16th International Workshop on Computational Electronics, June 4-7, 2013, Nara, Japan
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Open-Boundary-Condition Ballistic Quantum Transport using Empirical Pseudopotentials B. Fu and M. V. Fischetti Dept. of Materials Science and Engineering, UT Dallas, Richardson, Texas 75080, USA e-mail:
[email protected]
I NTRODUCTION
S IMULATION R ESULTS
Moore’s law aggressively pushes CMOS technology into the nanometer scale. Finite size and quantum confinement change the band structure, which deviates significantly from the bulk. Meanwhile, the study of electronic transport requires a quantum mechanical approach. Based our previous work on quantum transport using the Pauli Master Equation with effective mass [1], we developed an atomistic full band quantum transport model using empirical pseudopotentials to provide improved physical insight into novel nanodevices.
We simulate Si nanowires using local empirical pseudopotential parameters from Zunger [3]. These devices have a cross section of 1x1 cells (0.384 nm x 0.384 nm). Each unit cell, containing 9 Si atoms and 12 H atoms terminating the surface, is isolated by one cell of vacuum in the x-y plane. Figure 1 shows the band structure and the density of states. Injected and reflected states are chosen according to the complex band structure shown in Fig. 2. In order to reduce computational cost, in Fig. 3 we compare results obtained using different restricted subsets of reflected states, considering that states with a large imaginary component of their wavevector result in a small contribution to the charge. Keeping the cross section fixed, we have considered nanowires with length of 1, 3, 6, and 9 cells. As an illustration, we show in Fig. 4 a particular wavefunction in nanowire 3-cell long. For these devices, source, channel and drain each shares 1/3 of the total length. The charge density is set to 106 cm−1 in the S/D region and 100 cm−1 in the channel. In Fig. 5 we show the selfconsistent potential for a 9-cell long device and in Fig. 6 we compare the characteristics for devices of different length.
M ETHOD The Schr¨odinger equation with lattice potential V lat (r) and external potential V ext (r) can be written as [
−
] ¯ 2 ∇2 h + V lat (r) + V ext (r) ψ(r) = Eψ(r), (1) 2m
where the wave function ψ(r), in order to account for the effect of the spatially varying external potential, can be approximated by the envelop function ϕkG (r), ∑ ψk (r) = ϕkG (r)eiGr . (2) G
C ONCLUSION
Substituting Eq. (2) into Eq. (1), we have ∑ {[ G′
We have presented a nontrivial extension of QTBM to account for the full band structure and ] ¯h i¯h G ∇ ¯h G − − + + V ext (r) δGG′ improved the computational efficiency by ignoring 2m m 2m a subset of evanescent waves. For the application, } k k +VGG′ ϕG′ (r) = E(k)ϕG′ (r). we have shown simulation results of Si nanowires. R EFERENCES (3) 2
∇2
2
′
2
′
2
Equation (3) can be solved accounting for the open boundary conditions with a nontrivial extension of the Quantum Transmitting Boundary Method (QTBM) [2].
[1] B. Fu, and M.V. Fischetti, IWCE13 (2009) [2] C.S. Lent and D.J. Kirkner, Journal of Applied Physics 67, 6353 (1990). [3] S.B. Zhang, C. Yeh, and A. Zunger, Phys. Rev. B 48, 11204 (1993).
c 2013 Society for Micro- and Nanoelectronics 978-3-901578-26-7 ⃝
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4
2
Si [100] 2 Ry Zunger Size (x,y,z) =1×1×1 1 Vacuum Cell
ENERGY (eV)
ENERGY (eV)
4
0
−2 (a)
−4 0
0.5 kz (π/a0)
1
2
Si [100] 2 Ry Zunger Size (x,y,z) =1×1×1 1 Vacuum Cell
0
−2 (b) −4 0 1 2 DOS (109 eV−1 cm−1)
Fig. 1. Empirical pseudopotential band-structure (a) and density of states (b) for a [100] Si nanowire.
Fig. 4. Squared amplitude of a propagating wavefunction in a 3-cell long Si nanowire calculated using empirical pseudopotenitals.
POTENTIAL ENERGY (eV)
0.1
Fig. 2. Complex band structure of a Si [100] nanowire. Black crosses and red circles denote real and imaginary wavevectors, respectively, while the blue symbols denote fully complex wavevctors. The green planes show the projections of the dispersion onto the real and imaginare kz-planes.
4
∆ z=a /2
K
∆ z=a /18
K
2
Im(Kz) (π/a0)
|φ (z)|2
0.9
0
(a) 0.5 1 LENGTH (a0)
2.5
LR
K
RR
2 0
−2 0.8
0 0.05V 0.1V 0.15V 0.2V 0.25V 0.3V
1
Si [100] 2 Ry Zunger Size (x,y,z) =1×1×9 1 Vacuum Cell
2
3 4 5 6 LENGTH (a0)
7
8
9
Fig. 5. Self-consistent potential energy for a 9-cell Si nanowire as a function of the applied bias.
IDS (× 10−6 A)
0
1
−0.3 0
RI
0
0
−0.2
K
∆ z=a /6 ∆ z=a /36
−0.1
LI
0
1.1
0
−4
(b)
Size Size Size Size
(z) (z) (z) (z)
= = = =
1 3 6 9
1.5 1 0.5
−5 0 5 Re(Kz) (π/a0)
0 0
Si [100] 2 Ry Zunger Size (x,y) =1×1 1 Vacuum Cell
0.1
0.2
0.3
VDS (V)
Fig. 3. Evolvement of propagating waves (a) under different choices of real space discretization ∆z and reflected states kz (b) at an incoming energy E=-1.5V.
Fig. 6.
I-V characteristics of 1, 3, 6 and 9-cell Si nanowires.
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