14-1

Chapter 14 Low-Dimensional Nanostructures

Contents

Chapter 14 Low-Dimensional Nanostructures ...... 14-1 Contents ...... 14-1 14.1 Low-Dimensional Systems ...... 14-2 14.1.1 Quantum Well (2D) ...... 14-3 Example 14.1 Energy Levels of a Quantum Well ...... 14-9 14.1.2 Quantum Wires (1D)...... 14-10 14.1.3 Quantum Dots (0D)...... 14-15 14.1.4 Thermoelectric Transport Properties of Quantum Wells ...... 14-17 14.1.5 Thermoelectric Transport Properties of Quantum Wires...... 14-19 14.1.6 Proof-of-Principle Studies ...... 14-22 14.1.7 Size effects of Quantum Well on Lattice Thermal Conductivity ...... 14-25 Problems ...... 14-29 References ...... 14-30

14-2

The field of thermoelectrics advanced rapidly in the 1950s when the basic science of thermoelectric materials became well established, the important role of heavily doped semiconductors as good thermoelectric materials became accepted, the thermoelectric material bismuth telluride was discovered and developed for commercialization, and the thermoelectric industry was launched. Over the following three decades, 1960 to 1990, only incremental gains were made in increasing ZT, with bismuth telluride, remaining the best commercial material at ZT ≈ 1. During that three- decade period, the thermoelectrics field received little attention from the worldwide scientific research community. Nevertheless, the thermoelectric industry grew slowly but steadily by finding niche applications for space missions and medical applications, where cost and efficiency were not as important as energy availability, reliability, and predictability. The present interest in low-dimensional thermoelectric materials was prompted by the theoretical work of Hicks and Dresselhaus (1993)[1-3], stimulating the research community to once again become active in this field and to find new research directions that would have an impact on future developments and lead to thermoelectric materials with better performance. As a result of this stimulation, two different research approaches were taken for developing the next generation of thermoelectric materials, one using new families of bulk materials and the other using low-dimensional materials systems.

14.1 Low-Dimensional Systems

As the dimensionality is decreased from 3D crystal solids to two-dimensional quantum wells (2D) to one-dimensional (1D) quantum wires, and finally to zero-dimensional quantum dots (0D), new physical phenomena are introduced and new opportunities arise to vary the thermoelectric transport coefficients (, and k), independently. For example, it is known that we can have a transition from a semimetal (bismuth) to a semiconductor by increasing the conduction band edge with quantum confinement. Furthermore, the introduction of many interfaces offers the opportunity to increase phonon scattering more than electron scattering so that the electrical 14-3

conductivity is not changed much while the thermal conductivity is much reduced by interface scattering processes.

14.1.1 Quantum Well (2D)

Quantum effects arise in systems which confine electrons to regions comparable to their de Broglie wavelength. When such confinement occurs in one dimension only (say, by a restriction on the motion of the electron in the z-direction), with free motion in the x- and y-directions, a two-dimensional system is created, which is shown in Figure 14.1 (a).

L L z

y x d

(a)

E d U0 = ? z n = 3

n = 2 n = 1

U0 = 0 (b)

Figure 14.1 (a) Schematic presentation of a quantum well (2D), and (b) the quantum numbers of an electron in the quantum well.

The dispersion relation for electrons in a three-dimensional (3D) system is given by 14-4

2  k 2 k 2 k 2  E   x  y  z  (14.1) 2  m m m   x y z 

In a two-dimensional system with the quantum well width d in Figure 14.1 (a), the energy of an electron is

2  k 2 k 2  E   x  y   E   c,n (14.2) 2  mx my 

where Ec,n is the confined energy in the z-direction. The Schrödinger equation in the z- direction is given by

2   2    U0 z  Ez (14.3)  2m 

The solution for a zero potential of the quantum well with an infinite quantum-well potential of barriers is shown in Figure 14.1 (b) (see also Equation (10.41) for the detail). The wavenumber in the z-direction is

n k  , n  1, 2, ... (14.4) z d

The confined nth subband energy in the z-direction is expressed by

2 2 2 2 2  kz  n  Ec,n   2 (14.5) 2mz 2mzd

14-5

From Equation (14.2), the total kinetic energy of an electron is then expressed by

2  k 2 k 2  2n2 2 E   x  y     2 (14.6) 2  mx my  2mzd

Density of States in 2D

We here develop the density of states in a two-dimensional system. The total energy of an electron is expressed by introducing arbitrary k’ and m’ as

2k2 2  k2 k2  2n2 2 E    x  y   (14.7)      2 2m 2  m m  2mzd

We can relate the arbitrary parameters to the original parameters from Equation (14.6) as

2 2 kx kx mx  , which leads to kx  kx (14.8) mx m m

2 2 (14.9) ky ky my  , which leads to ky  ky my m m

In the strictly two-dimensional system, we have

m m m m (14.10) dk  dk dk  x y dkdk  x y 2kdk x y m x y m

14-6

The area of the smallest wavevector in a 2D crystal is 2 L2 . The number of states between k and k + dk in the perfect two-dimensional space is then obtained as

(14.11) 2  2k mxmy N(k)dk  2 dk  2  m    L  where the factor of 2 accounts for the electron spin (Pauli Exclusion Principle). Now the density of states per unit area is

(14.12) N(k)dk k mxmy g(k)dk   dk L2  m

From Equation (14.7), we have

1 1 (14.13) 2m2 E 2 k  

Differentiating this gives

1 1 (14.14) dk 2m2   E 2 dE 2

Inserting Equations (14.13) and (14.14) into (14.12), the 2D density of states per unit area for each allowable kxy series is

 (14.15) md g0 (E)  2 14-7

 where md  mxmy , which is called the 2D DOS effective mass. The confined energy E0 from

Equation (14.5) for n = 1 is defined by

2 2 E0  2 (14.16) 2mzd

For energy E < E0, there are no states (see Figure 14.2). For energy E0< E < 4E0, the density of

 2 states per unit area is just for a perfect two-dimensional electron, namely g0 (E)  md 

(Equation (14.15)). For energy 4E0< E < 9E0, the density of states per unit area is 2g0 . For energy 9E0< E < 16E0, the density of states per unit area is 3g0 , and so forth.

To convert this g0 , which is the density of states per unit area, to a density of states per unit volume, one must divide by an appropriate length in the z-direction, in this case by the wall width

 2 d. This three-dimensional DOS then rise in step of md  d , as shown in Figure 14.3, where it is also compared with the ordinary bulk (3D) DOS [4].

The 2D density of states per unit volume is finally defined by

 (14.17) md g2D (E)  2d where . 14-8

Figure 14.2 Energy versus wavenumber k for a quasi-quantum well [4].

Figure 14.3 Density of states for a quasi-quantum well. The corresponding density of states for an unconfined 3D system is also shown for comparison (broken line). E0 is defined in Equation (14.16) [4].

The most popular strategy for design is to take advantage of the enhanced density of states for electrons near the Fermi energy due to the reduced dimensionality. A quantum well can be 14-9

formed by sandwiching a thin film between two other materials (see Figure 14.4). For example, a thin layer of GaAs can be sandwiched between two AlAs layers. Both GaAs and AlAs are semiconductors. AlAs has a larger bandgap (2.17 eV) than GaAs (1.42 eV). Quantum confinement of an electron within the thin layer of GaAs will happen if its energy is below that of the conduction band edge in AlAs. This is a compositional quantum well. The barriers prevent the transmission of low energy electrons, and allow only high energy electrons. This may result in an enhancement in the power factor [5].

Figure 14.4 Band-edge diagram for a typical GaAs/AlAs quantum well.

Example 14.1 Energy Levels of a Quantum Well

A quantum well of GaAs (gallium arsenide) with the DOS effective mass of 0.07 me is used for a laser, estimate the first three subband energy levels at k = 0 for the quantum width of 60 Å assuming the infinite quantum-well barriers and the electron mass in the z-direction is equal to the DOS effective mass (for a cubic crystal).

14-10

Solution:

 From Equation (14.6) at k = 0 with mz  md , the nth subband energy with the infinite quantum- well barriers is

2n2 2 E   2 2md d

For n = 1

2 6.6261034 Js 2  12  2 E  2  0.149eV 20.07 9.10931031 kg601010 m 1.60211019 J / eV 

Likewise,

E = 0.597 eV for n = 2

E = 1.343 eV for n = 3

Comments: It is seen that the conduction band edge of the quantum well is lifted up 0.149 eV from the conduction band edge of a bulk material. This lift-up may be useful in enhancing the thermoelectric performance. The first three subband energy levels are calculated with the assumption of the infinite barrier potential. The lowest subband is most important due to the closeness of the Fermi energy. If the barrier potential around the quantum well is finite, the energy levels may be more complicated than given by Equation (14.6).

14.1.2 Quantum Wires (1D)

14-11

Quantum effects in systems which confine electrons to regions comparable to their de Broglie wavelength. When such confinement occurs in two dimensions only (say, by two restrictions on the motion of the electron in the z- and y-directions), with free motion in the x-direction, a one- dimensional electron is created, which is shown in Figure 14.5.

x z L d

y

Figure 14.5 Schematic presentation of a quantum wire.

In a one-dimensional system for a quantum wire, the total energy is

2 2 2 2 2 2 2 2  kx   n   l E   2  2 (14.18) 2mx 2myd 2mzd where n = 1, 2….. and l = 1, 2, …

Density of States in 1D

We here develop the density of states in a strict one-dimensional system. The total energy of an electron is

2 2 2 2 2 2 2 2 2 2  k  kx   n   l E    2  2 (14.19) 2m 2m 2myd 2mzd

14-12

We can relate the arbitrary parameter to the original parameter in Equation (14.18).

2 2 kx kx mx  , which leads to kx  kx (14.20) mx m m

In the strictly one-dimensional system, we have

m (14.21) dk  x dk m x

The length of the smallest wavevector in a 1D crystal is 2 L . The number of states between k and k + dk in the one-dimensional space is then obtained as

2 m (14.22) N(k)dk  x dk  2  m x    L  where the factor of 2 accounts for the electron spin (Pauli Exclusion Principle). The density of states per unit length is

N(k)dk 1 m (14.23) g(k)dk   x dk L  m x

From Equation (14.19), we have

1 1 (14.24) 2m2 E 2 k  

14-13

Differentiating this gives

1 1 (14.25) dk 2m2   E 2 dE 2

Inserting Equations (14.24) and (14.25) into (14.23), we have

1  1 (14.26) 1  2m  2  g (E)   d  E 2 0  2  2   

 where md  mx . The confined energy E0 from Equation (14.18) for n = l = 1 is

2 2 2 2 (14.27) E0  2  2 2myd 2mzd

The 1D density of states per unit volume with considering (n+l=2) is finally obtained by

1  1 (14.28) 1  2m  2  g (E)   d  E 2 1D 2  2  d    where , which is called the 1D DOS effective mass.

14-14

Figure 14.6 Energy versus wavenumber k for a quasi-quantum wire.

Figure 14.7 Density of states for a quasi-quantum wire. The corresponding density of states for an unconfined 3D system is also shown for comparison (broken line).

Figure 14.7 shows the density of states for such an ideal quantum wire, showing the characteristic singularity in E-1/2 which was derived for 1D as shown in Equation (14.26). In a quantum wire, such a singularity will occur at each energy of quantization in the x-direction. For 14-15

real quantum wires, the spacing of the quantized energies and the corresponding wavefunctions, will depend on the precise shape of the potential U0 in Equation (14.3).

14.1.3 Quantum Dots (0D)

Electrons can be confined in all three dimensions in a dot (see Figure 14.8). The situation is analogous to that of a hydrogen atom: only discrete energy levels are possible for electrons trapped by such a zero-dimensional potential. The spacing of these levels depends on the precise shape of the potential. The development and application of quantum dot systems is an increasingly important research topic for a number of reasons, both technological and theoretical. Quantum dots, where a confinement potential replaces the potential of the nucleus, are fascinating objects. On the other hand, these systems are thought to have vast potential for future technological applications, such as possible applications in memory chips, quantum computation, quantum-dot lasers, and so on.

When an electron motion is confined in all directions, one gets a zero-dimensional system as

2 2n2  1 1 1  (14.29) E      n 2   2d  mx my mz  which, for the lowest subband energy n = 1, leads to

2 2 (14.30) E0   2 2mc d where (14.31)

1 1  1 1 1          mc 3 mx my mz  14-16

which is called the conductivity effective mass (see also Equation (12.55)). The density of states is a series of -function peaks as shown in Figure 14.9.

g0D E   E  E0  (14.32)

z

x

y d

Figure 14.8 Schematic presentation of a quantum dot.

Figure 14.9 Density of states for a quasi-quantum dot. The corresponding density of states for an unconfined 3D system is also shown for comparison (broken line).

14-17

14.1.4 Thermoelectric Transport Properties of Quantum Wells

The thermoelectric transport properties are sought assuming that electrons occupy only the lowest (n = 1) subband of a quantum well. A power law model is used to calculate the thermoelectric transport properties of a quantum well. The electron relaxation time is expressed by

r (14.33)    0 E where r is the scattering parameter. The density of states for the quantum well from Equation (14.17) is given by

 (14.34) md g2D (E)  2d

For simplicity, the Fermi integral for the quantum well is defined by

 s (14.35) E  Fs   0 dE  E EF  0 e 1

 where the reduced energy is E  E kBT , the reduced Fermi energy of the quantum well

0 2 2 2 EF  EF  E0  kBT and the lowest subband energy E0    2mzd .

The electron concentration n in thermal equilibrium is

14-18

   (14.36) md kBT  n  g2D Ef0 EdE  2  0 dE   E EF  0 0  d e 1 which is reduced to

N  2mk T  (14.37) n  v  d B F  2  0 2d   

The electrical conductivity for the quantum well is expressed by

 N  2m k T   e F  (14.38)   v  d B F e 0 r 1 r  ne   2  0       x 2d      mc F0 

 1 2  1 where md  mxmy  and mc  0.51 mx 1 my  . The Seebeck coefficient for the quantum well is expressed by

 kB r  2Fr 1 0       EF  (14.39)  e  r 1Fr 

The Lorentz number for the quantum well is obtained by

2  2   kB  r  3Fr 2  r  2Fr 1   Lo         (14.40) e r 1 F  r 1 F       r    r   and the electronic thermal conductivity of the quantum well is expressed as

14-19

ke   T  Lo (14.41)

The dimensionless figure of merit for the quantum well is obtained as

2 r  2F  r 1 r 1  E0 F  r 1F F  r  r  (14.42) ZT2D  2 2 1 r  2 Fr 1  r  3Fr 2   r 1Fr

where  is the material parameter for the quantum well as

N  2mk T  k 2T   v  d B  B 0 2D  2   (14.43) 2d    mc kl

Equations (14.38) to (14.43) become identical to the formulae of Hicks and Dresselhaus (1993)[2] when the scattering parameter is assumed to be r = 0.

14.1.5 Thermoelectric Transport Properties of Quantum Wires

The thermoelectric transport properties are sought assuming that electrons occupy only the lowest (n = 1) subband of quantum well. The power law model is used to calculate the thermoelectric transport properties of quantum wires. The electron relaxation time is expressed by

r (14.44)    0 E where r is the scattering parameter. The density of states for the quantum wires from Equation (14.28) is given by 14-20

1  1 (14.45) N  2m  2  g (E)  v  d  E 2 1D 2  2  d   

 where md  mx .

For simplicity, the Fermi integral for the quantum wire is defined by

 s (14.46) E  Fs   0 dE  E EF  0 e 1

 where the reduced energy is E  E kBT , the reduced Fermi energy of the quantum wire

0 EF  EF  E0  kBT and the lowest subband energy for n = 1 (assumption) as

2 2 2 2 (14.47) E0  2  2 2myd 2mzd

The electron concentration n in thermal equilibrium is

1 (14.48)    2 1 1 Nv  2md    kBT  2 2 n  g2D E f0 EdE  2  2  E kBT   0 dE     E EF  0 0 d    e 1

(editor note: in the above equation, change g2D –> g1D) which is reduced to

14-21

1 2 N  2m k T  (14.49) n  v  x B  F 2  2  1 2 d   

The electrical conductivity for the quantum wires is expressed by

 1 2  (14.50) N  2m k T  e  1  Fr1 2     v  x B  F e 0 2r     ne d 2  2  1 2 m 2 F x      x   1 2 

The Seebeck coefficient for the quantum wire is expressed by

 3   r  Fr 1 2   kB   2  0      EF  (14.51)  e  1   r  Fr 1 2  2  

The Lorentz number for the quantum wire is obtained by

2   5    3    2  r  F 3  r  F 1   r  r   kB    2  2   2  2   Lo       (14.52)  e   1    1   r  F  r  F    2 r 1 2 2 r 1 2         and the electronic thermal conductivity of the quantum wire is expressed as

ke   T  Lo (14.53)

The dimensionless figure of merit for the quantum wire is obtained as

14-22

2  3   r  Fr 1 2   1   2  0 r    EF  Fr 1 2  2  1   r  Fr 1 2  2   ZT1D  2 (14.54)  3  2 r   Fr 1 2 1  5   2   r  F 3  r    2  2  1  r  Fr 1 2  2 

where  is the material parameter for the quantum wire,

1 2 2N  2k T  k 2T   v  B  B 0 (14.55) 1D 2  2  d   mx  kl

Equations (14.50) to (14.55) become identical to the formulae of Hicks and Dresselhaus (1993)[3] when the scattering parameter is assumed to be constant as r = 0.

14.1.6 Proof-of-Principle Studies

For a specific value of the material parameter , the dimensionless figure of merit within the quantum well or wire is optimized by varying the electron concentration (or equivalently the Fermi energy) of the system (Figure 14.10). The higher the  value, the higher the optimal ZT value. Therefore, the quantity  provides a guideline for selecting good thermoelectric materials and for designing optimum quantum wire thermoelectric materials.

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Figure 14.10 The dimensionless figure of merit versus electron concentration for bulk Bi2Te3.

Figure 14.11 Calculated dependence of ZT at room temperature within the quantum well or within the quantum wire (width d) for Bi2Te3 at the optimum doping concentration for transport in the highest mobility direction. Also shown is the ZT for bulk Bi2Te3 calculated using the corresponding bulk model. This plot is based on Hicks (1996) [6].

14-24

The following data were used in Figure 14.11. Bi2Te3 has a trigonal structure, which can be expressed in terms of a hexagonal unit cell of lattice parameters a0 = 4.3 Å and c0 = 30.5 Å. The compound has an anisotropic effective mass tensor, with components mx = 0.02 me, my = 0.08 me, and mz = 0.32 me. The lattice thermal conductivity is kl = 1.5 W/mK and the electron mobility is

2 x = 1200 cm /Vs. The degeneracy of valleys is Nv = 6 [7]. Using these simple assumptions, a substantial enhancement was calculated for ZT within the quantum well for 2D systems having small quantum well widths relative to their bulk sizes. To make these calculations more useful, we show in Figure 14.11, the enhancement of ZT within a

Bi2Te3 quantum well as a function of d, and an even greater enhancement in ZT is predicted for

Bi2Te3 when prepared as a 1D quantum wire [2, 3, 6]. The results of Figure 14.11 suggest that a good thermoelectric material in 3D might be expected to exhibit even higher ZT values in reduced dimensions. To make a fair comparison between 3D and lower dimensions, all ZT values in Figure 14.11 are given for the optimum electron concentration.

2 Figure 14.12 S n results (S is  in this book) for PbTe/Pb0.927Eu0.073Te multiple-quantum-well (MQW) structures (full circles) as a function of well thickness d at 300 K. For comparison, the best experimental bulk PbTe value is also shown. Calculated results for optimum doping using the model are shown as a solid line. Reprinted from Hicks et al. (1996)[8] with permission. 14-25

According to the model, the increase in Z due to 2D effects arises mainly from the power factor

2  n, while the lattice thermal conductivity kl is assumed to be unchanged from the bulk value. In fact, it is also assumed that mobility in the quantum well is the same as in the bulk. So any increase in Z would arise through the power factor, where n is the electron concentration in the quantum well. Therefore, according to the model, we should be able to observe an increase in 2n as the well width is narrowed. The reason for making the comparison between theory and experiment for 2n rather than 2 is to test the validity of the theoretical model in terms of intrinsic phenomena rather than phenomena sensitive to materials processing conditions that more strongly influence the electron mobility. Samples were grown by molecular-beam epitaxy (MBE). Details of the sample preparation are given in Harman et al. (1996) [9]. Superlattice samples with 100 periods of the multiple-quantum- well structures were grown, with PbTe well widths varying between 17 and 55 Å, separated by wide Pb0.927Eu0.073Te barriers of about 450 Å. The data points in Figure 14.12 show an increase in 2n as the well width d is narrowed, and the well 2n may reach several times the bulk value for small well widths. This result is predicted by the theoretical model and therefore gives qualitative support to the idea that MQW structures may be used to improve Z over bulk values.

14.1.7 Size effects of Quantum Well on Lattice Thermal Conductivity

When the electron or phonon mean free paths are comparable to or larger than the thickness of a thin film, the electrons or phonons will collide more with the boundaries. In this section, we consider transport parallel to boundaries, such as thermal conduction along a thin film. Consider now the schematic diagram in Figure 14.13 where heat is transported across a film of thickness d with temperature T1 and T2 at the two boundaries. When the thickness of a film d is much larger than the mean free path, d >> , a microscopic temperature gradient is established and the thermal conductivity is well defined. However, when the two length scales are comparable, d ~ , a temperature gradient cannot be established and therefore the thermal conductivity cannot be 14-26

defined. In the limiting case, with no phonon scattering with in the film, the heat flux across the film is given as radiative heat transfer. This is commonly known as the Casimir (1938) [10] limit for which the thermal conductivity cannot be defined. In this limit, phonons scatter only at boundaries, which restores local thermodynamic equilibrium. Heat conduction changes from a diffusive to a ballistic transport phenomenon, which is shown in Figure 14.13 (a). Two cases of heat transport along and across a thin film are considered.

T2 T2

Thin film d L L

Phonon

T T 1 1 (a)

(b)

Figure 14.13 (a) Schematic diagram of temperature profiles in a thin film for two limiting cases for d >>  and d ~ . (b) Coordinate system for phonon in a thin film.

In-Plane Phonon Heat Conduction

We start from the steady-state Boltzmann equation for a two-dimensional problem. 14-27

(14.56) f f Fx f f  f0 vx  vz    x z m vx 

We introduce a deviation function of g such as

g  f  f0 (14.57)

Equation (14.56) can be written as

f0 g0 f0 g Fx f0 Fx g g (14.58) vx  vx  vz  vz     x x z z m vx m vx 

In the above equation, f0 z  0 because f0 is a function of x only for transport along the film.

We can also set g0 vx  0 on the basis that the spatial size effect does not affect f in the momentum space. Along the x-direction, we will use the same approximation as we did in the

f x term deriving the diffusion equations and keep only the f0 x term while neglecting g x term, which is justified since we assumed that the length in this direction is long. This approximation leads to

f0 g Fx f0 g (14.59) vx  vz    x z m vx  or

g Fx f0 f0 (14.60) vcos  g   vx  S0 x z m vx x

14-28

which is a first-order differential equation and only one boundary condition is needed. This can be solved with the boundary conditions. Details of the derivation are found in Majumdar (1993)[11] and Chen (1997)[12]. If this is the case, how will the two surfaces, such as the two surfaces of a film in Figure 14.13 affect the transport? The answer lies in that we need one boundary condition for all the velocity components, that is, for all directions  in Equation (14.60). Taking a boundary point at z = 0, we need the boundary condition for phonons both coming toward the boundary and leaving the boundary. The boundary conditions are often specified for only the phonons leaving the boundary and at both z = 0 and z = d, each covering half the space, and this is equivalent to specifying a boundary condition at one boundary for the entire space.

For a freestanding single layer thin film and partially specular and partially diffuse boundaries, the lattice thermal conductivity for a quantum well can be calculated from

푘 3(1 − 푝) 1 1 − 푒푥푝(− 휉⁄휇) (14.61) 2퐷 = 1 − ∫ (휇 − 휇3) 푑휇 푘퐵푢푙푘 2휉 0 1 − 푝푒푥푝(− 휉⁄휇) where

d (14.62)    which is called the acoustic thickness  and its inverse is called the phonon Knudsen number

Kn   d.  is the directional cosine   cos . Figure 14.14 compares the modeling results with experimental data of the in=plane thermal conductivity for GaAs/AlAs superlattices, based on frequency-dependent phonon relaxation time.

14-29

Figure 14.14 Thermal conductivity as a function of the layer thickness for GaAs/AlAs superlattices of equal layer thickness at room temperature [12].

Problems

14.1. Derive the 2D density of states of Equation (14.17).

14.2. Bi (bismuth) is a very attractive material for low-dimensional thermoelectricity. With the

DOS effective mass of 0.023 me, estimate the first three subband energy levels at k = 0 for the quantum width of 100 Å assuming infinite quantum-well barriers and electron mass in the z-direction equal to the DOS effective mass (for a cubic crystal).

14.3. Derive the 1D density of states of Equation (14.28).

14-30

14.4. Construct Figure 14.11 by developing a Mathcad program for Bi2Te3. The program may −14 involve only ZT for bulk, 2D, and 1D. Hint: Assume 휏0 = 3.24 × 10 푠 (r = 0) and find the optimum reduced Fermi energy of quantum wells and wires for the ZTs.

References

1. Hicks, L.D., T.C. Harman, and M.S. Dresselhaus, Use of quantum-well superlattices to obtain a high figure of merit from nonconventional thermoelectric materials. Applied Physics Letters, 1993. 63(23): p. 3230. 2. Hicks, L. and M. Dresselhaus, Effect of quantum-well structures on the thermoelectric figure of merit. Physical Review B, 1993. 47(19): p. 12727-12731. 3. Hicks, L. and M. Dresselhaus, Thermoelectric figure of merit of a one-dimensional conductor. Physical Review B, 1993. 47(24): p. 16631-16634. 4. Barnham, K. and D. Vvedensky, Low-dimensional semiconductors structures. 2001, Cambridge: Cambridge University Press. 5. Martín-González, M., O. Caballero-Calero, and P. Díaz-Chao, Nanoengineering thermoelectrics for 21st century: Energy harvesting and other trends in the field. Renewable and Sustainable Energy Reviews, 2013. 24: p. 288-305. 6. Hicks, L.D., The effect of quantum-well superlattices on the thermoelectric figure of merit, in DEpartment of Physics. 1996, Massachusetts Institute of Technology. p. 99. 7. Goldsmid, H.J., Thermoelectric Refrigeration. 1964, New York: Plenum Press. 240. 8. Hicks, L., et al., Experimental study of the effect of quantum-well Thermoelectrics figure of merit. Physical Review B, 1996. 53(16): p. R10 493-496. 9. Harman, T.C., D.L. Spears, and M.J. Manfra, High thermoelecrtic figures of merit in PbTe Quantum wells. Journal of Electronic Materials, 1996. 25(7): p. 1121-1127. 10. Casimir, H.B.G., Note on the conduction of heat in crystals. Physica, 1938. V(6): p. 495- 500. 11. Majumdar, A., Microscale heat conduction in dielectric thin films. Journal of Heat Transfer, 1993. 115: p. 7-16. 12. Chen, G., Size and interface effects on thermal conductivity of superlattices and periodic thin-film structures. Journal of Heat Transfer, 1997. 119: p. 220-229.