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First of all, in considering
thermoelectric effects, we have to realize that we are dealing with a non-equilibrium situation. |
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A general theory of non-equilibrium is beyond our
means, suffice it to say that Lars Onsager, with a paper entitled "Reciprocal relations in irreversible
processes" induced some fundamental insights as late as
1930; he received the Nobel price for his contribution to
non-equilibrium thermodynamics in 1968 - for chemistry, of all things. |
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However, what we should be aware of,
is the essential statement of non-equilibrium theory: |
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As long as there is no equilibrium, we always
have currents of something trying to
establish equilibrium by reducing a gradient in something else that is the
actual cause of the non-equilibrium. A gradient in the electrical potential,
e.g., causes our well-known electrical currents, and a gradient in a
concentration causes diffusion currents. |
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But we must abstract even more, and consider
things like entropy currents as well as all kinds of combinations of gradients
and currents. |
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While Onsager discovered some quite
general relations between gradients and currents, we will not delve into
details here, but only look a bit more closely at what causes the Seebeck
effect. |
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For that, we still treat the
thermoelectric effect with equilibrium thermodynamics, simply assuming that
locally we are not very far from
equilibrium and thus can still use band structure models with a Fermi energy
(which is only a well defined quantity for equilibrium) and resulting carrier
distributions. |
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In the simplest possible case, what we will get
for a long bar of metal, hot at one end and cold a the other, is something like
this: |
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At the hot end, the Fermi
distribution is "soft", and we have a noticeable concentration of
electrons well above the Fermi energy. At the cold end, the Fermi distribution
is sharp, and we have fewer electrons above the Fermi energy. |
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The drawing, of course, grossly exaggerates the
real situation. Note also that the total concentration of electrons a both ends
is the same - even so the drawing does not show this because the holes below
the Fermi energy are not included. |
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Note too, that the Fermi energy is constant
throughout the material (we neglect any possible effects of the temperature on
the Fermi energy, as we have it, for example, in
doped
semiconductors). |
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As always, electrons go to where the energy is
lower; the electrons would tend to move from the hot end to the cold end,
thereby transporting energy and thus equilibrating the temperature eventually.
Equilibrium, with a constant temperature everywhere will be achieved. |
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An equally valid alternative interpretation just
looks at the concentration gradient of the electrons in energy space, which
would automatically drive a kind diffusion current until the concentration (and
thus the temperature) is equalized. |
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Yet another way of looking at it is to consider
that the average momentum of the electrons at the hot end is larger than that
of the electrons at the cold end. They would therefore "run away"
faster (taking energy with them) than the electrons from the other end would
"run in". |
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However, since we keep the temperature difference constant, all this cannot happen. We will have to maintain constant but
different temperatures and therefore different energy distributions at both
ends of the metal bar. |
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If nothing happens, we will loose the electrons
with large momentum faster than we gain electrons with smaller momentum; and a
temperature gradient cannot be maintained. The only way to change that, is to
lower the potential at the hot end somewhat, i.e. make the ends positively
charged, and to raise it at the cold end. |
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The potential difference must built up until it
is large enough to exactly counteract the net loss of "hot" electrons
due to momentum imbalance. |
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This is essentially the reason why we
find a thermovoltage. |
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Note that the junction is not directly essential.
However, if you just plug a wire from one material into your Voltmeter and heat
up the middle part, leaving the two ends cold (and at the same temperature),
your potential along the wire may change, but at the two ends you have the same
potential, and it is the potential difference between the two ends you measure
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Plucking the hot end into your Voltmeter is a bit
unpractical, so you necessarily end up with a junction to some other material.
The other material now will also have a hot end and a cold end, and thus
develop a potential difference. |
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Since the potential at the other end can only
have one value, you will now get a
potential difference between the two cold
ends which depends, of course, somehow on the choice of materials. |
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Still, there is a potential
difference between the hot and cold end of one piece of material, and even so
it cannot be measured directly, we can measure it indirectly somehow and
tabulate the values. |
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We can do this, with somewhat more involved but
similar reasoning not only for metals, but also for semiconductors. The table
below gives some absolute values and shows that semiconductors are good
candidates for actual thermocouples, because their Seebeck voltage is fairly large. The values
are for about room temperature, or about 700 oC for the last
three materials |
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| Material |
Al |
Cu |
Ag |
W |
(Bi,Sb)2Te3 |
Bi2(Te,Se)3 |
ZnSb |
InSb |
Ge |
TiO2 |
Seebeck voltage [µV/K]
(Vhot - Vcold) |
-0,20 |
+3,98 |
+3,68 |
+5,0 |
+195 |
-210 |
+220 |
-130 |
-210 |
-200 |
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Surprise! There are positive (as expected) and negative valued of the voltage. What does it
mean? |
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Simply that you are looking at positively charged carriers being reponsible for the
Seebeck effect - holes, in other
words. |
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Not so surprising for semiconductors, perhaps,
but somewhat unexpected for Al. But, as we should know, conduction in
Al relies heavily on holes, as evidenced, e.g., in its positive Hall
coefficient while most other metals have a negative one |
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Understanding qualitatively the
Seebeck effect does not help much to understand the Peltier effect. |
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Again, lets look at some simple junction, this
time an ohmic conduct of a metal to a semiconductor. |
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Shown is an equilibrium situation, where the
Fermi energy is constant throughout, and the flow of electrons across the
junction must be equal in both directions. Note that only the high-energy end
electrons oft the Fermi distribution in the metal makes it across the junction,
whereas all electrons of the semiconductor can flow into the metal. |
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The electrons of the metal thus also transport
some thermal energy out of the metal, but in equilibrium exactly the same
amount is gained by the semiconductor electrons, which are high-energy
electrons in the metal. |
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Now consider some external voltage
driving some net current through the junction in either direction. |
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If this current is an electron current flowing
from the metal into the semiconductor, it still transports some thermal energy out of the
metal, but since it is now much larger than the electron current flowing back,
we have a net transport of thermal energy
out of the metal, which therefore must cool
down. The semiconductor part heats up, of course. |
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If the current is reversed, the flow of thermal
energy reverses, too, and now the semiconductor part cools down. |
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It is conceivable then (also far from clear) that
the total effect in terms of temperature differences is proportional to the
current I flowing. |
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Note, however, that as a completely
independent process, you always have ohmic heating (or Joule heating) which is
simply given by the total power Pdumped into the system via |
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with U = voltage applied,
R = total serial resistance of the system. |
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Since this general heating of the whole device is
proportional to I2, it can easily overwhelm any
cooling effect that might be there. |
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If you want to use the Peltier effect as an
elegant way of cooling something, you must not only choose your materials very
carefully, but also optimize your system design and working points. |
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That this is possible is evidenced by the
successful marketing of Peltier cooling
elements, mostly for scientific applications. Here is a table with
technical data from a major supplier (EURECA Messtechnik GmbH, Am Feldgarten 3
D-50769 Köln, GERMANY ) |
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Micro Peltier Elements
| Module |
Imax
[A] |
Qmax
[W] |
Umax
[V] |
dTmax
[K] |
Dimensions |
Unit Price
[Euro] |
| A [mm] |
B [mm] |
C [mm] |
D [mm] |
H [mm] |
| TECM-4-4-1b/69 |
1,4 |
0,7 |
0,9 |
69 |
4,3 |
4,3 |
4,3 |
4,3 |
2,95 |
28,75 |
| TECM-4-5-1/67 |
0,7 |
0,4 |
1,0 |
67 |
3,4 |
3,4 |
3,4 |
5,0 |
2,30 |
29,50 |
| TECM-5-7-1/67 |
0,7 |
0,9 |
2,2 |
67 |
5,0 |
5,0 |
5,0 |
6,6 |
2,30 |
38,50 |
| TECM-7-8-2/67 |
0,7 |
1,7 |
3,9 |
67 |
6,6 |
6,6 |
6,6 |
8,3 |
2,30 |
52,50 |
| TECM-9-12-4/67 |
0,7 |
3,5 |
8,0 |
67 |
9,1 |
9,9 |
9,1 |
11,5 |
2,30 |
66,25 |
| TECM-12-6-4/69 |
1,7 |
4,4 |
4,3 |
69 |
6,0 |
12,0 |
6,0 |
12,0 |
2,75 |
57,50 |
In this class you will find elements with various geometries and electrical
parameters. For this reason, these elements are suited for very different and
partly exotic requests as you have in the research. Our support will help you
with the selection and the startup of the elements in consideration of your
particular requests.
Generally you will receive these elements in small quantities from stock.
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© H. Föll (Electronic Materials - Script)
To index
2.3.3 Thermoelectric Effects