semiconductor thermodynamics: peltier effect at a pÐn...

Semiconductor thermodynamics:

Peltier effect at a p–n junction

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

(corrected and extended version)

Motivation: Peltier effect in lock-in thermography (LIT)

investigations of photovoltaic devices

a) LIT image of a shunted mc-Si cell

Simple quantitative interpretation

possible: local heating power directly

proportional to the current strength,

P = UbiasIlocal (energy conservation)

1 mK

2a Retreat 2009, Weimar

Motivation: Peltier effect in lock-in thermography (LIT)

investigations of photovoltaic devices

b) LIT image of a CSG module showing

strong edge recombination and cooling

at the contacts

Quantitative interpretation only possible

after correction for the Peltier effect:

heat transfer from contacts to edge

0 125 250 375 500

0

100

200

300

400

Pixel

-0.75 -0.50 -0.25 0.00 0.25 0.50 0.75

_350mV_dat0heatingcooling

a) LIT image of a shunted mc-Si cell

Simple quantitative interpretation

possible: local heating power directly

proportional to the current strength,

P = UbiasIlocal (energy conservation)

1 mK

2b Retreat 2009, Weimar

Peltier effect: “heat current” (not: “flow”!) accompanying an electric current,

not directly observable (in contrast to temperature-gradient-driven heat flow)

General theory: thermoelectricity basics




– heat current density, ! – Peltier coefficient, – electric current density,

" – heat conductivity (all local quantities, i.e., dependent on position)

Generalized Fourier law: (!: heat energy per charge carr.)


r

Qjrj

= ! " #$r r

Qj j T







Temperature change caused by local heating or a change in heat current

c – specific heat capacity, # – mass density, p – heating power density

Heat conduction equation: “heat tone”


!" = #$ %!

r

Q

Tc p j

t

r

Qjrj

= ! " #$r r

Qj j T

3c Retreat 2009, Weimar






Temperature change caused by local heating or a change in heat current

c – specific heat capacity, # – mass density, p – heating power density

Heat conduction equation: “heat tone”

Seebeck effect (electric field caused by a temperature gradient): $ – electric conductivity, E – (applied) electric field, % – Seebeck coefficient

Generalized Ohm’s law: (%: voltage per kelvin)


!" = #$ %!

r

Q

Tc p j

t

r

Qj

( )= ! " #$r rj E T

rj

= ! " #$r r

Qj j T

3d Retreat 2009, Weimar

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)

Isothermal conditions (stationary):

! “Heat tone” (observable effect):


! = "T

( )!" # = !" # $ = ! #"$ !$" #r r r r

Qj j j j

0! = " = # $ %!

r r

QT 0 j j T








Effect: heat exchange at inhomogeneities of ! (e.g. jump at interfaces);

both signs (heating / cooling) are possible


! = "T

( )!" # = !" # $ = ! #"$ !$" #r r r r

Qj j j j

0! = " = # $ %!

r r

QT 0 j j T








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?


! = "T

( )!" # = !" # $ = ! #"$ !$" #r r r r

Qj j j j

0! = " = # $ %!

r r

QT 0 j j T


!"

Peltier heat transfer mechanisms:

– Thermal energy (excited states) of charge carriers ! !cc

– Stream of phonons being dragged along by the electric current ! !ph

Microscopic interpretation of the local Peltier coefficient





(n-type semicond.: “electron drag”, p-type semicond.: “hole drag”)

Roughly: (long-wavelength phonons ! sample size!)



free

ph

ph

cc

vL! "

µ




(n-type semicond.: “electron drag”, p-type semicond.: “hole drag”)

Roughly: (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



F

cc

(E) f (E) dE(E1

e (E) f (E) d

)

E

E !"# =

!"

$%%

free

ph

ph

cc

vL! "

µ

Non-degenerate semiconductor:

Boltzmann distribution, scattering time approximation (& ~ Er)

(r depends on scattering mechanism; r = –! for acoustic phonon scattering)

! effective band-structure energy contribution: above the

band edge,



eff 5band B2E ( r)k T= +





band edge, the latter relative to EF: , qe/h = ±e

!

n-type p-type



( )eff

e / h e / h band e / hE q! = " +

! = "e / h C / V F

E E


VE

CE

FE

eff

bandE

e!

e!

VE

CE

FE

eff

bandE h

!h!





band edge, the latter relative to EF: , qe/h = ±e

!

n-type p-type

Interpretation: EF is the free energy F; isothermal condition: Eint = F + TS

! The excess energy is heat; % = !/T: entropy per charge carrier



( )eff

e / h e / h band e / hE q! = " +

! = "e / h C / V F

E E


VE

CE

FE

eff

bandE

e!

e!

VE

CE

FE

eff

bandE h

!h!

Degenerate semiconductor: (e.g. solar cell emitter!)

– Large doping:

Fermi level inside the band (impurity deionization relevant)




– Large doping:


– Charge carrier contribution !cc small but not negligible (very roughly:

a few kBT/e); only band-structure energy relative to EF relevant




– Large doping:




– 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)




– Large doping:




– 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)



p–n junction: spatially varying Peltier coefficients

Zero bias, no illumination:

Consider also minority carriers!

metal

metal

constant ! !min determined by

! Minority carrier Peltier coefficient increased by Udiff compared to maj. carrier


VE

CE

e!

FE

h!

min

h!

min

e!

ohmic contact

FE

ohmic contact

Udiff

! = "e / h C / V F

E Eeff 5band B2E ( r)k T= +

Zero bias, no illumination:

Consider also minority carriers!

metal

metal

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”)

p–n junction: spatially varying Peltier coefficients


VE

CE

e!

FE

h!

min

h!

min

e!

ohmic contact

FE

ohmic contact

0!" #

Udiff

p–n junction: bias-dependent Peltier coefficients

Diode operation: forward bias, no illumination

! Carrier injection and recombination

metal

metal


VE

CE

e!

FE

h!

min

h!

min

e!

FE

ohmic contact

F,eE

F,hE

bias voltage; net heating

ohmic contact

p–n junction: bias-dependent Peltier coefficients

Diode operation: forward bias, no illumination

! Carrier injection and recombination

metal

metal

also in recombination regions

Minority carrier Peltier coefficients close to the junction change with bias

Recombination heat (non-radiative or radiative) contains Peltier heat


VE

CE

e!

FE

h!

min

h!

min

e!

FE

ohmic contact

0!" #

F,eE

F,hE

bias voltage; net heating

ohmic contact

CSG module: interpretation of Peltier contributions in LIT image

– Peltier heat exchangeproportional to local current

– Contacs: Peltier cooling

– Edge:a) p–n junction, but no Peltier cooling visible!b) Defects, leading to recombination (heating)

! combined effect

0 125 250 375 500

0

100

200

300

400

Pixel

-0.75 -0.50 -0.25 0.00 0.25 0.50 0.75

_350mV_dat0heatingcooling

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

6 m

m

LIT image:

! Cooling at the p–n junction observable by LIT only if laterally separated

from recombination heat sources


Si solar cell operation: asymmetric doping (n+–p) and BSF (p+), full illumination

! Photocurrent in reverse direction

metal

metal

Shunts in solar cells: Peltier-enhanced recombination heat


VE

CE

e!

FE

h!

min

e!

FE

F,eE

F,hE

generated

voltage

ohmic contact

–

+

base

recomb.

photo-gen. e–h pair




metal

metal

Additional forward current due to generated voltage

! additional recomb. losses at nonlinear shunts in the depletion region


VE

CE

e!

FE

h!

min

e!

FE

F,eE

F,hE

generated

voltage

ohmic contact

–

+depletion

region recombination

base

recomb.


defect

level




metal

metal

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!


VE

CE

e!

FE

h!

min

e!

FE

F,eE

F,hE

generated

voltage

ohmic contact

–

+depletion

region recombination

base

recomb.


effective

shunt

heating

voltage

defect

level

Sample: bifacial Si solar cell (surface measurement, 4-point probe)

Idea: separate Joule and Peltier contributions by reversing the current

p region (1016 cm–3): ! # 350 mV, ca. 1/3 from !ph

n region (1020 cm–3): ! # –70 mV, no !ph part

Recent example: quantitative interpretation

of Peltier contributions in a LIT measurement


Integration method:

! = –U "QP / " QJ

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 separatedfrom recombination heat sources

– For a solar cell, the “internal heating” at shunts is larger than according tothe 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!


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)


semiconductor thermodynamics: peltier effect at a pÐn...

Documents