Ballistic thermoelectric transport in structured nanowires

Thermoelectric (TE) devices are solid-state energy converters that can be used for power generation through the Seebeck effect and TE cooling through the Peltier effect. Nanostructures give great opportunities to engineer TE energy conversion efficiency. In this work, we investigate TE transport properties in structured nanowires (NWs) in the ballistic transport regime, where the NWs are bent, kinked, stubbed and segmented like a superlattice nanowire using the Green’s function method and the Landauer–Büttiker formula. A large Seebeck coefficient is found when the transmission gap appears due to the quantum interference effect of electrons. The sign of the Seebeck coefficient can be controlled by the geometries of these structured NWs. This finding is helpful for the design of nanoscale TE devices, such as thermocouple, with the same type of material doping rather than those comprised of n-type and p-type materials.


Introduction
Solid-state thermoelectric (TE) devices can be used for both power generation and refrigeration based on the two fundamental TE effects, the Seebeck effect and the Peltier effect [1,2]. The energy conversion efficiency of TE material is characterized by the dimensionless figure of merit (ZT): the interference between propagating electron states and confined electron states, i.e. the Fano resonance. This Fano resonance effect has been experimentally observed by Wu et al [27] in InAs NWs with diameter modulations. Zhou and Yang investigated the ballistic TE transport properties of double-bend NWs [7], where quantum confinement is not necessary for the strong energy dependence of transmission probability. The interference of electron wave in such structured NW is analogous to the interference of electromagnetic wave in a waveguide which can be explained as follows: the transmission probability is the modular square of the total transmission amplitude, which is the sum of transmission amplitudes associated with particular Feynman paths [25]. The Feynman paths are very sensitive to the geometry of NWs. If the transmission amplitudes are in phase, the transmission probability shows a peak. In contrast, if the transmission amplitudes are out of phase, the transmission probability shows a gap. This feature provides a pathway to control the transmission spectrum and TE transport properties by changing the geometry of NWs. Both above researchers pointed out that the Seebeck coefficient could be positive and negative without changing the doping type in NWs due to the ballistic transport. Here we investigate systematically the geometry effect on the ballistic TE transport in NWs with various structures such as bent [28], kinked [19], stubbed [29] and superlatticed [30] NWs, where such crystalline structured NWs have been experimentally synthesized. This paper is organized as follows. We present the tight-binding model and the Green's function method in section 2. The calculation results obtained from the Landauer-Büttiker formula for a variety of structured NWs are presented in section 3. Section 4 concludes the paper.

Model
The tight-binding approach is used to describe the electrons in NWs. The Hamiltonian H with the nearest-neighboring approximation is written as: where m and n denote discrete lattice sites that are located at (x, y) = (ma, na) ≡ [m, n] with m, n being integers, and a standing for the lattice constant of grid. There is no variable z in this Hamiltonian since only the lowest subband in the z-direction is considered in our model. ε m n , denotes the on-site energy with , where a hard wall boundary condition is applied by choosing = V m n ( , ) 0 inside the NW and → ∞ V m n ( , ) outside the NW. t is the hopping energy between the nearest-neighboring sites with = −ℏ * t m a /2 2 2 , where * m is the effective mass of electrons and ℏ is the Planck constant. + a m n , and a m n , are the creation and annihilation operators of electron on site [m, n], respectively.
In order to calculate the TE transport properties in the ballistic regime, the Green's function method is used, which is written as where + G is the retarded Green's function, E is the electron energy, I is identity matrix, Σ + L and Σ + R are the self-energy functions due to attached left and right leads, respectively. When the NW is long, the computation could be very costly since inverting a large matrix as shown in equation (3) requires tremendously large memory. We use the recursive Green's function method [31], which divides the NW layer by layer along the transport direction, to address this computational challenge.
The total transmission probability T E ( ) tot of electron can then be calculated using obtained Green's function as [25] Γ Γ = taking the imaginary part of self-energy functions, − G is the advanced Green's function of the NW.
The electrical conductance G and the Seebeck coefficient S are calculated using the Landauer-Büttiker formalism [25,32,33]: where e represents the absolute value of the unit electron charge, k B is the Boltzmann constant, μ is the chemical potential and f is the Fermi-Dirac distribution function, respectively.

Results and discussions
We perform numerical calculations for the bent, kinked, stubbed and superlatticed NWs as shown in figures 1(a)-(d) to study their TE properties along the x-direction, when the electron transport is in the ballistic regime. The y-direction is confined by a hard wall, which determines various geometries. In our study, only the lowest subband along the z-direction is considered. The NWs are connected to two electron reservoirs through two ideal leads with perfect Ohmic contact. The parameters used in calculation are chosen as follows: m* = 0.067 m e , which is the effective mass of conduction band in GaAs, where m e is the free electron mass, and a = 1.906 nm, which results in a hopping energy t = −1/4 eV. In order to simplify the discussion, we fix the width W of NWs to be 7a throughout the paper. All the calculations are performed when temperature is at T = 10 K, which ensures the electron transport is in the ballistic regime. Higher temperatures, such as room temperature, would smear the variation of transmission probability [7], which makes the ballistic effect on TE transport properties difficult to observe [25]. The main objective of this paper is the geometry effect. Therefore, the temperature dependences of the electrical conductance, the Seebeck coefficient and the power factor are not presented here while interested readers are referred to [7]. Figure 1(a) shows the schematics of a bent NW along the x-direction while the transverse direction is confined by hard walls. The bend angle is θ, the width is W and the inter-bend length is h. Figure 2(a) shows the transmission probability T tot plotted as a function of electron energy E in bent NWs with bend angle θ varying from 0°to 90°when h = 18a. Oscillation of transmission probability with multiple gaps occurs when the bend angle increases. These transmission gaps are the results of anti-resonance of electron wave near the bent areas [7]. The variation of transmission probability changes slightly when the bend angle varies from 0°to 30°. When the bend angle increases to beyond 30°, the transmission gaps become steeper and more significant, which indicates a much stronger quantum interference in bent NW with sharper bend angle. Figures 2(b)-(d) show the electrical conductance, the Seebeck coefficient and the power factor varying with chemical potential μ for different bend angles when h = 18a. Similar to the transmission probability, the electrical conductance also exhibits oscillating behavior, which becomes more significant with the increase of bend angle as shown in figure 2(b). In figure 2(c), the sign change of the Seebeck coefficient is observed when the bend angle increases. The maximum value of the absolute Seebeck coefficient, denoted as |S| max , also increases as the bend angle increases, since the transmission probability varies more sharply for larger bend angle as shown in figure 2(a). Typical |S| max is larger than 200 μV K −1 when the bend angle is over 80°. Figure 2(d) shows the peaks of power factor obtained when the chemical potential is close to the edge of the transmission gap. The power factor reaches about 2 k 2 B /h when the bend angle is 80°. Figure 3(a) shows the transmission probability T tot versus electron energy E in bent NWs for different h when bend angle is 80°. Larger h results in more transmission peaks and gaps. There are two transmission gaps when h = 14a, three gaps when h = 18a and four gaps when h = 22a. The reason is that the transmission probability would be close to 1 when resonance occurs once an integer multiple of electron wavelength matches h. The transmission probability Schematic diagrams of NWs of width W connected to electron reservoirs through two ideal leads with perfect Ohmic contact. (a) Bent NW whose bend angle is θ and inter-bend length is h. (b) Kinked NW whose bend angle is θ, length of each serration is 2L, and the number of periods is p (p = 2 as an example in the figure). (c) Stubbed NW whose length is L, and the height and width of the stub part are h and W. (d) Superlatticed NW with periodic potential modulations whose barrier height is u. The width of potential barrier along the x-direction is A, the distance between adjacent potential barriers is B and the number of periodic structure is N (N = 3 as an example in the figure).

Bent NWs
reaches its minimum when anti-resonance occurs once a half-integer multiple of electron wavelength matches h. The Seebeck coefficient describes the average entropy carried by each charge carrier and is approximately proportional to the slope of transmission probability at low temperature [7]. The resonance and anti-resonance enable us to control both the sign and the amplitude of the Seebeck coefficient by modulating the inter-bend length h.
show the electrical conductance, the Seebeck coefficient and the power factor as a function of chemical potential μ for different bend lengths h when the bend angle is 80°. Figure 3(b) shows that there are more gaps of electrical conductance when h increases. This is a consequence of the appearance of more transmission gaps. Moreover, according to equation (5b), a large positive Seebeck coefficient can be found when the chemical potential is close to the lower edge of the transmission gap since only the electrons whose energies are smaller than the chemical potential ( μ − < E 0) contribute to transport. Similarly, large negative Seebeck coefficient could be found when the chemical potential is close to the upper edge of the transmission gap since only the electrons whose energies are larger than the chemical potential ( μ − > E 0) contribute to transport. In figure 3(c), the sign of the Seebeck coefficient changes more frequently when chemical potential changes with the increase of h. The largest |S| max of 250 μV K −1 is obtained when h = 18a. In figure 3(d), the highest peak of power factor appears when μ ∼ 0.14 eV and h = 18a. The variation of the transmission probability and the sign change of the Seebeck coefficient due to the geometric interference obtained above are similar to the TE transport in molecules [34][35][36], which is largely determined by the electronic structure of the molecule. Similar behavior can also be found for other NWs studied in this paper. Figure 1(b) shows the schematics of a kinked NW. The width of NW is W, the length of each serration is 2L and the bend angle is θ. This kinked NW is a periodic structure with the number of periods p. We fix W = 7a and L = 18a to focus on the influences of the bend angle θ and the number of periods p. Figure 4(a) shows the transmission probability T tot as a function of electron energy E in kinked NWs with bend angles 30°, 45°and 60°when there is only one period, i.e. p = 1. A transmission gap is obtained when the bend angle is 30°because of anti-resonance of electron wave at bend areas. When the bend angle in kinked NW becomes large, the electron transmission probability decreases rapidly since more wave function is reflected back due to stronger interference. Figures 4(b)-(d) respectively show the electrical conductance, the Seebeck coefficient and the power factor as a function of chemical potential μ for different bend angles when p = 1. Figure 4(b) shows that the electrical conductance has a similar gap as transmission probability showed in figure 4(a). In figure 4(c), we find a sign change of the Seebeck coefficient in kinked NW. The largest |S| max of 160 μV K −1 is obtained when the bend angle is 60°in comparison with 30°and 45°. However, the power factor shown in figure 4(d) is not a large value around μ = 0.06 eV and θ = 60°even though the corresponding Seebeck coefficient is maximized. The maximum power factor reaches 1 k 2 B /h when bend angle is 30°at μ = 0.11 eV. Figure 5(a) shows the transmission probability T tot plotted as a function of electron energy E in kinked NWs for different p with fixed bend angle θ = 45°. Figure 5(a) shows that a larger p leads to more transmission gaps. As p increases, the transmission gaps become deeper and fine structures of transmission probability appear. When E = 0.08 eV, there is a major transmission gap for all three cases. Figures 5(b)-(d) respectively show the electrical conductance, the Seebeck coefficient and the power factor as a function of chemical potential μ for different p with bend angle 45°. Figure 5(b) shows that there are minimum electrical conductance when μ ∼ 0.08 eV for all three cases. This is a consequence of transmission gaps near E ∼ 0.08 eV as shown in figure 5(a). In figure 5(c), when p increases, |S| max is enhanced remarkably when p > 1 because of the fine structures of the transmission probability. |S| max reaches 600 μV K −1 when μ ∼ 0.08 eV. There are multiple peaks of power factor as shown in figure 5(d) due to the fine structures of the transmission probability. It is found that the power factor is not maximized even though the Seebeck coefficient reaches maximum values when μ = 0.08 eV. This is mainly because the electrical conductance decreases. After all, kinked structure enables us to obtain a large absolute Seebeck coefficient with reasonable large power factor. Figure 1(c) shows the schematics of a stubbed NW. The height of stub is h and the width is W, which is the same as the width of NW. Figure 6(a) shows the transmission probability T tot plotted versus electron energy E in stubbed NWs for different h, which demonstrates resonant behavior. Moreover, the existence of transmission gaps means that incident electrons with certain energy can be reflected back due to the anti-resonance occuring due to the stub. When the height increases, the number of transmission gaps increases since more anti-resonance modes appear: there are two transmission gaps when h = 5a, three transmission gaps when h = 9a and four transmission gaps when h = 14a. Figures 6(b)-(d) respectively show the electrical conductance, the Seebeck coefficient and the power factor as a function of chemical potential μ for different heights of stub h. Figure 6(b) shows the gaps of conductance when the chemical potential approaches the transmission gap. The larger h is, the more gaps appear. In figure 6(c), two extremes of the Seebeck coefficient are obtained when chemical potential is close to the edge of the transmission gap. The positions of the extremes of the Seebeck coefficient are symmetric to the center of the transmission gap. The peak of the positive Seebeck coefficient at low energy is close to the lower edge of the transmission gap. The gap of the negative Seebeck coefficient at high energy is close to the upper edge of the transmission gap. |S| max is larger than 140 μV K −1 when h = 5a. With increasing h, more peaks and gaps of the Seebeck coefficient that corresponds to the transmission gap can be obtained: there are two peaks and gaps when h = 5a, three peaks and gaps when h = 9a, and four peaks and gaps when h = 14a. In figure 4(d), the peaks of the power factor could be larger than 1 k 2 B /h. Each peak is related to an extreme of the Seebeck coefficient. It is not surprising that more peaks show up when h is 9a and 14a because of the existence of more anti-resonance modes.

Superlatticed NWs with periodic potential modulations
There has been great interest in growing superlatticed NWs for TE applications. For example, NWs with multiple heterostructures have been synthesized in Bi 2 Te 3 based materials [37][38][39]. Wu et al fabricated Si/SiGe superlatticed NWs [30]. Theoretically, Lin and Dresselhaus [21] have calculated the diffusive TE transport properties of superlatticed NW using the Boltzmann transport equation. TE transport properties in NWs with several quantum dots serially embedded have been studied in the Coulomb blockade regime [40][41][42]. The Coulomb interaction was found to be important. Here, we study the ballistic TE transport in superlatticed NWs that differs from the above theoretical studies. We focus on the transmission gap induced by Bragg reflection when the propagating wavelength of electron and the period length of NW satisfy Bragg's law. Figure 1(d) shows the schematics of a superlatticed NW along the x-direction while the y-direction is confined by hard walls. The superlatticed NW is modulated by a periodic potential, which can be written as V(x): where n = 0, 1, 2 … N, N is the number of periods of modulation, u is the height of potential barrier, A is the width of the potential barrier and B is the distance between two adjacent potential barriers. The total length of NW along the x-direction is denoted as N(A + B). Figure 7(a) shows the total transmission probability T tot as a function of electron energy E in superlatticed NWs with different numbers of modulations N = 5, 10 and 20 when A = B = 5a and u = 0.01 eV. The transmission probability exhibits a gap near E ∼ 0.07 eV. An increasing number of potential barriers (larger N) leads to a deeper transmission gap. This effect comes from the well-known Bragg scattering in periodic potential [43]. Larger N implies stronger Bragg scattering. Figures 7(b)-(d) respectively show the electrical conductance, the Seebeck coefficient and the power factor as a function of chemical potential μ for different numbers of potential barriers when A = B = 5a and u = 0.01 eV. Figure 7(b) shows that the electrical conductance decreases drastically when the chemical potential gets closer to the transmission gap. Therefore, there is a conductance gap that corresponds to the transmission gap for all three cases. Figure 7(c) shows that there are two extremes of the Seebeck coefficient when the chemical potential is close to the transmission gap, one extreme is positive when chemical potential is smaller than the energy of transmission gap and the other is negative when the chemical potential is larger than the energy of transmission gap. |S| max becomes larger as N increases, since the gap of transmission probability becomes deeper. |S| max reaches a maximum 250 μV K −1 when N = 20. At the same time, the peak values of power factor become larger with N increasing as shown in figure 7(d). The largest power factor could reach 1.8 k 2 B /h. Figure 8(a) shows the total transmission probability T tot plotted as a function of electron energy E in superlatticed NWs with different widths of potential barriers when N = 20 and u = 0.01 eV. The transmission gap shifts from E ∼ 0.08 eV to 0.068 eV and 0.06 eV when the width of potential barrier increases from A = B = 4a to 5a and 6a, respectively. This is analogous to the shift of band gap due to the energy band theory. Figures 8(b)-(d) respectively show the electrical conductance, the Seebeck coefficient and the power factor as a function of chemical potential μ. In figure 8(b), the gap of electrical conductance shifts along with the transmission gap. In figure 8(c), the chemical potential corresponding to extreme values shifts to lower energy with increasing width of the potential barriers. |S| max is as large as 250 μV K −1 . The peaks of power factors shift to lower chemical potentials while the peak values change little, as shown in figure 8(d). Figure 9(a) shows the total transmission probability T tot plotted as a function of electron energy E in superlatticed NWs with different heights of potential barriers when N = 20 and A = B = 5a. The width of transmission gap increases with increasing barrier height because of stronger modulation effect. Figures 9(b)-(d) respectively show the electrical conductance, the Seebeck coefficient and the power factor as a function of chemical potential μ. The gap of electrical conductance shown in figure 9(b) becomes wider when u increases, due to the widening transmission gap as shown in figure 9(a). Figure 9(c) shows that |S| max increases as u increases. When = u 0.005eV, |S| max is about 100 μV K −1 , when = u 0.01eV, |S| max increases to about 300 μV K −1 and when = u 0.015eV, |S| max is about 400 μV K −1 . This is because the edge of the transmission gap grows steeper with increasing u. Figure 9(d) shows the properties of the power factor for different u. When u = 0.005 eV, the maximum value of the power factor is about 1.10 k 2 B /h at μ = 0.06 eV with the positive Seebeck coefficient (100 μV K −1 ), and reaches 1.14 k 2 B /h at μ = 0.067 eV with the negative Seebeck coefficient (−98 μV K −1 ). The largest power factors exist when u = 0.01 eV rather than u = 0.015 eV, even though the Seebeck coefficient for the latter case is much larger than the former one. This is because when the energy gap in transmission probability becomes wide enough, the reduction of electrical conductance is more significant than the increase of Seebeck coefficient.

Conclusion
We have investigated the TE transport properties of structured NWs in the ballistic regime with special geometries such as bent, kinked, stubbed and superlatticed NWs by using the Green's function method and the Landauer-Büttiker formula. In these structured NWs, one can tune the sign and the amplitude of Seebeck coefficient by controlling the structure parameters such as bend angle, inter-bend length, stub height and potential barrier height. A large Seebeck coefficient and power factor can be obtained in these structured NWs by utilizing the transmission gap induced by geometry-related quantum interference of electron wave. Our study provides a promising way to design nanoscale TE devices with the same type of material in which both positive and negative Seebeck coefficients can be obtained, which differs from the traditional TE devices that consist of both p-type and n-type materials.