Tailoring of Seebeck coefficient with surface roughness effects in silicon sub-50-nm films
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Abstract
The effect of surface roughness on the Seebeck coefficient in the sub-50-nm scale silicon ultra thin films is investigated theoretically using nonequilibrium Green's function formalism. For systematic studies, the surface roughness is modelled by varying thickness periodically with square wave profile characterized by two parameters: amplitude (A_{0}) and wavelength (λ). Since high Seebeck coefficient is obtained if the temperature difference between the ends of device produces higher currents and higher induced voltages, we investigate how the generated current and induced voltage is affected with increasing A_{0} and λ. The theoretical investigations show that pseudoperiodicity of the device structure gives rise to two effects: firstly the threshold energy at which the transmission of current starts is shifted towards higher energy sides and secondly transmission spectra of current possess pseudobands and pseudogaps. The width of the pseudobands and their occupancies determine the total generated current. It is found that current decreases with increasing A_{0} but shows a complicated trend with λ. The trends of threshold energy determine the trends of Seebeck voltage with roughness parameters. The increase in threshold energy makes the current flow in higher energy levels. Thus, the Seebeck voltage, i.e. voltage required to nullify this current, increases. Increase in Seebeck voltage results in increase in Seebeck coefficient. We find that threshold energy increases with increasing A_{0} and frequency (1/λ). Hence, Seebeck voltage and Seebeck coefficient increase vice versa. It is observed that Seebeck coefficient is tuneable with surface roughness parameters.
Keywords
Threshold Energy Seebeck Coefficient Roughness Parameter Schrodinger Equation Cold Side1. Introduction
2. Theory
The expression for the current has been calculated using nonequilibrium Green's function formalism (NEGF) [14]. For the devices in the sub-50-nm scale quantum effects become important and quantum transport model is required for description of transport phenomenon. In this regards, NEGF formalism, in which quantum effects are inherent, provides an efficient framework to model the electron transport in thermoelectric devices. A brief description is given below and detailed development can be found in [15].
The real part of ∑'s just lifts up the energy eigenvalues of the channel. The second term that gets introduced in the Schrodinger equation is {S}, it is called the source term and is like an actual source driving the system.
A_{1}(E) and A_{2}(E) represent density of states due to coupling of the channel with contacts 1 and 2, respectively. The probability of filling of these states is determined by the Fermi Dirac distribution function, f^{1}(E) and f^{2}(E), of the contacts 1 and 2, which are at temperature T and T+ ΔT respectively.
where n_{ z } is the subband index.
3. Results and discussion
The above trends of I_{net} and V_{Seebeck} with the roughness A_{0} and 1/λ can be understood by examining the current flowing through each energy level I(E). As seen from Equation 7, I(E) is given as the product of transmission T(E) and $\left({f}_{1\mathrm{D}}^{\mathsf{\text{cold}}}\left(E\right)-{f}_{1\mathrm{D}}^{\mathsf{\text{hot}}}\left(E\right)\right)$, i.e. difference in the occupancy of cold ${f}_{1D}^{\mathsf{\text{cold}}}\left(E\right)$ and hot junctions ${f}_{1D}^{\mathsf{\text{hot}}}\left(E\right)$. The transmission T(E) gives the maximum current through the energy level 'E' and depends on the density of states A(E) and velocity, i.e. rate of flow of electrons in and out of that energy level. As such the geometry of the device does not affect the occupancy and hence $\left({f}_{1\mathrm{D}}^{\mathsf{\text{cold}}}\left(E\right)-{f}_{1\mathrm{D}}^{\mathsf{\text{hot}}}\left(E\right)\right)$ but it affects the transmission T(E) of electrons from energy level E through density of states A(E). Thus, the trends of I_{net} and V_{Seebeck} with A_{0} and λ can be understood by examining how the transmission gets affected by roughness.
3.1. Effect of roughness on transmission spectra
The presence of pseudobands and pseudogaps, a feature of only rough surfaces, is also due to alternate regions of low and high potential energy seen by the electron while crossing the channel. The energy eigenvalues for an electron in such a periodic potential energy profile are given by Kronig Penny model. The solution of the Schrodinger equation for such a potential energy profile shows the presence of energy bands and energy gaps. The investigations on the bandwidth (BW) of the pseudobands with respect to roughness parameters is important as the width of the pseudoband determines the area under the transmission curve and play an important role in determining the total current I_{net}. It is seen from Figure 4a, b that BW of the pseudobands decreases with increasing A_{0} and λ. These trends of BW on roughness parameters can be understood by looking, again, at the Kronig Penny model. According to Kronig Penny model, if the barrier heights and widths of the periodic potential profile are high, then the BW's are small. This is so because higher barrier heights and widths give smaller tunnelling probability. As the tunnelling probability becomes smaller the BW reduces. Along the same lines increasing A_{0}, which corresponds to higher barrier heights, and increasing λ, which corresponds to larger barrier widths of the pseudoperiodic potential profile, would give rise to smaller BW's. Thus BW decreases with increasing A_{0} and λ.
3.2. Effect of roughness on total current I_{net}
As I_{net} represents the current only due to temperature gradient with V_{applied} = 0, it is given by the intercept with the y-axis of I-V characteristics. It is observed from Figure 3a that I_{net} decreases with increasing amplitude but Figure 3b shows that there is no definite trend with λ. These features can be explained by considering that total current depends on the bandwidth of the pseudobands and their occupancy. For increasing A_{0} as seen from Figure 4b, threshold energy increases and BW of the pseudobands decreases. The smaller bandwidths along with their presence at higher energy sides result in the decrease in occupancy of these bands. This, in turn, decreases the total no. of carriers contributing to the current. Thus, I_{net} decreases with increasing A_{0}. The trends of I_{net} with λ are complicated. As seen from Figure 4a though the pseudobands are present at lower energy sides with increasing λ but their BW decreases. Thus, on one hand their presence at lower energy sides increases the occupancy on the other hand smaller bandwidths decrease the total occupancy which is obtained by summing the occupancy from all energy levels. These two competing features complicate the trends of I_{net} with λ. Figure 4a shows that BW for λ = 3.3 nm is smaller than λ = 2.5 nm but since the band is present at lower energy sides its occupancy is more. This results in more current for λ = 3.3 nm than for λ = 2.5 nm. For λ = 5 nm the BW is so small that the current is the least.
3.3. Effect of roughness on Seebeck voltage V_{Seebeck}and Seebeck coefficient
Variation of Seebeck coefficient for different roughness amplitudes
A_{0} (nm) | S(μV/K) |
---|---|
Smooth | 470 |
0.1 | 475 |
0.3 | 537 |
0.5 | 740 |
Variation of Seebeck coefficient for different roughness wavelengths
λ (nm) | S(μV/K) |
---|---|
Smooth | 470 |
5.0 | 545 |
3.3 | 646 |
2.5 | 740 |
For the devices if A_{0} is increased at constant λ it is observed that though I_{net} decreases but still V_{Seebeck} increases. This happens because BW decreases and threshold energy increases. Smaller BW's at higher energy sides reduce occupancy and hence I_{net}. Though the current reduces, it is made up of electrons present in higher energy levels and as already explained higher voltages are required to nullify the current from higher energy states.
The discussions show that it is the increase in threshold energy which causes the current to flow in the higher energy levels and hence results in increase in Seebeck coefficient. This result implies that any physical geometry whether periodic or aperiodic which results in an increase in threshold energy will show an increase in Seebeck coefficient. It also implies that if the threshold energy dependence on roughness parameters become feeble then Seebeck coefficient will saturate, i.e. it will not change much with the change in roughness parameters. The detailed investigations on the above two implications are in progress.
4. Conclusions
The surface roughness causes the net current I_{net}, produced due to the temperature gradient, to decrease with increasing roughness A_{0} but trends with λ are complicated. The voltage V_{Seebeck}, required to nullify this I_{net} shows a definite trend. It increases with increasing A_{0} and 1/λ. Increase in V_{Seebeck} results in increase in Seebeck coefficient. The reasons for the above trends have been attributed to the varying periodic thickness seen by the electron crossing the channel. The periodically varying thickness corresponds to varying confinement and hence periodically varying subband energy. The subband energy corresponds to the potential energy seen by the electron. For the electron moving in a pseudoperiodic potential, the transmission spectra shows the following features: (1) The threshold energy at which the transmission of current starts is shifted towards higher energy sides. It increases with increasing A_{0} and 1/λ. (2) There are regions of pseudobands and pseudogaps. The BW of the pseudobands decreases with increasing A_{0} and λ.
The above features of transmission spectra result in following trends for I_{net} and V_{Seebeck}: (1) It is observed that BW decreases and shift towards the higher energy sides for increasing A_{0}. Both these features result in reduced occupancy of the band and hence total carrier concentration. Since I_{net} depends on the total carrier concentration thus I_{net} reduces with increasing A_{0}. (2) It is observed that pseudobands are present at lower energy sides with increasing λ but their BW decreases. Thus, on one hand the presence of bands on the lower energy side increases the occupancy and on the other hand the smaller bandwidth reduces the total occupancy. These two competing features give rise to complicated trends of I_{net} with λ. (3) The trends of threshold energy determine the trends of V_{Seebeck} with roughness parameters. The increase in threshold energy results in the electrons to occupy higher energy states. Thus the net current I_{net} flowing from hot to cold end is made up from higher energy levels. To nullify this current, the occupancy of higher energy states at the cold end is increased by applying higher voltages. Thus higher threshold energy requires higher applied voltages, V_{Seebeck}, to nullify the current. Since threshold energy increases with increasing A_{0} and 1/λ, hence V_{Seebeck} and Seebeck coefficient increases vice versa. (4) The dependence of Seebeck coefficient on roughness parameters suggest that it can be tailored by choosing the appropriate roughness parameters.
Notes
Acknowledgements
SN acknowledges the financial support from Department of Atomic Energy, BRNS, Mumbai, India (grant no.2010/34/47/BRNS/2317). MK thanks Prof. Arun Kumar (IIT Delhi) for help and support during the preparation of manuscript.
Supplementary material
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