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current topics in solid state physics
Wigner ensemble Monte-Carlo simulation of nano-MOSFETs in degenerate conditions D. Querlioz1, J. Saint-Martin1, V.-N. Do1,2, A. Bournel1, and P. Dollfus1 1 2
Institut d’Electronique Fondamentale, Université Paris-Sud, CNRS, 91405 Orsay, France Theoretical Department, Institute of Physics, VAST, P.O. Box 429 Bo Ho, Hanoi 10000, Vietnam
Received 9 July 2007, revised 22 August 2007, accepted 22 August 2007 Published online 23 October 2007 PACS 02.70.Uu, 05.60.Gg, 73.23.-b, 85.30.De, 85.30.Tv Wigner quasi-distribution function is an appropriate quantum mechanics formulation to study the transition from semiclassical to quantum transport in nano-devices since it can accurately describe quantum transport including the decoherence due to scatterings. We have recently developed an efficient approach to solving the Wigner transport equation using a Monte Carlo (MC) algorithm that has been applied to Reso-
nant Tunnelling Diodes and nano-MOSFET simulation. The approach is here extended to incorporate degeneracy effects that are important in highly doped MOSFETs. The calculation is compared with Non Equilibrium Green’s Function and the semi-classical Boltzmann equation. Relative importance of quantum transport and decoherent scattering is discussed at low and room temperatures.
phys. stat. sol. (c) 5, No. 1, 150 – 153 (2008) / DOI 10.1002/pssc.200776565
T N I R P E R
solidi
status
pss
physica
phys. stat. sol. (c) 5, No. 1, 150– 153 (2008) / DOI 10.1002/pssc.200776565
c
www.pss-c.com
current topics in solid state physics
Wigner ensemble Monte-Carlo simulation of nano-MOSFETs in degenerate conditions D. Querlioz*,1, J. Saint-Martin1, V.-N. Do1,2, A. Bournel1, and P. Dollfus1 1 2
Institut d’Electronique Fondamentale, Université Paris-Sud, CNRS, 91405 Orsay, France Theoretical Department, Institute of Physics, VAST, P.O. Box 429 Bo Ho, Hanoi 10000, Vietnam
Received 9 July 2007, revised 22 August 2007, accepted 22 August 2007 Published online 23 October 2007 PACS 02.70.Uu, 05.60.Gg, 73.23.-b, 85.30.De, 85.30.Tv *
Corresponding author: e-mail
[email protected], Phone: +33 1 69 15 41 05, Fax: +33 1 69 15 40 20
Wigner quasi-distribution function is an appropriate quantum mechanics formulation to study the transition from semiclassical to quantum transport in nano-devices since it can accurately describe quantum transport including the decoherence due to scatterings. We have recently developed an efficient approach to solving the Wigner transport equation using a Monte Carlo (MC) algorithm that has been applied to Reso-
nant Tunnelling Diodes and nano-MOSFET simulation. The approach is here extended to incorporate degeneracy effects that are important in highly doped MOSFETs. The calculation is compared with Non Equilibrium Green’s Function and the semi-classical Boltzmann equation. Relative importance of quantum transport and decoherent scattering is discussed at low and room temperatures.
© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
1 Introduction The occurrence of quantum transport effects has often been considered as a fundamental limit to the miniaturization of Metal Oxide Semiconductor Field Effect Transistors (MOSFETs), the key foundation of modern integrated electronics. In contradiction with these predictions, research labs now demonstrate functional transistors with gate lengths smaller than 10 nm [1–4]. The quantum character of transport is known to play an important role in such short MOSFETs. Understanding their physics is however difficult since carriers in these devices still experience many decoherent scatterings (due e.g. to phonons and interfaces roughness) that disrupt their quantum behavior. Non Equilibrium Green’s Functions (NEGF) from the many body physics are often used to understand the physics of devices operating in the quantum regime [5–10]. However it is difficult to include realistic scattering models in the NEGF formalism due to theoretical and computational difficulties. In particular, NEGF modeling cannot readily use all the work that has been accomplished to model scatterings in classical electron devices. The Wigner quasi-distribution function (WF) can provide an original alternative to NEGF. WF is defined as the center-of-mass Fourier transform of the density matrix of the carriers, and is most interesting because it shares many
similarities with a distribution function, although it is not positive-definite [11]. In fact it tends to the Boltzmann’s equation distribution function in a semi-classical transport situation. Besides, scattering effects can be incorporated into a WF calculation in an approach similar to the Boltzmann’s equation. WF thus provides a rich formalism to study the transition from semi-classical to quantum transport in nano-scaled electron devices. Recently, we have developed an efficient approach to solving the Wigner transport equation in the presence of scattering using a Monte Carlo (MC) algorithm, inspired by [12] and that has been applied to Resonant Tunneling Diodes [13] and nano-MOSFET simulation [14, 15]. In this paper, we extend our approach to incorporate degeneracy effects that are important in highly doped MOSFETs. 2 Theory 2.1 Fundamental equations We study a Double Gate nano-MOSFET inspired by recent technological realizations like [1, 4], and presented in Fig. 1. The silicon body constitutes a 2-D Electron Gas (2DEG) thus leading to different conduction channels (subbands). In our approach, the transport of carriers from the source (S) to drain (D) is described by a Wigner function fw , determined © 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
Original Paper phys. stat. sol. (c) 5, No. 1 (2008)
151
for each individual subband. The motion equation of fw (Wigner Transport Equation, WTE) reads:
∂ fw k ∂ fw + = Q fw + C fw , ∂t m* ∂x
1 2π
∫
d k ' Vw ( x, k ') f w ( x, k + k ')
∫
G
x
EOT = 1 nm TSi = 3 nm
S
ND = Undoped 1020 cm-3 O2 EOT =Si1nm
(2)
G
ND = 1020 cm-3
Quantization axis
D
z
LG = 6nm
where Vw ( x, k ) =
0
(1)
where Q f w is the quantum evolution term resulting from the non-local effect of potential: Q f w ( x, k ) =
Wigner transport
⎡ ⎛ x'⎞ x ' ⎞⎤ ⎛ d x ' sin ( k x ') ⎢ En ⎜ x + ⎟ − En ⎜ x − ⎟ ⎥ (3) 2⎠ 2 ⎠⎦ ⎝ ⎣ ⎝
Figure 1 Simulated Double Gate MOSFET structure. Highly doped Source and Drain regions are 15 nm long.
referred as the Wigner potential, is calculated according to the profile of the sub-bands En ( x ) . The term C f w in Eq. (1) encodes the effect of electron scattering (typically by phonons, impurities or oxide roughness) on the Wigner function. In a one-particle approach, and within an instantaneous scattering approach (that neglects quantum collision effects like collision broadening and intra-collisional field effect [16]), the collision Hamiltonian leads to the standard collision term of the Boltzmann equation of semi-classical transport [17] C f w ( x, k ) =
∑∫ i
⎡ ⎢ d k' f w ( x, k' ) Si ( k', k ) − ⎣
∫
⎤ d k' f w ( x, k ) Si ( k , k ' ) ⎥ (4) ⎦
(i refers to the type of scattering mechanism and the Si are the associated scattering rates). This term, derived in a oneparticle approach, does not satisfy the Pauli exclusion principle. It is thus only valid in non-degenerate conditions. To take Pauli exclusion into account, instead of Eq. (4), we use a generalized term C f w ( x, k ) =
∑ ⎡⎢⎣∫ d k' f ( x, k') (1 − f ( x, k )) S ( k', k ) . w
w
i
i
(5)
∫
⎤ − d k' f w ( x, k ) (1 − f w ( x, k ') ) Si ( k , k ') ⎥ ⎦
A derivation of a similar term using many body Green’s functions can be found in Ref. [18]. This approach to the exclusion principle is not general and is valid in situations that are not too far from semi-classical transport, meaning that the Green’s function G<(x,k;E) has non zero values only around E ∼ En ( x ) + k 2 2m* . This is believed to be accurate in nano-MOSFETs, as illustrated in the observation of Green’s function in Ref. [19]. All the work on scattering mechanisms developed for traditional electron devices modeling can be adapted to this approach. This is a major advantage of Wigner’s formalism. Intra- and inter-subband scatterings by phonons, ionized impurities and surface roughness are thus considered in this work as in Refs. [20, 21]. www.pss-c.com
Figure 2 Sub-band profiles and cartography of the squared transverse wave-functions associated with the sub-bands.
2.2 Solution To solve the WTE we use a particle Monte-Carlo interpretation inspired by [12] and described in [13, 14]. The Wigner function is seen as a sum of “pseudo-particles” localized in both x and k space and weighted by a parameter called affinity. The affinity contains the quantum information on the particles and can take negative values. Such pseudo-particles have no direct physical meaning, but constitute a very ingenious way to solve the WTE: their x and k coordinates evolve exactly as that of classical particles, while their affinity is updated according to the Qfw term of the WTE. Scatterings are included using the standard MC algorithm, which constitutes the strength of the method. To treat Pauli exclusion, as mentioned above, we use an approach inspired by previous works on semi-classical MC simulation [22] where the scattering rates are simply multiplied by 1-f. 3 Results and discussion The simulated structure is presented in Fig. 1. Compared to the structure studied in Ref. [14], this one has higher doping in the source and drain regions. The inclusion of the Pauli principle enables us to study its operation in a large range of temperatures. WTE, Poisson’s equation and lateral Schrödinger’s equations (to compute the subbands) are solved selfconsistently, as described in [14]. Examples of subband profiles, along which the transport takes place, are plotted in Fig. 2, together with cartographies of the squared lateral wave function of each type of subbands considered in the simulation. The fact that the lateral wave-functions do not
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D. Querlioz et al.: Wigner ensemble Monte-Carlo simulation of nano-MOSFETs
Drain current ID (µA/µm)
Drain current ID (µA/µm)
evolve strongly along the device justifies the 2DEG approach [23]. 104 1000
T = 300 K VDS = 0.7 V
a)
100 10 Ballistic Green Wigner Semi-classical
1 0.1 -0.2
0
0.2
0.4
0.6
Gate voltage VGS (V)
0.8
104 1000
T = 77 K VDS = 0.7 V
b)
100 10 Ballistic Green Wigner Semi-classical
1 0.1 -0.2
0
0.2
0.4
0.6
fact that the subthreshold slope is not proportional to the temperature in nanoMOSFETs is well supported by experimental data [4, 24] and can be considered as a distinctive feature of source-to-drain tunneling. Additionally source-todrain tunneling directly causes the threshold voltage shift between semi-classical and quantum transport calculations. Figures 4(a) and 4(b) compare the Wigner function fw in the subthreshold regime with the distribution function from the semi-classical calculation at the same bias. Generally, both functions exhibit the same form with two high value regions in the phase space which correspond to electrons in the source and drain regions. Besides, a long mouse taillike curve is seen in the channel which describes the flow of ballistic carriers from S to D. However, while the distribution function remains positive-definite, there exist small domains around the tail in Fig. 4(a) where the WF oscillates and assumes negative values. This illustrates the impact of source-to-drain tunneling and emphasizes that the Wigner function cannot be interpreted as a probability function in the presence of quantum transport, consistently with Heisenberg uncertainty principle [11, 14].
a)
0.8
Gate voltage VGS (V) Figure 3 a) Room temperature (300 K) and b) low temperature (77 K) drain current ID versus gate voltage VGS using Wigner, Boltzmann and ballistic Green’s function simulations, at high VDS.
Figures 3(a) and 3(b) show the drain current versus gates voltage at 300 K and 77 K. They are compared with that obtained from semi-classical Boltzmann MC simulation including multi-subband description to emphasize the impact of quantum transport. They are also compared with the currents resulting from a ballistic Green’s function calculation (quantum transport model with no scattering) to emphasize the impact of scattering. All three models use an effective mass approach to the electronic band structure. In the subthreshold regime (low VGS) the results based on the Wigner’s function are very close to the ballistic Green’s function ones. However, the currents are significantly higher than that obtained using semi-classical simulation, especially at 77 K. This shows that scattering has a weak impact in this regime and that the subthreshold current is strongly enhanced by direct source-to-drain tunnelling through the high potential barrier in the channel [14]. We notice that in the semi-classical transport approach, the subthreshold slope S = ⎡ ∂ log10 ( I D I 0 ) ∂VGS ⎤ −1 is approximately ⎣
⎦
proportional to the temperature (S = 100 mV/dec. at 300 K vs. 30 mV/dec. at 77 K), whereas it is weakly temperaturedependent in the quantum transport approaches (S = 120 mV/dec. at 300 K vs. 110 mV/dec. at 77 K). The
© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
b)
Figure 4 a) Cartography of the Wigner function of the first unprimed subband in the sub-threshold regime (VGS = 0.25 V, VDS = 0.7 V) at T = 77K. b) Distribution function from a semiclassical calculation at same bias and temperature.
Above the threshold (high VGS), the behavior is very different. The ballistic approach strongly overestimates the current due to the efficiency of scattering and the Wigner current become similar to the current resulting from the semiclassical calculation. Contrary to the case of Refs. [14, 15], the Wigner current is only slightly smaller than the semiclassical current at high VGS. The quantum reflection effect due to the strong drop of the potential profile that causes this current reduction thus does not significantly affect the device characteristics. Due to high doping in the access regions
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Original Paper phys. stat. sol. (c) 5, No. 1 (2008)
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the barrier at the entrance of the channel is higher than in [14, 506844) and IP PULLNANO (IST 4-026828), and by the French 15] so that tunneling through the barrier remains the main ANR, through project MODERN. quantum transport effect at high VGS. References Figure 5 examines in more detail the effects of scattering [1] H. Lee, L. E. Yu, S. W. Ryu et al., in: Symposium on VLSI above the threshold. Drain current versus gate voltage is Technology, Digest of Technical Papers, p. 58 (2006). [2] F. L. Yang, D. H. Lee, H. Y. Chen et al., in: VLSI Technolplotted in linear scale. Ballistic Green’s calculation and ogy, Dig. Tech. Papers, p. 196 (2004). Wigner’s calculation are plotted, as well as a Wigner’s cal[3] B. Yu, L. Chang, S. Ahmed et al., in: IEDM Tech. Dig., p. culation where scattering has been artificially deactivated in 251 (2002). the channel, but not in the access regions. Comparison of [4] V. Barral, T. Poiroux, M. Vinet et al., Solid-State Electron. Green’s calculation with the two Wigner calculations shows 51, 537 (2007). that even at this doping level, access regions scattering has [5] R. Venugopal, M. Paulsson, S. Goasguen et al., J. Appl. an important impact on the characteristic. Comparison of Phys. 93, 5613 (2003). the two Wigner calculations shows that even for this 6 nm[6] M. Bescond, K. Nehari, J. L. Autran et al., in: IEDM Tech. long channel, scattering in the channel has a small but sigDig., p. 617 (2004). nificant impact: it reduces the transconductance [7] A. Svizhenko, M. P. Anantram, T. R. Govindan et al., J. g m = ∂I D ∂VGS by 20%. Such a result is consistent with Appl. Phys. 91, 2343 (2002).
Drain current ID (µA/µm)
the detailed analysis of ballistic effects of DG-MOSFETs as a function of gate length in [25]. Surprisingly, at 77 K scattering in the channel still reduces the transconductance by 14% (not shown). Although phonons have little effect at 77 K, the main scattering mechanism is indeed interface roughness scattering [15], which is not strongly temperature dependent. Wigner, all scatterings Wigner, ballistic channel Ballistic Green
5000 4000 3000 2000
T = 300 K VDS = 0.7 V
1000 0
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Gate voltage VGS (V) Figure 5 Room temperature current at VDS = 0.7 V using Wigner including or not scattering in the channel. Comparison with Green's function calculation (no scattering anywhere).
4 Conclusion This work examines the physics of an ultra-scaled nano-MOSFET thanks to an efficient Wigner Monte Carlo simulator extended to incorporate degeneracy effects. Quantum transport dominates the subthreshold behavior of the transistor and is responsible for the weak temperature dependence of the subthreshold slope, whereas scatterings have negligible impact in this regime. In contrast, quantum transport plays a moderate role above the threshold, even at low temperature. Device operation at such bias is dominated by scatterings, occurring both in the access regions and in the channel of the transistor. Our original algorithm appears as a rich method to study the transition from semi-classical to quantum transport in nano-devices.
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Acknowledgements This work was partially supported by the European Community 6th FP, through NoE SINANO (IST www.pss-c.com
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