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 Ez (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 2k2 2 k2 k2 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 ky my , which leads to ky ky my m m In the strictly two-dimensional system, we have m m m m (14.10) dk dk dk x y dkdk x y 2kdk x y m x y m 14-6 The area of the smallest wavevector in a 2D crystal is 2 L2 . The number of states between k and k + dk in the perfect two-dimensional space is then obtained as (14.11) 2 2k 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) 2m2 E 2 k Differentiating this gives 1 1 (14.14) dk 2m2 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 6.6261034 Js 2 12 2 E 2 0.149eV 20.07 9.10931031 kg601010 m 1.60211019 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.