The information about how does Inet, i.e. the current due to the temperature gradient, and, VSeebeck, i.e. the voltage required to nullify it, depend on A0 and λ, is obtained by plotting the current voltage (I-V) characteristics for devices with different surface roughness parameters. Figure gives the I-V characteristics for (a) different amplitudes: A0's = 0.1 nm, 0.3 nm and 0.5 nm and λ = 2.5 nm and (b) different wavelengths: λ's = 2.5 nm, 3.3 nm and 5 nm and A0 = 0.5 nm. The plot for smooth surface is given in Figure for comparison. The temperature gradient of ΔT = 20 K is applied across the device. The graphs are obtained for the devices in the sub-50-nm scale by considering the parameters: length of the channel Lc = 10 nm and average thickness Tb = 2 nm. As Inet is the current generated only due to temperature gradient with Vapplied = 0, it is given by the intercept with the y-axis. It is observed from Figure that Inet decreases with increasing A0 but with λ there is no definite trend. VSeebeck is the voltage applied to make the current due to temperature gradient equal to zero and is obtained by the intercept with the x-axis. It is observed from Figure that VSeebeck increases with increasing A0 and 1/λ.
Figure 3 Current-voltage characteristics for the 2D Si structure. Lc = 10 nm, Tb = 2 nm, transport mass mx = 0.19 m0, quantization mass mz = 0.91 m0 and ΔT = 20 K. For (a) λ = 2.5 nm &A0's = 0.1 nm, 0.3 nm, 0.5 nm and for (b) A0 = 0.5 nm (more ...)
The above trends of Inet
with the roughness A0
can be understood by examining the current flowing through each energy level I
). As seen from Equation 7, I
) is given as the product of transmission T
, i.e. difference in the occupancy of cold
and hot junctions
. The transmission T
) gives the maximum current through the energy level 'E
' and depends on the density of states A
) 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
but it affects the transmission T
) of electrons from energy level E
through density of states A
). Thus, the trends of Inet
can be understood by examining how the transmission gets affected by roughness.
3.1. Effect of roughness on transmission spectra
Figure gives the plots of T
) for (a) roughness wavelength: λ
's = 2.5 nm, 3.3 nm and 5 nm with A0
= 0.5 nm and (b) roughness amplitude: A0
's = 0.1 nm, 0.3 nm and 0.5 nm with λ
= 2.5 nm. The following trends are observed: (i) threshold energy, i.e. the 'onset' energy at which the transmission starts, increases with increasing A0
. (ii) There are regions of psuedobands and pseudogaps. The width of the pseudobands decreases with increasing A0
. Within the pseudobands, there are peaks in transmission at some energy. The phenomenon of increase in threshold energy has also been observed experimentally in quantum wells with thickness < 4 nm [16
]. We would like to mention here that it has been observed only for thicknesses below 4 nm because surface roughness effects become prominent for ultra thin films. The criterion for the thickness below which surface roughness effects become prominent is discussed in [16
]. So, our discussions can be experimentally validated only for ultra thin films. We will analyse the implications of the above trends on Inet
but before that we discuss in brief the presence of the above trends.
Figure 4 Energy resolved transmission spectra at zero applied voltage. For (a) A0 = 0.5 nm &λ's = 2.5 nm, 3.3 nm, 5 nm and for (b) λ = 2.5 nm &A0's = 0.1 nm, 0.3 nm, 0.5 nm. Temperature gradient ΔT = 20 K for both (a, b) (more ...)
The increase in threshold energy occurs due to the varying thickness seen by the electron while crossing the channel. The regions of the channel where the thickness is large, the confinement is small and the regions where the thickness is small, the confinement is high as seen from Figure . Since the lowest value of εz, which corresponds to the subband energy, depends on the confinement it alternates with the same periodicity as the thickness as shown in Figure . The value of εz, i.e. subband energy is large for high confinement and vice versa. The subband energy corresponds to the potential energy seen by the electron. The electron crossing the channel experiences this varying potential energy landscape which was not the case when the surface was smooth. As the electron crossing the channel moves from the region of low potential energy to the high potential energy it feels an additional confinement effect which was not present for smooth surface. This additional confinement effect causes the increase in lowest energy which the electron can take. Thus, the threshold energy, which corresponds to the lowest energy eigenvalue, increases. The confinement effect will be more for increasing A0 as the electron while moving from low potential energy to high potential energy region sees higher barrier heights. Thus, the lowest energy eigenvalue, i.e. threshold energy, increases with increasing A0. The increase in threshold energy with increasing 1/λ can be explained on the similar lines. The longer λ with the same A0 corresponds to the wider regions of the low potential energy with the same barrier height. This means the confinement effect reduces with increasing λ. Hence, the threshold energy decreases with increasing λ. In other words threshold energy increases with increasing 1/λ.
Suband energy profile along the length of the device. Roughness amplitude (A0) = 0.5 nm and periodicity (λ) = 2.5 nm. The regions of channel at which confinement is high, subbands energies are high and vice versa.
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 Inet. It is seen from Figure that BW of the pseudobands decreases with increasing A0 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 A0, 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 A0 and λ.
3.2. Effect of roughness on total current Inet
As Inet represents the current only due to temperature gradient with Vapplied = 0, it is given by the intercept with the y-axis of I-V characteristics. It is observed from Figure that Inet decreases with increasing amplitude but Figure 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 A0 as seen from Figure , 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, Inet decreases with increasing A0. The trends of Inet with λ are complicated. As seen from Figure 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 Inet with λ. Figure 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 VSeebeck and Seebeck coefficient
As already mentioned VSeebeck
increases with increasing A0
. This trend can be understood by examining the circumstances for which Itotal
) becomes zero. Itotal
is the sum of current from all energy levels, i.e. I
. Figure gives I
) versus E
= 2.5 nm and A0
= 0.5 nm. It is seen that below Enull
the current is +ve and above Enull
the current is -ve. Hence, the total current obtained by the sum of current from all energy levels is small. We need to check when does it become zero. As seen from Equation (7), I
) is product of transmission T
, i.e. difference in the occupancy of cold
and hot junctions
. Since T
) ≥ 0 for all energies therefore current changes sign whenever
changes sign as seen from Figure . For those energy levels in which the number of electrons at the cold side are more than the number of electrons at the hot side, the net current flows from cold to hot and
is +ve and I
) is +ve. These are the energy levels below Enull
. It is vice versa for the energy levels above Enull
. At Enull
= 0. This Enull
is a function of Vapplied
as occupancies of energy states at cold side,
and hot side,
, depend on Vapplied
shifts towards higher energy sides with increasing Vapplied
. We apply voltage to shift Enull
to such an extent that total current coming from energy levels below Enull
balances the total current coming from energy levels above Enull
so that the total current becomes zero.
Figure 6 Energy resolved transmission, Fermi Dirac occupancy and current for λ = 2.5 nm and 3.3 nm with A0 = 0.5 nm. Energy resolved transmission is shown in (a, d), difference in the Fermi Dirac occupancy of cold and hot ends is shown in (b, e) and current (more ...)
To understand how does Enull
shifts towards higher energy states with increasing Vapplied
, we observe that as the threshold energy increases the electrons occupy higher energy states and the current contributions come from higher energy levels. The net current Inet
, which flows from hot side to cold side is also made of electrons flowing in higher energy levels. We need to apply voltage, VSeebeck
, to nullify this current Inet
. Thus, we need to increase the potential of the cold side so that electrons at the cold side occupy higher energy states to balance this current. Thus, as the threshold energy increases we need to apply higher VSeebeck
to balance the net current. Also, since the occupancy of higher energy states at the cold side increases with increasing VSeebeck
, for which
, shifts towards higher energy sides. Since the threshold energy increases with increasing A0
, hence Seebeck voltage VSeebeck
increases with the increasing roughness amplitude and frequency. Since Seebeck coefficient is obtained by taking the ratio of VSeebeck
with temperature difference ΔT
, i.e. S
, therefore the increase of VSeebeck
for fixed ΔT
causes the increase in Seebeck coefficient with the similar trends. It is seen from Tables and that Seebeck coefficient increases with increasing A0
. The dependence of Seebeck coefficient on roughness parameters suggest that it can be tailored by choosing the appropriate roughness parameters.
Variation of Seebeck coefficient for different roughness amplitudes
Variation of Seebeck coefficient for different roughness wavelengths
For the devices if A0 is increased at constant λ it is observed that though Inet decreases but still VSeebeck increases. This happens because BW decreases and threshold energy increases. Smaller BW's at higher energy sides reduce occupancy and hence Inet. 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.