as for Seebeck coefficient)
7c
Retreat 2009, Weimar
Microscopic interpretation of the local Peltier coefficient
Degenerate semiconductor: (e.g. solar cell emitter!)
– Large doping:
Fermi level inside the band (impurity deionization relevant)
– Charge carrier contribution !cc small but not negligible (very roughly:
a few kBT/e); only band-structure energy relative to EF relevant
– Phonon contribution !ph negligible:
(i) reduced free path (more dopants ! more scattering centers)
(ii) back-transfer of momentum from phonons to electrons (“phonon drag”,
as for Seebeck coefficient)
Metal: ! ! 0 (compared to semiconductors)
7d
Retreat 2009, Weimar
p–n junction: spatially varying Peltier coefficients
Zero bias, no illumination:
Consider also minority carriers!
ohmic contact
EF
!e
! min
e
Udiff
EC
ohmic contact
metal
EF
metal
!h
EV
! min
h
eff
Eband
= ( 52 + r)k B T constant ! !min determined by !e / h = EC / V " EF
! Minority carrier Peltier coefficient increased by Udiff compared to maj. carrier
8a
Retreat 2009, Weimar
p–n junction: spatially varying Peltier coefficients
! min
e
Zero bias, no illumination:
Consider also minority carriers!
ohmic contact
EF
!e
Udiff
ohmic contact
metal
EC
EF
metal
!h
EV
! min
h
!" # 0
at the contacts and at the junction
! Heat exchange at junction: cooling for “forward” current (carriers “go up”),
heating for “reverse” current (carriers “go down”)
8b
Retreat 2009, Weimar
p–n junction: bias-dependent Peltier coefficients
Diode operation: forward bias, no illumination
! Carrier injection and recombination
ohmic contact
metal
bias voltage; net heating
EF,h
metal
ohmic contact
EC
!e
EF,e
EF
!h
EV
! min
h
9a
!
min
e
Retreat 2009, Weimar
EF
p–n junction: bias-dependent Peltier coefficients
Diode operation: forward bias, no illumination
! Carrier injection and recombination
ohmic contact
!
min
e
EC
!e
metal
EF,e
EF
ohmic contact
bias voltage; net heating
EF,h
metal
!h
EV
! min
h
!" # 0 also in recombination regions
Minority carrier Peltier coefficients close to the junction change with bias
Recombination heat (non-radiative or radiative) contains Peltier heat
9b
EF
Retreat 2009, Weimar
CSG module: interpretation of Peltier contributions in LIT image
CSG (crystalline silicon on glass) module: many long (module width) but narrow
stripes (6 mm) of polycrystalline p–n Si layers (2 "m) connected in series
LIT image:
– Peltier heat exchange
proportional to local current
0
– Contacs: Peltier cooling
6 mm
100
200
300
400
0
125
250
375
500
– Edge:
a) p–n junction, but no Peltier
cooling visible!
b) Defects, leading to
recombination (heating)
Pixel
cooling
-0.75
-0.50
-0.25
0.00
0
0.25
0.50
0.75
heating
! combined effect
_350mV_dat
! Cooling at the p–n junction observable by LIT only if laterally separated
from recombination heat sources
10
Retreat 2009, Weimar
Shunts in solar cells: Peltier-enhanced recombination heat
Si solar cell operation: asymmetric doping (n+–p) and BSF (p+), full illumination
! Photocurrent in reverse direction
–
min
!e
ohmic contact
!e
EF
EF,e
generated
voltage
EV
metal
EF,h
metal
!h
EC
base
recomb. +
photo-gen. e–h pair
11a
Retreat 2009, Weimar
EF
Shunts in solar cells: Peltier-enhanced recombination heat
Si solar cell operation: asymmetric doping (n+–p) and BSF (p+), full illumination
! Photocurrent in reverse direction
–
min
!e
ohmic contact
!e
EF
EF,e
defect
level
generated
voltage
EV
depletion
region recombination
metal
EF,h
metal
!h
EC
base
recomb. +
photo-gen. e–h pair
Additional forward current due to generated voltage
! additional recomb. losses at nonlinear shunts in the depletion region
11b
Retreat 2009, Weimar
EF
Shunts in solar cells: Peltier-enhanced recombination heat
Si solar cell operation: asymmetric doping (n+–p) and BSF (p+), full illumination
! Photocurrent in reverse direction
–
min
!e
ohmic contact
!e
EF
effective
shunt
heating
voltage
metal
EV
EF,e
defect
level
generated
voltage
!h
depletion
region recombination
EC
metal
EF,h
base
recomb. +
photo-gen. e–h pair
Additional forward current due to generated voltage
! additional recomb. losses at nonlinear shunts in the depletion region
Shunt heating at the p–n junction larger than due to generated voltage!
11c
Retreat 2009, Weimar
EF
Recent example: quantitative interpretation
of Peltier contributions in a LIT measurement
Sample: bifacial Si solar cell (surface measurement, 4-point probe)
Idea: separate Joule and Peltier contributions by reversing the current
Integration method:
! = –U "QP
/ " QJ
p region (1016 cm–3): ! # 350 mV, ca. 1/3 from !ph
n region (1020 cm–3): ! # –70 mV, no !ph part
12
Retreat 2009, Weimar
Summary
– The Peltier effect leads to a redistribution of heat (isothermally)
– Heat exchange occurs at inhomogeneities of the Peltier coefficient !
– ! = !cc + !ph
– For a diode, the Peltier coefficient changes (heat exchange occurs)
at the contacts, at the p–n junction, and in recombination regions
– Cooling at the p–n junction observable by LIT only if laterally separated
from recombination heat sources
– For a solar cell, the “internal heating” at shunts is larger than according to
the generated voltage
– For the quantitative interpretation of measured Peltier values of Si, the
“electron drag” effect must be taken into account even at room temp.
Outlook: direct observation of junction cooling in cross-section geometry;
Peltier coefficients for reverse bias?
Thanks for your attention!
13
Retreat 2009, Weimar
References
– Peltier effect at a p–n junction:
K. P. Pipe et al., “Bias-dependent Peltier coefficient and internal cooling in bipolar devices”,
Phys. Rev. B 66, 125316 (2002)
– Contributions to the Peltier coefficient (! = !cc + !ph):
G. S. Nolas et al., “Thermoelectrics: basic principles and …” (Springer, 2001);
C. Herring, “Theory of the thermoelectric power of semiconductors”, Phys. Rev. 96, 1163
(1954)
– Phonon drag in Si at room temperature:
L. Weber et al., “Transport properties of silicon”, Appl. Phys. A 53, 136 (1991)
– General theory:
Solar cells: P. Würfel, “Physics of Solar Cells” (Wiley, 2005);
Transport: M. Lundstrom, “Fundamentals of carrier transport” (Cambridge, 2000);
Irreversible thermodynamics: H. B. Callen, “Thermodynamics …” (Wiley, 1985);
Radiation: C. E. Mungan, “Radiation thermodynamics …”, Am J. Phys. 73, 315 (2005)
– “Internal heating” at shunts (but Eband missing):
M. Kaes et al., “Light-modulated Lock-in Thermography …”, Prog. Photovolt: Res. Appl. 12,
355 (2004)
– Sum/difference imaging and integration method:
H. Straube et al., “Measurement of the Peltier coefficient by lock-in thermography”
(manuscript in preparation)
14
Retreat 2009, Weimar