Patent Publication Number: US-2013247951-A1

Title: Thermoelectric material with high cross-plane electrical conductivity in the presence of a potential barrier

Description:
RELATED APPLICATIONS 
     This application claims the benefit of provisional patent application Ser. No. 61/613,015, filed Mar. 20, 2012, the disclosure of which is hereby incorporated herein by reference in its entirety. 
    
    
     FIELD OF THE DISCLOSURE 
     The present disclosure relates to a thermoelectric material and more specifically relates to a thermoelectric material having high cross-plane electrical conductivity. 
     BACKGROUND 
     The figure-of-merit (ZT) of a thermoelectric material is a dimensionless unit that is used to compare the efficiencies of various thermoelectric materials. The figure-of-merit (ZT) is determined by three physical parameters: thermopower α (also known as a Seebeck coefficient); electrical conductivity a; and thermal conductivity k=k e +k ph , where k e  and k ph  are thermal conductivities due to transport of electrons and phonons, respectively; and absolute temperature T: 
     
       
         
           
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     Significant research has been conducted to develop thermoelectric materials having a high figure-of-merit (ZT) value. Increasing this value to 2.0 or higher will disrupt existing technologies and will ultimately enable more widespread use of thermoelectric systems. 
     U.S. Patent Application Publication No. 2012/0055528, entitled THERMOELECTRIC MATERIALS, which was filed on Mar. 29, 2010 and is hereby incorporated herein by reference in its entirety, discloses a thermoelectric material that utilizes one or more potential barriers to provide an enhanced, or improved, Seebeck coefficient. From the equation above, it can be seen that enhancing the Seebeck coefficient provides an improved figure-of-merit (ZT) value for the thermoelectric material. More specifically, the Seebeck coefficient is defined as an electrical potential of a charge carrier over a temperature differential across which the charge carrier travels. As disclosed in U.S. Patent Application Publication No. 2012/0055528, a potential barrier provides a hot carrier skimming effect by which hot carriers (i.e., hot electrons or hot holes depending on the conductivity type of the thermoelectric material) are skimmed from one side of the potential barrier to the other side of the potential barrier. Thus, the hot carriers that pass across the potential barrier are at a high energy level and thus have a high electrical potential. Because these hot carriers have high electrical potential, the Seebeck coefficient of the thermoelectric material is enhanced. More specifically, by letting a thickness of a barrier material layer be approximately equal to a mean free path distance for charge carriers between scattering events at a desired temperature of the barrier material layer during operation of a corresponding thermoelectric device, ballistic transport of charge carriers through the barrier material layer is enabled, thereby increasing the Seebeck coefficient of the thermoelectric material and thus the figure-of-merit (ZT) value of the thermoelectric material. 
     What is desired is a thermoelectric material that has enhanced cross-plane electrical conductivity in the presence of one or more Seebeck coefficient enhancing potential barriers and a method of fabrication thereof. The enhanced cross-plane electrical conductivity would further improve the figure-of-merit (ZT) value of the thermoelectric material. 
     SUMMARY 
     Embodiments of a thermoelectric material having high cross-plane electrical conductivity in the presence of one or more Seebeck coefficient enhancing potential barriers and methods of fabrication thereof are disclosed. In one embodiment, a thermoelectric material includes a first matrix material layer, a barrier layer, and a second matrix material layer. The barrier layer is a short-period superlattice structure that includes multiple superlattice layers. Each superlattice layer has two different sub-bands, namely, a high energy sub-band and a low energy sub-band. In one preferred embodiment, the thermoelectric material is a Group IV-VI thermoelectric material grown in the [111] direction, and the high energy sub-band and the low energy sub-band are an oblique valley sub-band and a normal valley sub-band, respectively. For each superlattice layer, the energy level of the high energy sub-band of the superlattice layer is resonant with the energy level of the low energy sub-band of an adjacent superlattice layer and/or the energy level of the low energy sub-band of the superlattice layer is resonant with the energy level of the high energy sub-band of an adjacent superlattice layer. As a result, a resonant path, or resonant tunnel, for hot carriers is created through the barrier layer. The resonant path for hot carriers increases an electrical conductivity of the thermoelectric material and, as a result, improves a figure-of-merit (ZT) of the thermoelectric material. 
     Those skilled in the art will appreciate the scope of the present disclosure and realize additional aspects thereof after reading the following detailed description of the preferred embodiments in association with the accompanying drawing figures. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWING FIGURES 
       The accompanying drawing figures incorporated in and forming a part of this specification illustrate several aspects of the disclosure, and together with the description serve to explain the principles of the disclosure. 
         FIG. 1  illustrates a thermoelectric material including barrier material layers each having a short-period superlattice structure that provides high cross-plane electrical conductivity in the presence of a potential barrier according to one embodiment of the present disclosure; 
         FIG. 2  is a more detailed illustration of one of the barrier material layers of  FIG. 1  according to one embodiment of the present disclosure; 
         FIG. 3  graphically illustrates high energy level sub-bands and low energy level sub-bands of each superlattice layer in the short-period superlattice structure of the barrier material layer of  FIG. 2  where, for each superlattice layer, the high energy level sub-band is resonant with the low energy level sub-band of an adjacent superlattice layer such that cross-plane electrical conductivity through the barrier material layer is substantially improved according to one embodiment of the present disclosure; 
         FIG. 4  graphically illustrates the operation of the barrier material layer of  FIG. 2  according to one embodiment of the present disclosure; 
         FIG. 5  illustrates experimental data and theoretical fits to the data for the energy levels of a normal valley sub-band and an oblique valley sub-band of a Lead Strontium Selenide (PbSrSe)/Lead Selenide (PbSe)/PbSrSe superlattice versus quantum well width; 
         FIG. 6  graphically illustrates quantum well widths that can be utilized to create adjacent superlattice layers having resonant normal and oblique valley sub-bands according to one embodiment of the present disclosure; 
         FIG. 7  illustrates one embodiment of the barrier material layer of  FIG. 2  in which the superlattice layers are PbSrSe/PbSe/PbSrSe superlattice layers having quantum well widths selected such that, for each superlattice layer, the oblique valley sub-band is resonant with the normal valley sub-band of an adjacent superlattice layer according to one embodiment of the present disclosure; 
         FIG. 8  illustrates another embodiment of the barrier material layer of  FIG. 2  in which the superlattice layers are PbSrSe/PbSe/PbSrSe superlattice layers having quantum well widths selected such that, for each superlattice layer, the oblique valley sub-band is resonant with the normal valley sub-band of an adjacent superlattice layer according to one embodiment of the present disclosure; 
         FIG. 9  illustrates the thermoelectric material of  FIG. 1  in which one of the barrier layers of the thermoelectric material is the barrier layer of  FIG. 7  and the other barrier layer of the thermoelectric material is the barrier layer of  FIG. 8  according to one embodiment of the present disclosure; 
         FIG. 10  illustrates theoretical curves for the energy levels of a normal valley sub-band and an oblique valley sub-band of a PbSe/Lead Tin Selenide (PbSnSe)/PbSe superlattice versus quantum well width along with quantum well widths that can be utilized to create adjacent superlattice layers having resonant normal and oblique valley sub-bands according to one embodiment of the present disclosure; 
         FIG. 11  illustrates one embodiment of the barrier material layer of  FIG. 2  in which the superlattice layers are PbSe/PbSnSe/PbSe superlattice layers having quantum well widths selected such that, for each superlattice layer, the oblique valley sub-band is resonant with the normal valley sub-band of an adjacent superlattice layer according to one embodiment of the present disclosure; 
         FIG. 12  illustrates another embodiment of the barrier material layer of  FIG. 2  in which the superlattice layers are PbSe/PbSnSe/PbSe superlattice layers having quantum well widths selected such that, for each superlattice layer, the oblique valley sub-band is resonant with the normal valley sub-band of an adjacent superlattice layer according to one embodiment of the present disclosure; 
         FIG. 13  illustrates the thermoelectric material of  FIG. 1  in which one of the barrier layers of the thermoelectric material is the barrier layer of  FIG. 11  and the other barrier layer of the thermoelectric material is the barrier layer of  FIG. 12  according to one embodiment of the present disclosure; and 
         FIG. 14  is a flow chart that illustrates a process for designing and fabricating the thermoelectric material of  FIG. 1  according to one embodiment of the present disclosure. 
     
    
    
     DETAILED DESCRIPTION 
     The embodiments set forth below represent the necessary information to enable those skilled in the art to practice the embodiments and illustrate the best mode of practicing the embodiments. Upon reading the following description in light of the accompanying drawing figures, those skilled in the art will understand the concepts of the disclosure and will recognize applications of these concepts not particularly addressed herein. It should be understood that these concepts and applications fall within the scope of the disclosure and the accompanying claims. 
     Embodiments of a thermoelectric material having high cross-plane electrical conductivity in the presence of one or more Seebeck coefficient enhancing potential barriers and methods of fabrication thereof are disclosed. In this regard,  FIG. 1  illustrates a thermoelectric material  10  having high cross-plane electrical conductivity in the presence of Seebeck coefficient enhancing potential barriers according to one embodiment of the present disclosure. In this embodiment, the thermoelectric material  10  includes matrix material layers  12 - 1  through  12 - 3  (generally referred to herein collectively as matrix material layers  12  and individually as matrix material layer  12 ) and barrier material layers  14 - 1  and  14 - 2  (generally referred to herein collectively as barrier material layers  14  and individually as barrier material layer  14 ) arranged as shown. In general, a bandgap of each of the barrier material layers  14  is greater than a bandgap of the adjacent matrix material layers  12  such that the barrier material layers  14  provide potential barriers. Notably, a height of the potential barrier (i.e., the barrier height) created by the barrier material layer  14 - 1  and the adjacent matrix material layers  12 - 1  and  12 - 2  may be the same as or different than a height of the potential barrier created by the barrier material layer  14 - 2  and the adjacent matrix material layers  12 - 2  and  12 - 3 . While two barrier material layers  14 - 1  and  14 - 2  are illustrated in the embodiment of  FIG. 1 , the thermoelectric material  10  may include any number of one or more barrier material layers  14 . 
     As taught in U.S. Patent Application Publication No. 2012/0055528, which has been incorporated herein by reference in its entirety, the potential barriers created by the barrier material layers  14  enhance a Seebeck coefficient of the thermoelectric material  10 . More specifically, the Seebeck coefficient is defined as an electrical potential of a charge carrier over a temperature differential across which the charge carrier travels. Each of the potential barriers created by the barrier material layers  14  provides a hot carrier skimming effect by which hot carriers (i.e., hot electrons or hot holes depending on the conductivity type of the thermoelectric material  10 ) are skimmed from one side of the potential barrier to the other side of the potential barrier. Thus, the hot carriers that pass across the potential barrier are at a high energy level and, as such, have a high electrical potential. Because these hot carriers have high electrical potential, the Seebeck coefficient of the thermoelectric material  10  is enhanced. 
     Each of the barrier material layers  14  has a short-period superlattice (SPSL) structure (i.e., is a short-period superlattice) that enhances a cross-plane electrical conductivity of the barrier material layer  14  in the presence of the potential barrier created by the barrier material layer  14 . A superlattice is a periodic structure of alternating layers of two (or more) different materials. As used herein, a short-period superlattice is a superlattice in which a thickness of each individual layer of the superlattice is less than or equal to about 20 nanometers (nm). As illustrated in  FIG. 2 , each of the barrier material layers  14  has a short-period superlattice structure that includes multiple superlattice layers  16 - 1  through  16 -N, where N is greater than 1 and is more preferably greater than or equal to 3. Note that the number of superlattice layers  16 - 1  through  16 -N may be the same or different for different barrier material layers  14 . The superlattice layers  16 - 1  through  16 -N are more generally referred to herein collectively as superlattice layers  16  and individually as superlattice layer  16 . 
     As discussed below in detail, rather than having a continuum of allowable states for charge carriers, each of the superlattice layers  16  of the barrier material layer  14  includes two sub-bands at different energy levels, namely, a high energy sub-band and a low energy sub-band. As illustrated in  FIG. 3 , the energy levels of the high energy sub-bands (H) and the low energy sub-bands (L) of the superlattice layers  16  are selected such that the barrier material layer  14  provides a desired potential barrier to enhance the Seebeck coefficient of the thermoelectric material  10  while at the same time the barrier material layer  14  enhances, or improves, transport of charge carriers through the potential barrier. Enhanced transport of charge carriers, which in  FIG. 3  are electrons, through the barrier material layer  14  is achieved by configuring the superlattice layers  16  such that, for each superlattice layer  16 , the high energy sub-band of the superlattice layer  16  is resonant with the low energy sub-band of an adjacent superlattice layer  16  and/or the low energy sub-band of the superlattice layer  16  is resonant with the high energy sub-band of an adjacent superlattice layer  16 . 
     More specifically, as illustrated in  FIG. 3 , the superlattice layer  16 -X (X=(N+1)/2) has a maximum bandgap of the barrier material layer  14 . In other words, the energy level of the high energy sub-band of the superlattice layer  16 -X is the highest energy level among the high energy sub-bands of the superlattice layers  16 . The energy levels of the sub-bands of the superlattice layers  16 - 1  through  16 -(X−1) provide a stair-step increase from the bandgap of the adjacent matrix material layer  12  to the maximum bandgap of the barrier material layer  14  provided by the superlattice layer  16 -X. The high energy sub-band (H) of each of the superlattice layers  16 - 1  through  16 -(X−1) is resonant with the low energy sub-band (L) of an immediately succeeding superlattice layer  16  in the short-period superlattice structure. Specifically, the high energy sub-band (H) of the superlattice layer  16 - 1  is resonant with the low energy sub-band (L) of the superlattice layer  16 - 2 , the high energy sub-band (H) of the superlattice layer  16 - 2  is resonant with the low energy sub-band (L) of the superlattice layer  16 - 3 , and so on. Likewise, the energy levels of the sub-bands of the superlattice layers  16 -(X+1) to  16 -N provide a stair-step decrease from the maximum bandgap provided by the superlattice layer  16 -X to the bandgap of the adjacent matrix material layer  12 . The high energy sub-band (H) of each of the superlattice layers  16 -(X+1) through  16 -N is resonant with the low energy sub-band (L) of an immediately preceding superlattice layer  16  in the short-period superlattice structure. Specifically, the high energy sub-band (H) of the superlattice layer  16 -(X+1) is resonant with the low energy sub-band (L) of the superlattice layer  16 -X, the high energy sub-band (H) of the superlattice layer  16 -(X+2) is resonant with the low energy sub-band (L) of the superlattice layer  16 -(X+1), and so on. As used herein, two sub-bands are resonant when the bottom energy levels of the two sub-bands are equal. 
     The resonant sub-bands in the superlattice layers  16 -(X−1),  16 -X, and  16 -(X+1) provide a resonant path, or resonant tunnel, through the potential barrier created by the barrier material layer  14 . In addition, assuming that electron flow is from left to right in  FIG. 3 , the resonant sub-bands between the superlattice layers  16 - 1  through  16 -(X−1) enable more efficient transport of electrons through the potential barrier created by the barrier material layer  14 . Specifically, the resonant sub-bands of the superlattice layers  16 - 1  and  16 - 2  enable electrons to move from the high energy sub-band (H) of the superlattice layer  16 - 1  to the low energy sub-band (L) of the superlattice layer  16 - 2  without any loss of energy. In this manner, the electrons are efficiently moved from the superlattice layer  16 - 1  to the superlattice layer  16 - 2  and, therefore, are closer to moving through the potential barrier. As a result of the resonant path and the more efficient transport of electrons through the potential barrier, the cross-plane electrical conductivity of the thermoelectric material  10  is enhanced, or increased, which in turn improves the figure-of-merit (ZT) of the thermoelectric material  10 . 
     Before proceeding, it should be noted that the superlattice structure of the barrier material layer  14  of  FIGS. 2 and 3  is symmetrical. More specifically, the superlattice layers  16 - 1  and  16 -N are the same (i.e., have the same superlattice layer structure SL M), the superlattice layers  16 - 2  and  16 -(N−1) are the same (i.e., have the same superlattice layer structure SL M−1), the superlattice layers  16 - 3  and  16 -(N−2) are the same (i.e., have the same superlattice layer structure SL M−2), and so on. However, the barrier material layer  14  is not limited to being symmetrical and may alternatively be asymmetrical. 
     It should also be noted that the superlattice layers  16  may further be configured to reflect phonons and thereby decrease a thermal conductivity of the thermoelectric material  10  (and therefore increase the figure-of-merit (ZT) of the thermoelectric material  10 ) as discussed in U.S. Patent Application Publication No. 2013/0009132, entitled LOW THERMAL CONDUCTIVITY MATERIAL, which was filed on Jun. 29, 2012 and is hereby incorporated herein by reference in its entirety. More specifically, the superlattice layers  16  include, for each phonon wavelength to be reflected or blocked, multiple layers of one material composition each having a thickness approximately equal to a quarter of the phonon wavelength and multiple layers of another material composition each having a thickness approximately equal to a quarter of the phonon wavelength. Thus, the sub-layers within the superlattice layers  16  can be optimized to both provide resonant sub-bands as described above and to block multiple phonon wavelengths. 
     In one preferred embodiment, the thermoelectric material  10  is a Group IV-VI thermoelectric material grown in the [111] direction, and the high energy sub-bands and the low energy sub-bands of the superlattice layers  16  are oblique valley sub-bands and normal valley sub-bands, respectively. More specifically, in this preferred embodiment, each of the superlattice layers  16  is a Group IV-VI quantum well material having one or more quantum wells. Energy levels for electrons and holes in Group IV-VI semiconductor quantum well materials can be calculated using Schrödinger&#39;s one-dimensional time-independent equation: 
     
       
         
           
             
               
                 
                   
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     where Ψ(x) is a wavefunction describing the charge carrier, V(x) is a potential function describing the quantum well or superlattice layer, m is a mass of the charge carrier, and ç is Planck&#39;s constant. The equation above can be solved, with given boundary conditions and charge carrier masses, to calculate the energy levels, E, of the sub-bands in the Group IV-VI quantum well material (i.e., the Group IV-VI superlattice layer  16 ). It is known that quantum confinement in the [111] direction removes L-valley band degeneracy in Group IV-VI semiconductor materials resulting in charge carriers (i.e., electrons or holes) with two different effective masses and thus two different allowed energy levels. While not essential, for more information, the interested reader is directed to H. Z. Wu, N. Dai, M. B. Johnson, P. J. McCann, and Z. S. Shi, “Unambiguous Observation of Subband Transitions from Longitudinal Valley and Oblique Valleys in IV-VI Multiple Quantum Wells,” Applied Physics Letters, Vol. 78, No. 15, Apr. 9, 2011, pages 2199-2201. The low energy sub-band is for electrons, or charge carriers, in what is referred to as a normal valley or longitudinal valley, while the high energy sub-band is for electrons, or charge carriers, in what is referred to as a three-fold degenerate oblique valley. As such, for Group IV-VI, the low energy sub-band is more specifically referred to as a normal valley sub-band, and the high energy sub-band is more specifically referred to as an oblique valley sub-band. 
     As discussed below in detail, each of the superlattice layers  16  in the barrier material layer  14  includes one or more quantum wells. The energy levels of the normal and oblique valley sub-bands for each of the superlattice layers  16  are a function of a quantum well width of the individual quantum wells in that superlattice layer  16 . The quantum well thicknesses for the superlattice layers  16  are selected such that the barrier material layer  14  provides the desired potential barrier to enhance the Seebeck coefficient of the thermoelectric material  10  while at the same time enhancing, or increasing, transport of charge carriers through the potential barrier. More specifically, in a manner similar to that discussed above with respect to  FIG. 3 , the quantum well widths for the superlattice layers  16  are selected such that, for each of the superlattice layers  16 - 1  through  16 -(X−1), the oblique valley sub-band of the superlattice layer  16  is resonant with the normal valley sub-band of the immediately succeeding superlattice layer  16  in the short period superlattice structure of the barrier material layer  14  and, for each of the superlattice layers  16 -(X+1) through  16 -N, the oblique valley sub-band of the superlattice layer  16  is resonant with the normal valley sub-band of the immediately preceding superlattice layer  16  in the short period superlattice structure of the barrier material layer  14 . 
       FIG. 4  illustrates the operation of the superlattice structure of the barrier material layer  14  in more detail for one example of the barrier material layer  14 . In this example, the superlattice layers  16  are Group IV-VI superlattice lattice layers. Further, in this example, the thermoelectric material  10  is utilized in a waste heat harvesting or power generation application where electrons are thermally injected into the barrier material layer  14  from right to left. Note, however, that this discussion is equally applicable to cooling applications (Peltier effect applications). As illustrated, the quantum well widths of the superlattice layers  16  are selected such that the normal valley sub-bands (N) and the oblique valley sub-bands (O) are arranged as illustrated to thereby create a desired potential barrier while also improving cross-plane electrical conductivity. Due to thermal excitation, a large number of electrons occupy high energy levels in the right-most superlattice layers  16  (i.e., the superlattice layers  16 - 1  and  16 - 2 ). These high energy electrons pass over or through the potential barrier. Notably, a resonant path created from the oblique valley sub-band (O) of the superlattice layer  16 - 2  to the normal valley sub-band (N) of the superlattice layer  16 - 3  to the oblique valley sub-band (O) of the superlattice layer  16 - 4 . Electrons flow through this resonant path without any loss of energy, but do have a change in momentum upon moving from a normal valley sub-band (N) to an oblique valley sub-band (O) or vice versa. The resonant path improves the cross-plane electrical conductivity of the thermoelectric material  10 . 
     Notably, electrons are represented by solid arrows and phonons are represented by jagged or squiggly arrows. As electrons move from the oblique valley sub-band (O) of the superlattice layer  16 - 3  to the lower energy level oblique valley sub-band (O) of the superlattice layer  16 - 4 , a phonon is released. In a similar manner, a phonon is released when an electron moves from the oblique valley sub-band (O) of the superlattice layer  16 - 4  to the lower energy level oblique valley sub-band (O) of the superlattice layer  16 - 5  and again when the electron moves from the oblique valley sub-band (O) of the superlattice layer  16 - 5  to the lower energy level oblique valley sub-band (O) of the adjacent matrix material layer  12 . In this embodiment, the thicknesses of the individual sub-layers of the superlattice layers  16  are selected such that, in addition to providing the desired quantum well thicknesses, the superlattice layers  16  reflect phonons, which in turn decreases the thermal conductivity of the thermoelectric material  10  and, therefore, increases the figure-of-merit (ZT) of the thermoelectric material  10 . 
     In the embodiment of  FIG. 4 , the thicknesses of the superlattice layers  16  are approximately equal to a mean free path distance of electrons between scattering events. This allows efficient transport of electrons without loss of energy to the lattice. The bottom of  FIG. 4  is an energy versus momentum (k) diagram that illustrates the normal valley and oblique valley sub-bands of the superlattice layers  16 - 3  and  16 - 4 . As illustrated, electrons are transported from the superlattice layer  16 - 3  to the superlattice layer  16 - 4  by going from a state in the normal valley sub-band in the superlattice layer  16 - 3  to a resonant state in the oblique valley sub-band of the superlattice layer  16 - 4 . This transition requires a change in crystal momentum, but a phonon cannot provide this momentum change because there is no accompanying change in energy. Instead, the transition must be assisted by elastic scattering events such as electron-electron interaction. Purely elastic scattering is desirable for thermoelectric materials because it facilitates charge carrier transport from one superlattice layer  16  to the next without dissipating energy to the lattice in the form of phonon generation. The bottom of  FIG. 4  also illustrates that, when an electron moves from an oblique valley sub-band to a lower energy level normal valley sub-band (e.g., moves from the oblique valley sub-band of the superlattice layer  16 - 3  to the normal valley sub-band of the superlattice layer  16 - 3 ), a phonon is emitted. 
     As discussed above, the quantum well widths of the superlattice layers  16  determine the energy levels of the normal valley and oblique valley sub-bands of the superlattice layers  16 . As such, only certain combinations of quantum well widths in adjacent superlattice layers will result in the desired potential barrier as well as resonant normal and oblique sub-bands in adjacent superlattice layers  16 .  FIGS. 5 and 6  graphically illustrate a process by which appropriate combinations of quantum well widths for the superlattice layers  16  can be obtained. In  FIGS. 5 and 6 , the superlattice layers  16  include alternating layers of Lead Strontium Selenide (PbSrSe) and Lead Selenide (PbSe), where the PbSe layers correspond to quantum wells within the superlattice layers  16 . The thickness of the individual PbSe layer(s) (i.e., the quantum well(s)) in a superlattice layer  16  is the quantum well width for the superlattice layer  16 . 
     More specifically,  FIG. 5  illustrates experimental sub-band energy data and theoretical fits with effective mass as the only fitting parameter for PbSrSe/PbSe/PbSrSe quantum well materials versus quantum well widths. In this particular example, infrared transmission measurements of the oblique and normal valley sub-bands were obtained for PbSrSe/PbSe/PbSrSe quantum well materials having four different quantum well widths. Theoretical curves, or theoretical plots, for the normal and oblique valley sub-band energies versus quantum well width were then obtained using a theoretical fit of Schrödinger&#39;s equation to the measurements using effective mass as the only fitting parameter. 
     As illustrated in  FIG. 6 , the theoretical curves of  FIG. 5  can be used to determine combinations of quantum well widths that (1) provide the desired potential barrier and (2) give resonant, or the same, normal valley and oblique valley sub-band energy levels in adjacent superlattice layers  16  in the manner described above. As illustrated, a sequence of connected vertical lines  18 - 1  through  18 - 11  (more generally referred to herein collectively as vertical lines  18  and individually as vertical line  18 ) and horizontal lines  20 - 1  through  20 - 10  (more generally referred to herein collectively as horizontal lines  20  and individually as horizontal line  20 ) provides the combinations of quantum well widths that can be used for the superlattice layers  16 . The vertical lines  18 - 1  through  18 - 11  correspond to different quantum well widths. Each of the horizontal lines  20 - 1  through  20 - 10  illustrates the resonant normal and oblique valley sub-bands for quantum well materials that have the quantum well widths corresponding to the two vertical lines  18  connected by the horizontal line  20 . In one embodiment, the quantum well width that corresponds to a left-most vertical line  18 - 1  can be selected as the quantum well width of the superlattice layer  16 -X, the quantum well width that corresponds to the next vertical line  18 - 2  can be selected as the quantum well widths of the superlattice layers  16 -(X−1) and  16 -(X+1), the quantum well width that corresponds to the next vertical line  18 - 3  can be selected as the quantum well widths of the superlattice layers  16 -(X−2) and  16 -(X+2), and so on. 
       FIG. 7  illustrates one embodiment of the short period superlattice structure of the barrier material layer  14  where the superlattice layers  16  have quantum well widths selected using  FIG. 6 . In this embodiment, the barrier material layer  14  includes nine superlattice layers  16 - 1  through  16 - 9 . The quantum well width that corresponds to the vertical line  18 - 1  of  FIG. 6  is selected as the quantum well width for the superlattice layer  16 - 5 , the quantum well width that corresponds to the vertical line  18 - 2  of  FIG. 6  is selected as the quantum well widths for the superlattice layers  16 - 4  and  16 - 6 , the quantum well width that corresponds to the vertical line  18 - 3  of  FIG. 6  is selected as the quantum well widths for the superlattice layers  16 - 3  and  16 - 7 , the quantum well width that corresponds to the vertical line  18 - 4  of  FIG. 6  is selected as the quantum well widths for the superlattice layers  16 - 2  and  16 - 8 , and the quantum well width that corresponds to the vertical line  18 - 5  of  FIG. 6  is selected as the quantum well widths for the superlattice layers  16 - 1  and  16 - 9 . As a result, the barrier material layer  14  creates a desired potential barrier while at the same time the barrier material layer  14  is such that adjacent superlattice layers  16  have resonant sub-band energies that enhance cross-plane electrical conductivity, as described above. 
     Each of the superlattice layers  16 - 1  through  16 - 9  includes multiple periods of PbSe/PbSrSe. The individual thicknesses of the PbSe layers within the superlattice layers  16 - 1  through  16 - 9  are the quantum well widths of the corresponding superlattice layers  16 - 1  through  16 - 9 . In this embodiment, the number of periods within each superlattice layer  16  is selected such that a total thickness of that superlattice layer  16  is approximately equal to a mean free path distance of electrons between scattering events for a given temperature in a temperature gradient for which the thermoelectric material  10  is designed. Notably, the temperature gradient for which the thermoelectric material  10  is designed is a temperature gradient across the thermoelectric material  10  when incorporated into a thermoelectric device (e.g., a thermoelectric cooler or a thermoelectric power generator) under normal operating conditions. In this example, the thickness of the PbSe and PbSrSe layers within the superlattice layer  16 - 1  is 4.6 nm or 13 monolayers (ML) (which is the quantum well width that corresponds to the vertical line  18 - 5  of  FIG. 6 ), and the number of periods in the superlattice layer  16 - 1  is 4 such that the total thickness of the superlattice layer  16 - 1  is 36.9 nm; the thickness of the PbSe and PbSrSe layers within the superlattice layer  16 - 2  is 3.5 nm or 10 ML (which is the quantum well width that corresponds to the vertical line  18 - 4  of  FIG. 6 , and the number of periods in the superlattice layer  16 - 2  is 5 such that the total thickness of the superlattice layer  16 - 2  is 35.5 nm; and so on. Notably, the superlattice layers  16 - 1  through  16 - 9  also operate to reflect phonons with quarter-wavelength values equal to 4.6 nm, 3.5 nm, 2.5 nm, 2.1 nm, and 1.4 nm as taught in U.S. Patent Application Publication No. 2013/0009132. 
       FIG. 8  illustrates another embodiment of the short period superlattice structure of the barrier material layer  14  where the superlattice layers  16  have quantum well widths selected using  FIG. 6 . In this embodiment, the barrier material layer  14  includes 21 superlattice layers  16 - 1  through  16 - 21 . The quantum well width that corresponds to the vertical line  18 - 1  of  FIG. 6  is selected as the quantum well width for the superlattice layer  16 - 11 , the quantum well width that corresponds to the vertical line  18 - 2  of  FIG. 6  is selected as the quantum well widths for the superlattice layers  16 - 10  and  16 - 12 , the quantum well width that corresponds to the vertical line  18 - 3  of  FIG. 6  is selected as the quantum well widths for the superlattice layers  16 - 9  and  16 - 13 , the quantum well width that corresponds to the vertical line  18 - 4  of  FIG. 6  is selected as the quantum well widths for the superlattice layers  16 - 8  and  16 - 14 , the quantum well width that corresponds to the vertical line  18 - 5  of  FIG. 6  is selected as the quantum well widths for the superlattice layers  16 - 7  and  16 - 15 , the quantum well width that corresponds to the vertical line  18 - 6  of  FIG. 6  is selected as the quantum well widths for the superlattice layers  16 - 6  and  16 - 16 , the quantum well width that corresponds to the vertical line  18 - 7  of  FIG. 6  is selected as the quantum well widths for the superlattice layers  16 - 5  and  16 - 17 , the quantum well width that corresponds to the vertical line  18 - 8  of  FIG. 6  is selected as the quantum well widths for the superlattice layers  16 - 4  and  16 - 18 , the quantum well width that corresponds to the vertical line  18 - 9  of  FIG. 6  is selected as the quantum well widths for the superlattice layers  16 - 3  and  16 - 19 , the quantum well width that corresponds to the vertical line  18 - 10  of  FIG. 6  is selected as the quantum well widths for the superlattice layers  16 - 2  and  16 - 20 , and the quantum well width that corresponds to the vertical line  18 - 11  of  FIG. 6  is selected as the quantum well widths for the superlattice layers  16 - 1  and  16 - 21 . As a result, the barrier material layer  14  creates a desired potential barrier while at the same time the barrier material layer  14  is such that adjacent superlattice layers  16  have resonant sub-band energies that enhance cross-plane electrical conductivity, as described above. 
     Each of the superlattice layers  16 - 1  through  16 - 21  includes multiple periods of PbSe/PbSrSe. The individual thicknesses of the PbSe layers within the superlattice layers  16 - 1  through  16 - 21  are the quantum well widths of the corresponding superlattice layers  16 - 1  through  16 - 21 . In this embodiment, the number of periods within each superlattice layer  16  is selected such that a total thickness of the superlattice layer  16  is approximately equal to a mean free path distance of electrons between scattering events for a given temperature in the temperature gradient for which the thermoelectric material  10  is designed. In this example, the thickness of the PbSe and PbSrSe layers within the superlattice layer  16 - 1  is 20.2 nm or 57 ML (which is the quantum well width that corresponds to the vertical line  18 - 11  of  FIG. 6 ), and the number of periods in the superlattice layer  16 - 1  is 1 such that the total thickness of the superlattice layer  16 - 1  is 40.4 nm; the thickness of the PbSe and PbSrSe layers within the superlattice layer  16 - 2  is 15.6 nm or 44 ML (which is the quantum well width that corresponds to the vertical line  18 - 10  of  FIG. 6 , and the number of periods in the superlattice layer  16 - 2  is 1 such that the total thickness of that superlattice layer  16 - 2  is 31.2 nm; and so on. Notably, the superlattice layers  16 - 1  through  16 - 21  also operate to reflect phonons with quarter-wavelength values equal to 20.2 nm, 15.6 nm, 12.1 nm, 9.6 nm, 7.5 nm, 5.7 nm, 4.6 nm, 3.5 nm, 2.5 nm, 2.1 nm, and 1.4 nm as taught in U.S. Patent Application Publication No. 2013/0009132. 
       FIG. 9  illustrates one embodiment of the thermoelectric material  10  of  FIG. 1  in which the barrier material layer  14 - 1  is the barrier material layer  14  of  FIG. 7  and the barrier material layer  14 - 2  is the barrier material layer  14  of  FIG. 8 . In this particular embodiment, the barrier material layer  14 - 1  is near a cold side of the thermoelectric material  10  during operation whereas the barrier material layer  14 - 2  is near a hot side of the thermoelectric material  10  during operation. In this embodiment, the matrix material layer  12 - 1  includes a PbSe bulk layer  22  and a PbSrSe/PbSe superlattice layer  24  having a quantum well width (and thus sub-band energy levels) that is equal to that of the adjacent superlattice layer  16 - 1  in the barrier material layer  14 - 1 . The superlattice layer  24  effectively lowers a barrier height of the barrier material layer  14 - 1 , which is preferable near the cold side of the thermoelectric material  10 . The matrix material layer  12 - 2  includes a PbSrSe/PbSe superlattice layer  26  having a quantum well width (and thus sub-band energy levels) that is equal to that of the adjacent superlattice layer  16 - 9  in the barrier material layer  14 - 1  and a PbSe bulk layer  28 . The low band-gap of the PbSe bulk layer  28  increases a barrier height of the barrier material layer  14 - 2 , which is preferable near the hot side of the thermoelectric material  10 . In this example, the barrier height of the barrier material layer  14 - 1  is 23.7 meV, and the barrier height of the barrier material layer  14 - 2  is 81.5 millielectron volts (meV). 
       FIGS. 10 through 13  are substantially the same as  FIGS. 6 through 9  but for embodiments where PbSe/Lead Tin Selenide (PbSnSe)/PbSe quantum well materials are utilized for the barrier material layers  14  according to another embodiment of the present disclosure. In particular,  FIG. 10  graphically illustrates a process by which appropriate combinations of quantum well widths for the superlattice layers  16  can be obtained when using PbSe/PbSnSe/PbSe quantum well materials for the superlattice layers  16 . More specifically,  FIG. 10  illustrates sub-band energy curves obtained with different effective masses for normal and oblique valleys for PbSe/PbSnSe/PbSe quantum well materials versus quantum well widths. Theoretical curves, or theoretical plots, for the normal and oblique valley sub-band energies versus quantum well width were obtained using Schrödinger&#39;s equation with different effective masses for the normal and oblique valley sub-bands. 
     As illustrated, the theoretical curves of  FIG. 10  can be used to determine combinations of quantum well widths that (1) provide the desired potential barrier and (2) give resonant, or the same, normal and oblique valley sub-band energy levels in adjacent superlattice layers  16  in the manner described above. As illustrated, a sequence of connected vertical lines  30 - 1  through  30 - 10  (more generally referred to herein collectively as vertical lines  30  and individually as vertical line  30 ) and horizontal lines  32 - 1  through  32 - 9  (more generally referred to herein collectively as horizontal lines  32  and individually as horizontal line  32 ) provides the combinations of quantum well widths that can be used for the superlattice layers  16 . The vertical lines  30 - 1  through  30 - 10  correspond to different quantum well widths. Each of the horizontal lines  32 - 1  through  32 - 9  illustrates the resonant normal and oblique valley sub-bands for quantum well materials that have the quantum well widths corresponding to the two vertical lines  30  connected by the horizontal line  32 . In one embodiment, the quantum well width that corresponds to a left-most vertical line  30 - 1  can be selected as the quantum well width of the superlattice layer  16 -X, the quantum well width that corresponds to the next vertical line  30 - 2  can be selected as the quantum well widths of the superlattice layers  16 -(X−1) and  16 -(X+1), the quantum well width that corresponds to the next vertical line  30 - 3  can be selected as the quantum well widths of the superlattice layers  16 -(X−2) and 16-(X+2), and so on. 
       FIG. 11  illustrates one embodiment of the short period superlattice structure of the barrier material layer  14  where the superlattice layers  16  have quantum well widths selected using  FIG. 10 . In this embodiment, the barrier material layer  14  includes 13 superlattice layers  16 - 1  through  16 - 13 . The quantum well width that corresponds to the vertical line  30 - 1  of  FIG. 10  is selected as the quantum well width for the superlattice layer  16 - 7 , the quantum well width that corresponds to the vertical line  30 - 2  of  FIG. 10  is selected as the quantum well widths for the superlattice layers  16 - 6  and  16 - 8 , the quantum well width that corresponds to the vertical line  30 - 3  of  FIG. 10  is selected as the quantum well widths for the superlattice layers  16 - 5  and  16 - 9 , the quantum well width that corresponds to the vertical line  30 - 4  of  FIG. 10  is selected as the quantum well widths for the superlattice layers  16 - 4  and  16 - 10 , the quantum well width that corresponds to the vertical line  30 - 5  of  FIG. 10  is selected as the quantum well widths for the superlattice layers  16 - 3  and  16 - 11 , the quantum well width that corresponds to the vertical line  30 - 6  of  FIG. 10  is selected as the quantum well widths for the superlattice layers  16 - 2  and  16 - 12 , and the quantum well width that corresponds to the vertical line  30 - 7  of  FIG. 10  is selected as the quantum well widths for the superlattice layers  16 - 1  and  16 - 13 . As a result, the barrier material layer  14  creates a desired potential barrier while at the same time the barrier material layer  14  is such that adjacent superlattice layers  16  have resonant sub-band energies that enhance cross-plane electrical conductivity, as described above. 
     Each of the superlattice layers  16 - 1  through  16 - 13  includes multiple periods of PbSe/PbSnSe. The individual thicknesses of the PbSnSe layers within the superlattice layers  16 - 1  through  16 - 13  are the quantum well widths of the corresponding superlattice layers  16 - 1  through  16 - 13 . In this embodiment, the number of periods within each superlattice layer  16  is selected such that a total thickness of that superlattice layer  16  is approximately equal to a mean free path distance of electrons between scattering events for a given temperature in the temperature gradient for which the thermoelectric material  10  is designed. In this example, the thickness of the PbSe and PbSnSe layers within the superlattice layer  16 - 1  is 8.5 nm or 24 MLs (which is the quantum well width that corresponds to the vertical line  30 - 7  of  FIG. 10 ), and the number of periods in the superlattice layer  16 - 1  is 2 such that the total thickness of the superlattice layer  16 - 1  is 34 nm; the thickness of the PbSe and PbSnSe layers within the superlattice layer  16 - 2  is 6.4 nm or 18 MLs (which is the quantum well width that corresponds to the vertical line  30 - 6  of  FIG. 10 ), and the number of periods in the superlattice layer  16 - 2  is 3 such that the total thickness of the superlattice layer  16 - 2  is 38.2 nm; and so on. Notably, the superlattice layers  16 - 1  through  16 - 13  also operate to reflect phonons with quarter-wavelength values equal to 8.5 nm, 6.4 nm, 5.0 nm, 3.5 nm, 2.8 nm, 2.1 nm, and 1.4 nm as taught in U.S. Patent Application Publication No. 2013/0009132. 
       FIG. 12  illustrates another embodiment of the short period superlattice structure of the barrier material layer  14  where the superlattice layers  16  have quantum well widths selected using  FIG. 10 . In this embodiment, the barrier material layer  14  includes 19 superlattice layers  16 - 1  through  16 - 19 . The quantum well width that corresponds to the vertical line  30 - 1  of  FIG. 10  is selected as the quantum well width for the superlattice layer  16 - 10 , the quantum well width that corresponds to the vertical line  30 - 2  of  FIG. 10  is selected as the quantum well widths for the superlattice layers  16 - 9  and  16 - 11 , the quantum well width that corresponds to the vertical line  30 - 3  of  FIG. 10  is selected as the quantum well widths for the superlattice layers  16 - 8  and  16 - 12 , the quantum well width that corresponds to the vertical line  30 - 4  of  FIG. 10  is selected as the quantum well widths for the superlattice layers  16 - 7  and  16 - 13 , the quantum well width that corresponds to the vertical line  30 - 5  of  FIG. 10  is selected as the quantum well widths for the superlattice layers  16 - 6  and  16 - 14 , the quantum well width that corresponds to the vertical line  30 - 6  of  FIG. 10  is selected as the quantum well widths for the superlattice layers  16 - 5  and  16 - 15 , the quantum well width that corresponds to the vertical line  30 - 7  of  FIG. 10  is selected as the quantum well widths for the superlattice layers  16 - 4  and  16 - 16 , the quantum well width that corresponds to the vertical line  30 - 8  of  FIG. 10  is selected as the quantum well widths for the superlattice layers  16 - 3  and  16 - 17 , the quantum well width that corresponds to the vertical line  30 - 9  of  FIG. 10  is selected as the quantum well widths for the superlattice layers  16 - 2  and  16 - 18 , and the quantum well width that corresponds to the vertical line  30 - 10  of  FIG. 10  is selected as the quantum well widths for the superlattice layers  16 - 1  and  16 - 19 . As a result, the barrier material layer  14  creates a desired potential barrier while at the same time the barrier material layer  14  is such that adjacent superlattice layers  16  have resonant sub-band energies that enhance cross-plane electrical conductivity, as described above. 
     Each of the superlattice layers  16 - 1  through  16 - 19  includes multiple periods of PbSe/PbSnSe. The individual thicknesses of the PbSnSe layers within the superlattice layers  16 - 1  through  16 - 19  are the quantum well widths of the corresponding superlattice layers  16 - 1  through  16 - 19 . In this embodiment, the number of periods within each superlattice layer  16  is selected such that a total thickness of the superlattice layer  16  is approximately equal to a mean free path distance of electrons between scattering events for a given temperature in the temperature gradient for which the thermoelectric material  10  is designed. In this example, the thickness of the PbSe and PbSnSe layers within the superlattice layer  16 - 1  is 18.4 nm or 52 ML (which is the quantum well width that corresponds to the vertical line  30 - 10  of  FIG. 10 ), and the number of periods in the superlattice layer  16 - 1  is 1 such that the total thickness of the superlattice layer  16 - 1  is 36.8 nm; the thickness of the PbSe and PbSnSe layers within the superlattice layer  16 - 2  is 14.2 nm or 40 ML (which is the quantum well width that corresponds to the vertical line  30 - 9  of  FIG. 10 ), and the number of periods in the superlattice layer  16 - 2  is 1 such that the total thickness of that superlattice layer  16 - 2  is 28.3 nm; and so on. Notably, the superlattice layers  16 - 1  through  16 - 19  also operate to reflect phonons with quarter-wavelength values equal to 18.4 nm, 14.2 nm, 11.0 nm, 8.5 nm, 6.4 nm, 5.0 nm, 3.5 nm, 2.8 nm, 2.1 nm, and 1.4 nm as taught in U.S. Patent Application Publication No. 2013/0009132. 
       FIG. 13  illustrates one embodiment of the thermoelectric material  10  of  FIG. 1  in which the barrier material layer  14 - 1  is the barrier material layer  14  of  FIG. 11  and the barrier material layer  14 - 2  is the barrier material layer  14  of  FIG. 12 . In this particular embodiment, the barrier material layer  14 - 1  is near a cold side of the thermoelectric material  10  during operation whereas the barrier material layer  14 - 2  is near a hot side of the thermoelectric material  10  during operation. In this embodiment, the matrix material layer  12 - 1  includes a PbSnSe bulk layer  34  and a PbSnSe/PbSe superlattice layer  36  having a quantum well width (and thus sub-band energy levels) that is equal to that of the adjacent superlattice layer  16 - 1  in the barrier material layer  14 - 1 . The superlattice layer  36  effectively lowers a barrier height of the barrier material layer  14 - 1 , which is preferable near the cold side of the thermoelectric material  10 . The matrix material layer  12 - 2  includes a PbSnSe/PbSe superlattice layer  38  having a quantum well width (and thus sub-band energy levels) that is equal to that of the adjacent superlattice layer  16 - 13  in the barrier material layer  14 - 1  and a PbSnSe bulk layer  40 . The low band-gap of the PbSnSe bulk layer  40  increases a barrier height of the barrier material layer  14 - 2 , which is preferable near the hot side of the thermoelectric material  10 . In this example, the barrier height of the barrier material layer  14 - 1  is 24.4 meV, and the barrier height of the barrier material layer  14 - 2  is 59.7 meV. 
       FIG. 14  is a flow chart that illustrates a method of designing and fabricating the thermoelectric material  10  of  FIG. 1  for Group IV-VI materials according to one embodiment of the present disclosure. Note that the same or a similar process may be used to design and fabricate the thermoelectric material  10  in other material systems. First, measurements for intersubband transition energies for multiple quantum well material samples having different quantum well widths are obtained and normal and oblique valley sub-band energy levels are calculated (step  100 ). The sub-band energy levels are calculated in the conduction and valence bands assuming equal band edge discontinuities between the well and barrier materials. Next, using Schrödinger&#39;s equation, theoretical fits, or theoretical plots, of the energy levels for the normal and oblique valley sub-bands versus quantum well width are generated (step  102 ). More specifically, the effective masses for electrons and holes in the normal and oblique valley sub-bands are adjusted to determine theoretical plots that best fit the measurements obtained in step  100 . Next, the theoretical curves for sub-band energy versus quantum well width are used to determine combinations of quantum well widths that give resonant, or equal, normal and oblique valley sub-band energy levels in adjacent superlattice layers  16  in the short period superlattice structure of the barrier material layer  14  (step  104 ). Lastly, the combinations of the quantum well widths determined in step  104  are used to fabricate the thermoelectric material  10  (step  106 ). 
     Those skilled in the art will recognize improvements and modifications to the preferred embodiments of the present disclosure. All such improvements and modifications are considered within the scope of the concepts disclosed herein and the claims that follow.