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, 1999, 41, . 5 Low Dimensional Thermoelectric Materials M.S. Dresselhaus, G. Dresselhaus, X. Sun, Z. Zhang, S.B. Cronin, T. Koga Department of Electrical Engineering and Computer Science and Department of Physics, Massachusetts Institute of Technology, Cambridge MA 02139-4307 Francis Bitter Magnet Laboratory, Massachusetts Institute of Technology, Cambridge, MA 02139-4307 Department of Physics, Massachusetts Institute of Technology, Cambridge MA 02139-4307 Division of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138 E-mail: millie@mgm.mit.edu The promise of low dimensional thermoelectric materials for enhanced performance is reviewed, with particular attention given to quantum wells and quantum wires. The high potential of bismuth as a low dimensional thermoelectric material is discussed.

Professor Abram Ioffe and his collaborators at the Ioffe It is customary to express the usefulness of a thermoInstitute in St. Petersburg, Russia did pioneering work [1] electric material for use in refrigeration or power generation applications in terms of the dimensionless quantity ZT where in introducing semiconductors as promising thermoelectric T is the temperature (in degrees Kelvin) and Z is the materials and in showing how semiconductors could be used thermoelectric figure of merit in practical devices for cooling and for electrical power generation [1]. This early work led to a very active period S2 in thermoelectrics research in the 1950s and early 1960s, Z =. (1) when many new thermoelectric materials were discovered and investigated. At this early time, the high potential Here S is the thermoelectric power or Seebeck coefficient, of Bi2Te3 as a thermoelectric material was discovered by is the electrical conductivity and is the thermal conductivity. Large values of ZT require high S, high, H. J. Goldsmid and coworkers in the U.K. [2], and this and low. Since an increase in S normally implies a material system remains the basis for the thermoelectric decrease in because of carrier density considerations, and industry up to the present time [3]. Ioffes proposal to since an increase in implies an increase in the electronic employ semiconductor alloys rather than simple binary contribution to as given by the WiedemannFranz law, it compounds in order to lower the thermal conductivity of is very difficult to increase Z in typical thermoelectric matesemiconductors [4] proved very useful for thermoelectric rials. The best commercial 3D thermoelectric material is in applications, thereby leading to the industrial use of alloys the Bi2(1-x)Sb2xTe3(1-y)Se3y family with room temperature in the Bi2Te3, Bi2Se3, and Sb2Te3 family.

ZT 1 for Bi0.5Sb1.5Te3 [3].

For a 30 year period since the early 1960s, research Reduced dimensionality offers one strategy for increasing activity in the field of thermoelectricity has been greatly ZT relative to bulk values [6,7]. The use of low dimensional reduced, and only modest progress was made in improving systems for thermoelectric applications is of interest because the performance of thermoelectric materials [3]. Recently, low dimensionality provides: (1) a method for enhancing the study of thermoelectric materials has once again bethe density of states near EF, leading to an enhancement of come an active research field, in part due to the recent the Seebeck coefficient, (2) opportunities to take advantage demonstration of enhancement in the thermoelectric figure of the anisotropic Fermi surfaces in multi-valley cubic of merit of a two-dimensional PbTe quantum well system, semiconductors, (3) opportunities to increase the boundary relative to its three-dimensional (3D) bulk counterpart [5].

scattering of phonons at the barrier-well interfaces, without Calculations suggest that the thermoelectric performance of as large an increase in electron scattering at the interface, any 3D material should show an enhanced thermoelectric (4) opportunities for increased carrier mobilities at a given carrier concentration when quantum confinement conditions figure of merit, when prepared as a 2D multi-quantum are satisfied, so that modulation doping and -doping can be well superlattice, utilizing the enhanced density of states utilized.

at the onset of each electronic subband, and the increased scattering of vibrational waves at the boundary between the quantum well and the adjacent barrier of the superlattice. In 1. Theoretical Modeling addition, low dimensionality allows certain materials such as bismuth, which are poor thermoelectrics in 3D, to become In early models for thermoelectricity in 2D quantum well good thermoelectrics, in principle, in 2D quantum well or structures [69] it was assumed that the electrons in the 1D quantum wire structures. valence and conduction bands are in simple parabolic energy 1 756 M.S. Dresselhaus, G. Dresselhaus, X. Sun, Z. Zhang, S.B. Cronin, T. Koga bands and that the electrons occupy only the lowest subband to the power factor S2. Some of the systems that were of the quantum well. The electronic dispersion relations for studied include p-type PbTe, where power factors higher a 2D system are then given by than for n-type PbTe were reported, while at the same time using db/dW ratios reduced by a factor of more than two, 2 2 kx 2ky relative to n-type PbTe [14]. Another superlattice system that E2D(kx, ky) = + +, (2) 2mx 2my 2mzdW was studied was the Si/SiGe system, where an increase in the room temperature power factor of the superlattice relative to where dW is the width of the quantum well, and mx, my, and the bulk silicon value was predicted, and where even better mz are the effective mass tensor components of the constant performance could be expected at elevated ( 300 K) energy surfaces. It is further assumed that the current flows temperatures [15].

in the x direction and that quantum confinement is in the z As a second approach, emphasis was given to achieving direction. The corresponding relation used for a square 1D higher Z3DT for the whole superlattice sample (both for quantum wire is the quantum well and barrier regions) and to studying 2 2 2 kx 2 superlattices which do not show quantum confinement E1D(kx) = + +, (3) 2 effects [16]. By using the same basic model, as was 2mx 2mydW 2mzdW developed for the PbTe/Pb1-xEuxTe superlattice, to study where the current flow is also along the x direction, and the GaAs/AlAs superlattice, it was shown that when the dW quantum confinement occurs in the y and z directions.

and db values are sufficiently small, the electronic density Solutions of Boltzmanns equation were then obtained for of states is basically two-dimensional in both the quantum S,, and e (the electronic contribution to the thermal well and barrier regions [17], and an increase in power conductivity) for both the 2D and 1D systems [6,911].

factor relative to bulk values could be achieved. One great advantage of allowing conduction in both the barrier and 2. Experimental Proof-of-Prinsiple the quantum well region is that much shorter barrier widths can be used and a significant contribution to S2 from An early phase of the experimental work was devoted the barrier region can be obtained. The gains in Sto showing proof-of-principle [5], in order to confirm the from the quantum well region relative to bulk values more validity of the basic theoretical model for low dimensional than compensate for the somewhat lower contribution to thermoelectric materials [6,9]. PbTe was chosen as the S2 from the barrier region, so that S2 for the whole quantum well material for demonstrating proof-of-principle superlattice (denoted by Z3DT ) exceeds that of the bulk.

of an enhanced ZT in a 2D system because of its desirable In addition, a large reduction in thermal conductivity is thermoelectric and materials science properties [12,13]. Reexpected, due to the strong interface phonon scattering, so garding its thermoelectric properties, PbTe has a reasonably that significant increases in Z3DT are expected from this high ZT at 300 K in bulk form (ZT 0.4), reflecting its approach.

high carrier mobility, multiple anisotropic carrier pockets, Calculations suggest that another effect that may come and low thermal conductivity that can be achieved under into play relates to carrier pocket engineering, whereby by isoelectronic alloying. Measurements of S2 by Hicks et proper selection of dW, db and their ratio (db/dW ), it is al. [5] corroborated that quantum confinement could be possible to raise the energy of the lowest point subband so achieved in a PbTe/Pb1-xEuxTe superlattice, for x = 0.073, that it lies higher than the L and X point subbands, thereby by using large barrier widths db dW. Good agreement leading to an enhancement of Z3DT because of the high between experiment and theory was demonstrated in plots density of states for the L and X point subbands [17]. Simple of S2 as a function of carrier concentration and quantum calculations suggest that such carrier pocket engineering can well width, thereby corroborating the model, using no be carried out in the case of GaAs/AlAs superlattices with adjustable parameters (only literature values measured by dW and db in the 2030 range [17].

other techniques) [5]. High carrier mobilities were found for these superlattice samples [5].

Experimentally, enhancement in the power factor and Z3DT have been reported for several superlattice systems not exhibiting quantum carrier confinement effects, including the 3. Recent 2D Superlattice Studies Bi2Te3/Bi2Se3 [16], PbTe/PbSe1-xTex/Te (x = 0.02) [18], and Si/Ge systems [19]. In fact, the highest ZT value ever Having demonstrated experimental evidence in support reported for any thermoelectric material is Z3DT = 1.9 for of the basic theoretical model, the focus of research on the PbTe/PbSe1-xTex/Te system (x = 0.02) at 570 K [18].

low dimensional thermoelectricity has shifted to two new It is expected that we will see considerably more activity directions. One research direction focused on observations in the near future, both experimentally and theoretically, on of the enhancement of ZT in other 2D superlattice systems, the study of such composite superlattice systems, which may so that the basic phenomenon could then be studied in more detail, and the barrier widths could be reduced, noting that exhibit some type of lower dimensional electronic properties, the barrier regions contribute only to in Eq. (1) and not but do not exhibit significant carrier quantum confinement.

, 1999, 41, . Low Dimensional Thermoelectric Materials 4. Bismuth as a Low Dimentional Thermoelectric Bismuth is a very attractive material for low dimensional thermoelectricity because of the large anisotropy of the three ellipsoidal constant energy surfaces for electrons at the L point in the rhombohedral Brillouin zone (m = 0.00651m0, x m = 1.362m0, m = 0.00993m0), the very long mean free y z path of the L-point electrons, and the high mobility of the carriers ( = 3.5 104 cm2/Vs in the binary direction) [20].

Furthermore, the heavy mass of the Bi ions results in efficient phonon scattering and low phonon mean free paths. Since bulk bismuth is a semimetal, the contribution from the holes to the Seebeck coefficient approximately cancels that for the electrons, so that S is quite small, and high magnetic fields are needed to make bulk Bi interesting for thermoelectric applications [21].

Figure 2. Schematic energy band diagram showing the energies The situation for 2D Bi in a quantum well is, however, of the subband edges for the heavy and light electrons and for the quite different. As the quantum well width decreases, the holes for: (a) bulk Bi, (b) 100 nm diameter Bi nanowires along lowest bound state in the conduction band rises above the the bisectrix direction, and (c) 50 nm diameter Bi nanowires along highest bound state in the valence band, thereby leading to the bisectrix direction.

a semimetal-semiconductor transition at some critical value of dW [11]. If the 2D bismuth system is then doped to the optimum doping level, a large enhancement in Z2DT been made toward making proof-of-principle studies of this within the quantum well is predicted with decreasing dW, system. Small diameter Bi wires have been successfully as shown in the plot of Z2DT vs dW in Fig. 1 for a Bi fabricated by pressure-injecting liquid Bi into the cylindrical superlattice normal to the trigonal direction, for which all nano-channels of a porous anodic alumina template [23].

the electron carrier pockets are equivalent. Considerable Such anodic alumina templates with pore diameters ranging progress has been made recently by use of CdTe as a barrier from 13110 nm have been filled with liquid Bi to reach a material for synthesizing Bi quantum wells [22]. While density as high as 7 1010 parallel wires/cm2, with template study of 2D Bi superlattices continues, recent effort has thicknesses and wire lengths up to 100 m. As shown also been expended to study 1D bismuth nanowires [23,24].

by high resolution electron microscopy and selected area Calculations for transport in the trigonal direction indicate electron diffraction experiments, the individual Bi nanowires that 1D bismuth nanowires with very small wire diameter are essentially single crystals, with nearly the same crystal could be even better than 2D bismuth for thermoelectric structure and lattice parameters as bulk Bi. The individual applications (see Fig. 1).

wires are of uniform diameter along their entire length, and Although no enhancement in ZT has yet been demonthe nanowires in a given template have a similar crystalline strated experimentally with Bi nanowires, progress has orientation along their common wire axes [23]. The large bandgap of the anodic alumina host material ensures good carrier quantum confinement. Optical transmission [23] and magnetoresistance studies [2426] provide evidence for a semimetal to semiconducting transition as the wire diameter decreases and reaches the critical diameter dc, belowwhich a semiconducting gap is found. Our theoretical calculations, based on values of the effective masses for bulk bismuth, indicate on the basis of a simple parabolic band model for the electron states, that for the binary, trigonal and bisectrix directions, dc 30 nm, 45 nm and 81 nm, respectively [24].

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