Subbands and quantum confinement

Transcription

1 Ballistic nanotransistors Modeled on analysis by Mark Lundstrom C, Purdue. Unless otherwise indicated, all images are his. nteresting hybrid of classical transistor behavior, quantum confinement effects, and Landauer-Buttiker type ideas. evice parameters: Length: L > ~ 10 nm to avoid contributions of direct source-drain tunneling. irst look at effects of confinement and see what we can neglect. Subbands and quantum confinement his problem is essentially something we ve done several times before. One can solve the Schroedinger equation for neutral, noninteracting particles confined in two dimensions but etended in the third. his is effectively a 1d quantum system, with 1d subbands separated in energy by spacings that look like those between low-lying levels of a d particle in a bo. As usual, we recall that one can consider the envelope function of Bloch waves, and arrive at an identical equation for the single-particle levels for non-interacting electrons. 1

2 Subbands and quantum confinement ndividual 1d subbands are significantly higher in energy than the floor of the conduction band. Much weaker dependence ~0 on transverse position, too. Subbands and quantum confinement Must worry about energy vs. position of individual subbands, not just conduction band as in usual s.

3 Complete solution n generality, one must self-consistently solve the electrostatic and quantum mechanical problems. enerally this is requires numerical solution, but it is possible to come up with analytical epressions in certain limits. irst, we ll look qualitatively at what happens as gate and drain bias are varied. hen we ll consider three regimes: >> 0 ~ 0 > 0 -- nondegenerate carriers -- degenerate carriers -- general case. Nonequilibrium velocity momentum distributions Remember our Landauer formula discussions? We worked in 1d and considered the chemical potential at different places along a device: contact 1 contact 1 lead 1 lead µ µ 1 µ 1 µ µ 1 µ µ µ 1 µ 1 µ µ µ 1 µ 1 µ µ µ 1 Can do same thing here, but plot velocity momentum distributions as a function of position:

4 Nonequilibrium velocity momentum distributions n source at top of barrier near drain in drain Plots for case of high gate voltage and high drain voltage, finite temperature. Nonequilibrium velocity momentum distributions Now fi gate bias. amine distributions as function of drain bias at top of barrier. At low, carriers pass over the barrier in both directions. As is increased, higher fraction of carriers getting over barrier are from the source. At sufficiently large, no carriers from drain reach top of barrier. urther increases in don t change the distribution of carriers at the top of the barrier! elocity saturation near source, without pinchoff! 4

5 elocity saturation without scattering urther increases in don t change the distribution of carriers at the top of the barrier! elocity saturation near source, without pinchoff! Carrier distribution at top of barrier varies with, but total density in good still determined by gate voltage. Can use these pictures to derive quantitative model of ballistic. > 0 case, nondegenerate carriers or electrostatically nice which we can check if we want by solving Poisson eqn. everywhere, charge density in inversion layer controlled by -, even at top of barrier: tot C n d nd, n, e d right-moving carriers from source left-moving carriers from drain As increases, n d- decreases. o maintain the equality, for the source effectively increases - more carriers come in from the source. position along channel; 0 defined as top of barrier. 5

6 > 0 case, nondegenerate carriers Net current is, similarly, given by an epression familiar from our Landauer picture: e elocity distribution of right moving carriers is hemi- Mawellian: kb v v f M πm p > 0, p y * Mawell-Boltzmann distribution. Same argument works for left-moving carriers, so their average speed is essentially identical to that of the right-movers. > 0 case, nondegenerate carriers Resulting current density: 0 v W end 0 v end W en 1 n 1 n 0 n 0 n tot d d dv d d 0 0 Ahh, but we can figure out the ratio n d n d- : n n d d N d N d ep kb ep e k B where N m* kb πh d ffective density of states 6

7 > 0 case, nondegenerate carriers Result for current density: W en 1 n 1 n 0 n 0 n tot d d dv d d e 1 e e k B tot W endv e k B Plugging in our epression for carrier density in a nice gives: e 1 e WC v e 1 e k B k B > 0 case, nondegenerate carriers - saturation here is saturation at high, because all current is determined by charge density at top of barrier, where effective velocity saturates out to the hemi-mawell mean velocity. Unlike the standard MOS, sat is independent of : e 1 e WC v e 1 e or >> dsat, k B k B WC v sat sat kb e 7

8 > 0 case, nondegenerate carriers linear regime We can epand e 1 e WC v e 1 e k B k B for small e k B to find linear regime behavior: v WC kb e v sat So, channel conductance WC kb e kb e µ Regular MOS has WC L Since regular MOS can never be better than ballistic case, µ kb < v Upper limit on mobility. L e > 0 case, nondegenerate carriers linear regime v WC k e Note that channel conductance is finite even for ballistic case, as in Landauer picture. Here, it s a direct consequence of the thermionic emission model used here when eamined at small bias. Left-moving current down from right-moving current by ep-e k B B 8

9 9 ~ 0, degenerate case, linear regime Still have e e e * * * h h π m π m ew m k ew Assuming hard transverse walls, transverse modes spaced by πw in k space, Result: Wk h e π his is the Landauer epression, with M, the number of channels, given by W k M π h e M ~ 0, degenerate case, saturated regime f transistor is on all the way, current is just : * m k ew π h or d gas one vert. subband, many transverse modes, all the right-moving carriers must be due to gate: e C k n tot d 4 π Plugging in, π q C m WC sat 8 * h

10 ~ 0, degenerate case, saturated regime enerically, for ballistic, dsat varies like - α. or nondegenerate case, α 1. or degenerate case, α. Saturation happens when pulls right contact ermi level below bottom of conduction band. his happens here: sat e C ν d eneral finite temperature results: efining the general ermi-irac integral of order s as: s d s η 0 ep η 1 and the normalized drain voltage: U k e and the normalized ermi energy: we find: ewc where: v ~ v ~ kb πm 1 1 η U 1 η U 0 1 η η * 0 η ε 1 k B 1 η η 0 B 10

11 eneral finite temperature results: Saturation regime: sat ewc v ~ Linear regime: WC ~ v 1 kb e 0 η η Summary: Quantum confinement effects strongly affect transmission in ballistic nanoscale MOSs. gnoring source-drain tunneling, velocity saturation happens near source at high. or good electrostatic design, result is current determined just by and source properties. Can derive analytic epressions under these conditions for nondegenerate, degenerate, or arbitrary conditions. Conductance near zero source-drain bias is still finite, even when device is ballistic. A melding of classical MOS theory and a Landauer way of thinking about such problems. 11

12 Net time: Coulomb Blockade physics Single-electron devices as successors to MOSs? 1