Thermoelectric Cooling in Bulk and Quantum Well Semiconductors
December 18, 1996
a) Current address: Jack Baskin School of Engineering, University of California, Santa Cruz, CA 95064
(1) Introduction
(2) Electron transport in semiconductors
(3) Modeling of thermoelectric coolers: capacity, figure-of-merit, and COP
(4) Effect of various semiconductor parameters on TE figure-of-merit
(5) Current trends in quantum well and superlattice thermoelectrics
(6) References
(7) Useful links
In the last 25 years, there has been an extensive research in the area of artificial semiconductor structures. Various means of producing ultrathin and high quality crystalline layers (such as molecular beam epitaxy, or metalorganic chemical vapor deposition), have been used to alter the "bulk" characteristics of the materials. Drastic changes are produced by changing the crystal periodicity (by e.g., depositing alternating layers of different crystals), or by changing the electrons dimensionality (by confining the carriers in a plane (quantum well) or in a line (quantum wire),...).
Even though electrical and optical properties of these artificial crystalline structures have been extensively studied, there hasn't been much work for their thermal and thermoelectric (TE) properties. Recently, Hicks et al. (93), Mahan et al. (94) and Broido et al. (95) have look at the problem of thermoelectric cooling for lower dimensionality structures (multiquantum wells and wires).
In this report we consider the problem of electron transport in artificial semiconductor structures from a more general point of view: how we can tailor the band structure and the electron transport to maximize thermoelectric properties for specific applications. First we consider the case of band conduction by Boltzmann transport equation. In this transport regime, TE cooling can be characterized by the figure-of-merit (Z= S2.sig /ß ), where S is the thermopower, sig is electrical conductivity, and ß is the thermal conductivity which include both contributions of lattice and electrons. The crystal vibrations (phonons) are also very much affected by these artificial structures, one can also tailor the phonon dispersion and electron-phonon interaction to modify the lattice thermal conductivity, but we will not consider lattice contributions in our analysis.
Next, we will look at the electron transport beyond Boltzmann regime, and in particular describe advantages of using thermionic emission in band engineered heterostructures for refrigeration and power conversion applications.
(2) Electron transport in semiconductors
The transport of electrons in crystalline solids is described using the band theory, which is a consequence of the periodicity of the structure. This periodicity allows description of electron movement in the complicated potential of many atoms, using some effective parameters (bandgap, effective mass, etc.). In a point-particle picture, assuming localized scattering events (with acoustic and optical phonons, and with various impurities), and neglecting coherent scattering and many-body effects, the electron motion in the crystal can be modeled using Boltzmann transport equation (BTE).
BTE is a Master equation for the probability distribution function, f(r,k,t). fdr is the probability of finding an electron with crystal momentum p= hk/2¼ in a volume dr centered at r at time t. At steady state the total rate of change of the distribution function due to the acceleration by the electric field, should be equal to the change due to the collisions. i.e.
where e is the electron charge, v the electron velocity and F the electric field. The effect of the collisions can be taken into account using relaxation time approximation:
where T is the absolute temperature, Ef the Fermi energy, feq the equilibrium distribution function, and tau(k) the momentum-dependent relaxation time. From the solution to the linearized Boltzmann equation, one can calculate the "perturbed" electronic distribution function due to the external electric field and the temperature non-uniformity:
The electrical and thermal currents are then easily evaluated:
where the transport coefficients Ln are defined by the following integral:
.
From the expressions for J and JQ, various material parameters such as electrical conductivity (sigma), thermal conductivity (ße due to electrons), and the thermopower (or Seebeck coefficient, S) can be calculated. For simplicity we assume current flow and the temperature gradient to be both in x-direction:
(3) Modeling of thermoelectric coolers
In the previous section, we derived some of the macroscopic properties of solids from their microscopic electronic structure (scattering times, band structure, ...). To discuss the theory of thermoelectric refrigeration, we will consider a junction between two thermo-elements as shown in Figure 1. Even though thermoelectric properties are "bulk" effects, their manifestation to create temperature differences and to pump heat from one place to another require a junction of two dissimilar materials. If one considers a closed-loop of a homogeneous material, because of the symmetry, the current flow can not create a temperature difference.
Figure 1. Thermocouple used as refrigerator
A single element thermoelectric cooler of Figure 1 is composed of two branches n and p with respectively negative and positive Seebeck coefficients (e.g. n-type and p-type semiconductors). The two thermoelement are electrically in series and thermally in parallel. When the current is flowing from n to p (i.e. electrons moving from the p-branch to the metallic contact and from that to the n-branch), the heat is absorbed at the junctions p-metal and metal-n. Electrons in the branch with positive Seebeck coefficient, have an "average transport energy" smaller than the Fermi energy (this is due to a curious sign convention in the expression for J, i.e. using (-gradT ) instead of (+gradT )!), and the ones in n-branch have their average energy bigger than the Fermi one. Metals, with relatively very small Seebeck coefficients, can be considered as having their average transport energy of electrons equal to Ef. In a perfect "ohmic" conduction from p-branch to metal to n-branch, electrons should absorb energy (heat) to increase their average energy. The same argument can be applied to the bottom contacts where the heat is generated. This heat absorption (or generation) occurs at distances very close to the contacts, on the order of electron average velocity times its thermalization time constant in a band. One should also notice that this heat absorption or generation, so called Peltier effect, at the junction of two dissimilar materials, is a reversible thermodynamical phenomena depending to the direction of current flow.
Cooling capacity and COP
The cooling capacity, Q, at the cold junction of the thermocouple shown in Figure 1, is given by the Peltier cooling at the junctions minus the Joule heating generated in the branches n and p, and minus the amount of heat conduction from hot to cold junctions. From a more elaborate analysis of the temperature distribution within thermoelements and neglecting the Tompson effect, it can be shown that half of the Joule heating goes to the cold junction and the rest to the hot one, thus:
where S is the Seebeck coefficient, sigma and ß are the electrical and thermal conductivities, and l and s are each branch's length and cross section,
The coefficient of performance, COP, of a TE cooler is the ratio of cooling capacity to the amount of input power. This is given by,
For given material parameters (Sp,n,sigp,n,ßp,n), given the geometrical factor (l/s), and given temperature of the hot and cold junctions (Th,Tc), Q and COP have their optimum values at different currents. The current, Imax cool, gives the maximum cooling, and Imax COP the maximum coefficient-of-performance:
where:
In practical TE cooling applications, the current is chosen in the range [Imax COP, Imax cool], depending on the requirements for maximum efficiency or maximum cooling power.
From the expression Qmax cool it can be seen that a positive cooling effect can not be achieved if the temperature difference between the junctions is too great. The maximum temperature difference is found by setting Qmax cool=0, i.e.
It is seen that the parameter Z determines both the maximum temperature difference and the COPmax cop. Expressing Z of a single branch as S2.sig /ß, it describes how "good" the material is for thermoelectric cooling applications and it is called the thermoelectric figure-of-merit.
(4) Effect of various semiconductor parameters on TE figure-of-merit
Starting from the microscopic expressions of electrical and thermal conductivities and the thermopower, which was derived in section (2), we will try to develop an intuitive picture for the influence of various semiconductor parameters on TE figure-of-merit (Z =S2.sig /ß). This picture is used in subsequent sections to describe recently proposed quantum well devices, and to introduce a new class of devices with a potential improved TE properties.
Rewriting the expressions for electrical conductivity and the thermopower in form of integrals over electron energy we get:
where we introduced the "differential" conductivity:
.
sig(E) is a measure of the contribution of electrons with energy E to the total conductivity. The Fermi "window" factor (-¶feq/¶E) is a bell-shape function centered at E=Ef, having a width of ~kBT. The Fermi window can be interpreted as a direct consequence of Pauli exclusion principle; at a finite temperature only electrons near the Fermi surface contribute to the conduction process. In this picture, the thermopower is the "average" energy transported by the charge carriers. This so called diffusion thermopower may be enhanced by the coupling of electronic motion to other means of transport of energy (e.g. by phonons).
We are mainly concerned with various means of increasing thermoelectric cooling efficiency and capacity, using bandstructure engineering and its effects on electronic motion. As in most semiconductors the thermal conductivity at room temperature is dominated by the lattice contribution, in the expression for Z we only consider the electrical "power" product S2.sig ~ sig.|<E-Ef>|2. This means that sig(E), within the Fermi window, should be as big as possible, and at the same time, as asymmetric as possible with respect to the Fermi energy.
(5) Current trends in quantum well and superlattice thermoelectrics
Using the interpretation introduced in the last section, the advantage of going to lower dimensions becomes more transparent. The number of electronic states in each energy interval is increased when the dimensionality is reduced, and at the same time the DOS "accumulates" near the subband edges, which increases the asymmetry in sig(E) if a proper doping is chosen. Recent literature on quantum well and wire thermoelectrics (Hicks et al., Mahan et al., and Broido et al.) emphasis the increased density-of-states, but the symmetry is not mentioned and its consequences are buried in the calculations of optimum doping in these structures. The symmetry of sig(E)is the main cause of low thermopower in metals, even though they have a very large density-of-states.
Figure 2 Density-of-states as a function of electron energy for various dimensionalities (3d-circle, 2d-dotted, SL-plain, 1d-cross); only the contribution of the first quantized state is plotted.
Based on the picture given in the last section, we see that we have extra degrees of freedom in optimizing thermoelectric figure-of-merit by bandstructure engineering. Using heterostructures we can modify not only the DOS, but also the electron velocity and the relaxation times. Based on these ideas, electron transport perpendicular to the quantum well layers was proposed to reduce the mobility of "cold" electrons and to increase the thermopower (Moizhes et al., Rowe et al., and Whithlow et al.).
All of these concepts and primary calculations are based on linearized Boltzmann transport equation which is valid in band conduction regime and when the electronic distribution function is not changed considerably with respect to the Fermi distribution.
Hicks, L.D.; Dresselhaus, M.S. Effect of quantum-well structures on the thermoelectric figure of merit. Physical Review B (Condensed Matter), 15 May 1993, vol.47, (no.19):12727-31.
Hicks, L.D.; Dresselhaus, M.S. Thermoelectric figure of merit of a one-dimensional conductor. Physical Review B (Condensed Matter), 15 June 1993, vol.47, (no.24):16631-4.
Hicks, L.D.; Harman, T.C.; Dresselhaus, M.S. Use of quantum-well superlattices to obtain a high figure of merit from nonconventional thermoelectric materials. Applied Physics Letters, 6 Dec. 1993, vol.63, (no.23):3230-2.
Farmer, J.C.; Barbee, T.W., Jr.; Chapline, G.C., Jr.; Foreman, R.J.; and others. Sputter deposition of multilayer thermoelectric films: an approach to the fabrication of two-dimensional quantum wells. (Thirteenth International Conference on Thermoelectrics, Kansas City, MO, USA, 30 Aug.-1 Sept. 1994). AIP Conference Proceedings, 1995 (no.316):217-25.
Hicks, L.D.; Harman, T.C.; Sun, X.; Dresselhaus, M.S. Experimental study of the effect of quantum-well structures on the thermoelectric figure of merit. Physical Review B (Condensed Matter), 15 April 1996, vol.53, (no.16):R10493-6.
Harman, T.C.; Spears, D.L.; Manfra, M.J. High thermoelectric figures of merit in PbTe quantum wells. Journal of Electronic Materials, July 1996, vol.25, (no.7):1121-7.
Mahan, G.D.; Lyon, H.B., Jr. Thermoelectric devices using semiconductor quantum wells. Journal of Applied Physics, 1 Aug. 1994, vol.76, (no.3):1899-901.
Sofo, J.O.; Mahan, G.D. Thermoelectric figure of merit of superlattices. Applied Physics Letters, 21 Nov. 1994, vol.65, (no.21):2690-2.
Lin-Chung, P.J.; Reinecke, T.L. Thermoelectric figure of merit of composite superlattice systems. Physical Review B (Condensed Matter), 15 May 1995, vol.51, (no.19):13244-8.
Broido, D.A.; Reinecke, T.L. Effect of superlattice structure on the thermoelectric figure of merit. Physical Review B (Condensed Matter), 15 May 1995, vol.51, (no.19):13797-800.
Broido, D.A.; Reinecke, T.L. Thermoelectric figure of merit of quantum wire superlattices. Applied Physics Letters, 3 July 1995, vol.67, (no.1):100-2.
Broido, D.A.; Reinecke, T.L. Comment on "Use of quantum well superlattices to obtain high figure of merit from nonconventional thermoelectric materials" (Appl. Phys. Lett. 63, 3230 (1993)). Applied Physics Letters, 21 Aug. 1995, vol.67, (no.8):1170-1.
Whitlow, L.W.; Hirano, T. Superlattice applications to thermoelectricity. Journal of Applied Physics, 1 Nov. 1995, vol.78, (no.9):5460-6.
Elsner, N.B.; Ghamaty, S.; Norman, J.H.; Farmer, J.C.; and others. Thermoelectric performance of Si/sub 0.8/Ge/sub 0.2//Si heterostructures synthesized by MBE and sputtering. (Thirteenth International Conference on Thermoelectrics, Kansas City, MO, USA, 30 Aug.-1 Sept. 1994). AIP Conference Proceedings, 1995 (no.316):328-33.
Rowe, D.M.; Gao Min. Multiple potential barriers as a possible mechanism to increase the Seebeck coefficient and electrical power factor. (Thirteenth International Conference on Thermoelectrics, Kansas City, MO, USA, 30 Aug.-1 Sept. 1994). AIP Conference Proceedings, 1995 (no.316):339-42.
Wagner, A.V.; Foreman, R.J.; Summers, L.J.; Barbee, T.W., Jr.; and others. Multilayer thermoelectric films: a strategy for the enhancement of ZT. IN: Proceedings of the 30th Intersociety Energy Conversion Engineering Conference (IEEE Cat. No.95CH35829).Proceedings of 30th Intersociety Energy Conversion Engineering Conference, Orlando, FL, USA, 30 July-4 Aug. 1995). Edited by: Yogi Goswami, D.; Kannberg, L.D.; Mancini, T.R.; Somasundaram, S. New York, NY, USA: ASME, 1995. p. 87-92 vol.3.