Deterministic solution of the 1D Boltzmann transport equation: Application to the study of current transport in nanowire FETs

Microelectronics Journal 44 (2013) 20–25

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Deterministic solution of the 1D Boltzmann transport equation: Application to the study of current transport in nanowire FETs E. Gnani, A. Gnudi, S. Reggiani, G. Baccarani  ARCES and DEIS, University of Bologna, Via Toffano 2/2, 40125 Bologna, Italy

a r t i c l e i n f o

a b s t r a c t

Article history: Received 28 June 2010 Received in revised form 9 May 2011 Accepted 28 July 2011 Available online 26 August 2011

In this work we investigate the ballistic ratio and the backscattering coefficient in nanowire FETs operating under quasi-ballistic conditions. Starting from general expressions of the current–voltage characteristics worked out in a previous paper, we extract the above parameters and their functional dependence on inversion-layer charge and device length. The computation is based on a rigorous ¨ analytic solution of the BTE and on a numerical solution of the coupled Schrodinger–Poisson equations, by which multiple subbands are taken into account. We propose three different definitions of the ballistic ratio, clarify their meaning and compute their values against the gate voltage and the device length. As opposed to most phenomenological treatments addressing this subject for 2D nanoscale MOSFETs, the strength of our approach is that the aforementioned parameters can be computed from the knowledge of the scattering probabilities, without introducing any major simplifying assumptions. & 2011 Elsevier Ltd. All rights reserved.

Keywords: One-dimensional Boltzmann transport equation (BTE) Quasi-ballistic transport Silicon nanowire (NW) FET Ballistic ratio Backscattering coefficient

1. Introduction Ballistic and quasi-ballistic transport in silicon planar MOSFETs and nanowire (NW) FETs has been the subject of many investigations. Most papers tackle the problem numerically, often using transport models which address either the Boltzmann transport equation (BTE) by Monte Carlo [1,2] and deterministic ¨ techniques [3–5], or the open-boundary Schrodinger equation [6–12]. Recently, atomistic computations of nanowires based on the tight-binding approach [13] have been extended to NW-FETs [14], thus providing the most complete description of current transport under ballistic conditions in such devices. Some papers [15–21] have attempted an analytical approach based on the relaxation-time approximation (RTA) using McKelvey’s flux theory [22–24]. This methodology allowed analytical expressions of the current–voltage characteristics to be worked out [18], emphasizing the key role played by the backscattering coefficient r, i.e., the ratio between the negative flow of backscattered carriers and the positive flow of carriers injected into the channel at the virtual source. The coefficient r is then related with the carrier mean-free path and the gate length by a simple expression [16], modified in saturation by replacing the gate length with the kT-layer. Due to the lack of a theoretical expression for the carrier mean-free path and the coefficient r in 2D, these quantities are treated as phenomenological parameters

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E-mail address: [email protected] (G. Baccarani). 0026-2692/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.mejo.2011.07.010

which can only be extracted from the comparison with experimental data. In a recent paper [25] a rigorous analytical solution of the BTE in Si nanowire (NW) FETs operating under quasi-ballistic transport conditions has been worked out assuming dominant elastic collisions. Also, the device I–V characteristics are found to be in close agreement with a numerical model [4] accounting for acoustic phonons (AP), optical/intervalley phonons (OP) and surface-roughness (SR) scattering. The aim of this work is to take advantage from the analytical solution of the BTE and the resulting expression of the I–V characteristics to compute the ballistic ratio and the backscattering coefficient both in the linear and saturation regions. Our approach differs from previous ones in several respects: (i) we address 1D silicon NW-FETs, but most considerations apply to 2D FETs as well; (ii) a methodology for the computation of the coefficient r from the scattering probabilities is worked out; and (iii) we propose three different definitions of the ballistic ratio and clarify their meanings, thus removing an ambiguity which has not been resolved before. As expected, the resulting expression for r is nonlocal and reflects the physical and geometrical features of the NW-FET via the device length and a statistically averaged momentum-relaxation length [25]. The theory worked out here may be helpful for the extraction of physical parameters from experimental measurements. This paper is organized as follows: in Section 2 we derive the transmission and backscattering coefficient from a general expression of the current–voltage characteristics. Also, a procedure for their calculation from the averaged momentum-relaxation length is

E. Gnani et al. / Microelectronics Journal 44 (2013) 20–25

outlined. In Section 3 we propose three different definitions of the ballistic ratio, and compute their values against charge density in the linear and saturation regions. Section 4 is devoted to the discussion of the obtained results. Finally, conclusions are drawn in Section 5.

2. Nanowire drain current It has been shown in [25] that the current flowing within a single subband of a NW-FET can be expressed in two alternative and equivalent ways as follows: ID ¼

2qkB T Rt fF 0 ðZES ÞF 0 ðZED Þg h

ID ¼ Qinv ðxm ÞR0t vT

pfF 0 ðZES ÞF 0 ðZED Þg , F 1=2 ðZES Þ þ R0t F 1=2 ðZED Þ

where /  S denotes a statistical average over energy with the 1 Fermi function as a weight; L is the device length and lav ðEÞ is the 1 average of l ðEEc ðxÞÞ over the classical electron trajectory from contact to contact at constant energy E. Finally, l ¼ ux t, ux and t being the group velocity and the relaxation time, respectively. In order to compute the lav ðEÞ, the whole potential distribution within the NW-FET must be known, which involves the solution of Poisson’s equation. The coefficient R0t is instead R0t ¼

ð2Þ

where Qinv ðxm Þ is the charge density per unit length at the top of the barrier in the channel, often referred to as ‘‘virtual source’’; vT ¼ ð2kB T=pmnx Þ1=2 is the thermionic injection velocity under nondegenerate conditions; Rt and R0t are two differently defined transmission coefficients, as discussed below. Also, F 0 and F 1=2 are the Fermi integrals of order 0 and  12, respectively; ZES ¼ ðEFS Ec ðxm ÞÞ=kB T and ZED ¼ ðEFD Ec ðxm ÞÞ=kB T, where Ec ðxm Þ is the subband energy at the virtual source; finally, EFS and EFD are the Fermi energies at the source and drain contacts, respectively. If more than one subband contributes to the current, a sum of terms like (1) or (2) extended to all populated subbands must be considered. Eq. (1) basically represents Landauer’s expression for the nanowire current [26]. In this expression, the first factor does not depend upon the carrier effective mass. The motivation is that the effective-mass dependences of the density of states and group velocity cancel out in 1D systems, so that this factor turns out to be a universal constant for a given temperature. Likewise, the carrier density per unit length at the virtual source and the injection velocity are hidden in the third factor, which explicitly accounts for the current dependence on the source and drain voltages and, implicitly, the gate voltage via the subband energy at the virtual source. The second term Rt represents the probability that an electron injected into the channel with an energy exceeding the barrier height at the virtual source reaches the drain contact, and is referred to here as the transmission coefficient. Rt is related with the backscattering coefficient r by the simple relationship Rt ¼ 1r:

The transmission coefficient Rt is a function of a suitably defined momentum-relaxation length lp [25], i.e.,   2lp 2lav ¼ , ð5Þ Rt ¼ 2lp þ L 2lav þL

ð1Þ

pffiffiffiffi

ð3Þ

As opposed to Eq. (1), the charge density per unit length at the virtual source Qinv ðxm Þ appears in the drain-current expression (2). However, being the carrier density in 1D systems proportional to the Fermi integral of order  12, a compensating term shows up in the denominator of the third factor. The transmission coefficient R0t is thus defined for a constant Qinv ðxm Þ and is found to be

21

1r lp ¼ 1 þr lp þ L

ð6Þ

and the backscattering coefficient turns out to be r ¼ 1Rt ¼

1R0t L : ¼ 2lp þ L 1 þR0t

ð7Þ

All the above coefficients are nonlocal quantities which cannot be predicted with straightforward calculations: rather, the complete ¨ solution of the coupled Schrodinger–Poisson equations is required along with the analytical BTE solution, as well as the knowledge of the scattering probabilities.

3. Ballistic ratio and backscattering coefficient In principle, several definitions of the ballistic ratio are possible. The latter can be defined as: (i) the ratio between the FET drain current and the ballistic current with the same terminal voltages; (ii) alternatively, it can be defined for a constant carrier inflow at the virtual source and, finally, (iii) we may define it for a constant carrier density at the virtual source. Except for equilibrium conditions, the carrier density is generally a function of the backscattering coefficient r, so that turning the scattering probability on and off changes the carrier and potential profiles along the channel via Poisson’s equation. In order to keep the carrier inflow, or carrier density, constant at the virtual source under quasi-ballistic and ballistic conditions, the gate voltage has to be changed. Therefore, the three definitions lead to different qualitative and quantitative results. In order to avoid a terminological confusion, in what follows we define BR the ballistic ratio at constant terminal voltages and retain the symbols Rt and R0t for the above second and third definitions, respectively. We now express Eq. (2) in the linear and saturation regions. For VDS 5kB T=q, we find pffiffiffiffi pf0 ðZES Þ qV DS , ð8Þ ID ¼ Qinv ðxm Þð1rlin ÞvT F 1=2 ðZES Þ 2kB T

ð4Þ

where f0 ðZES Þ is the Fermi function. At large drain voltages, instead, Eq. (2) reduces to   pffiffiffiffi 1rsat pF 0 ðZES Þ , ð9Þ ID ¼ Qinv ðxm Þ vT F 1=2 ðZES Þ 1 þ rsat

This equation shows that, for a constant charge density Qinv ðxm Þ, Rt 0 is in fact reduced both by the lower positive flow of carriers reflected by the factor (1þr) at the denominator, and by the the negative contribution to the current of backscattered carriers represented by the factor (1 r). It should be noticed, however, that Qinv ðxm Þ is itself a function of the backscattering coefficient r and, hence, of R0t .

which reflects similar expressions worked out for 2D devices [18]. The use of different subscripts for the backscattering coefficient in the saturation and linear regions stems from the assumption of dissimilar quantitative values under different operating conditions. Some authors [21,28] identify the factor R0t ¼ ð1rsat Þ=ð1 þ rsat Þ as the ballistic ratio in saturation. This definition implicitly assumes Qinv to be independent of scattering under nondegenerate conditions,

R0t ¼

1r : 1þr

22

E. Gnani et al. / Microelectronics Journal 44 (2013) 20–25

i.e., when the last factor in Eq. (9) equals 1 and, when degeneracy occurs, it requires that the product of Qinv with the degeneracy factor be independent of scattering. As this product has no clear physical meaning and Qinv is a function of r, we do not consider the identification of the ballistic ratio with R0t to be appropriate. In order to analyze the relationship between the definitions of the ballistic ratio, we plot in Fig. 1 the carrier density per unit length Qinv at the virtual source of two NW-FETs with gate lengths Lg ¼10 nm and Lg ¼300 nm vs. VGS for VDS ¼5 mV (left plot) and VDS ¼1 V (right plot). The computation is carried out under ballistic and quasi-ballistic conditions accounting, in the latter case, for AP and SR scattering as in [25]. It may be seen that turning on and off the scattering probability at low VDS has a negligible impact on the carrier density at the virtual source. Moreover, Qinv is largely insensitive to the gate lengths, except for a very small region in weak inversion, indicating that the VT roll-off due to the short-channel effect is very weak in our gate-all-around structure. From the above plots we may deduce that, when the device operates in the linear region, the potential energy and the carrier density at the virtual source are uniquely related with the gate voltage, regardless of the backscattering coefficient. Hence, the ballistic ratio is simply given in this case by BRlin ¼ Rt ¼ R0t ¼ 1rlin :

ð10Þ

As opposed to the former case, it may be seen that deviations as large as 15% occur in saturation between the ballistic and quasiballistic conditions at the largest gate voltage. Moreover, we notice a moderate deviation between the long- and the shortchannel FETs predominantly due to the effect of drain-induced barrier lowering (DIBL). In this case, the imbalance between the carrier inflows from the source and drain contacts makes the distribution function asymmetrical, so that the carrier density is not just dependent upon the potential energy at the virtual source, but also depends on the scattering events suffered by electrons along the channel, which appreciably alter the negative part of the distribution function. In turn, the change in carrier concentration raises the potential energy at the virtual source via Poisson’s equation, acting as a negative feedback on this change.

4. Simulation results The data presented in this Section are based on complete simulations of n-type cylindrical NW-FETs with varying gate lengths, according to the procedure outlined in Ref. [25]. Therefore, no simplifying assumptions are taken in the computation of the barrier height at the virtual source, and multiple subbands are considered. In our simulations, the NW diameter D ¼5 nm, the electrical oxide thickness tox ¼1 nm and the source and drain regions are 10 nm long. Following [25], the computations account for AP and SR scattering, but neglect OP scattering. Also electronelectron and Coulomb scattering are disregarded due to the assumption of an undoped nanowire in the channel and to the small extension of source and drain regions. Hence, electron transport is considered to be quasi-ballistic everywhere in the nanowire from contact to contact. The procedure we use is as follows: first, we compute the device I–V characteristics under quasi-ballistic conditions by summing up the contributions of every populated subband. Next, we extract Rt for all subbands from Eq. (1). The global transmission coefficient is then computed as a weighted average of the individual (Rt)s. The backscattering coefficient is simply defined as r ¼1 Rt for every individual subband and globally for multiple subbands. Fig. 2 shows a comparison of the turn-on characteristics for a NW-FET with diameter D ¼5 nm and gate length L ¼13 nm, as obtained from a Monte Carlo simulation tool [27], a deterministic BTE solver [4,27], and the semi-analytical model worked out in [25] and used in this work. While the two former models account for acoustic- and optical/intervalley-phonon scattering, the latter model only accounts for acoustic-phonon and surface-roughness scattering. Nevertheless, the figure shows that the agreement of the three models is excellent across the whole range of gate voltages, encompassing both the subthreshold and the stronginversion regions. Fig. 3 shows the ballistic ratios in the linear region vs. gate voltage for different channel lengths. The three ratios equal each other as shown in Eq. (10). Fig. 4 shows instead the ballistic ratio (top) and the transmission coefficients Rt (middle plot) and R0t (lower plot) vs. gate voltage for different channel lengths at VDS ¼1 V. As expected, BRsat, Rt and R0t differ both quantitatively and qualitatively, and exhibit slightly different trends against VGS.

3

101

102

Solid: AP+SR

D=5 nm

Dashed:ballistic

2.5

100

100

40

LG=13 nm

10−2

2 L =10 nm L =300 nm

1.5

10−3

10

1

−4

10−2

30

10−4 20

10−6

analytical: AP+SR numerical: AP+OP+SR MC: AP+OP+SR

10−8

10

0.5

−5

0.5 0.6 0.7 0.8 0.9 1.0

10

V =1.0 V

V =5 mV

10

Drain current (µA)

7

−1

Qinv (10 cm )

VDS=1.0 V 10−1

0.5 0.6 0.7 0.8 0.9

1

0 1.1

Gate voltage (V)

−10

10−12

0

0.2

0.4

0.6

0.8

1

0

Gate voltage (V) Fig. 1. Linear and log plots of the electron density per unit length at the virtual source of a 5.0 nm diameter NW-FET vs. VGS for VDS ¼5 mV (left) and VDS ¼ 1 V (right). Two gate lengths are considered, namely Lg ¼10 nm and Lg ¼300 nm. The computation is carried out under ballistic conditions (dashed lines) and accounting for AP and SR scattering (solid lines).

Fig. 2. Comparison of the turn-on characteristics for a NW-FET with diameter D¼ 5 nm and gate length L¼ 13 nm at VDS ¼ 1 V, as obtained from a Monte Carlo simulation tool [27], a numerical solution of the 1D BTE [27] and the semianalytical solution outlined in Ref. [25] and used in this work.

E. Gnani et al. / Microelectronics Journal 44 (2013) 20–25

1

23

VDS=5 mV

30

LG=10 nm LG=30 nm LG=50 nm LG=100 nm LG=300 nm

VDS=5 mV BR = Rt = Rt’

20

LG=10 nm LG=30 nm LG=50 nm LG=100 nm LG=300 nm

Iscat/Ibal

0.6

10

λp (nm)

0.8

0 VDS=1 V

40

0.4

30 20

0.2

10 0

0 0.5

0.7

0.9

1.1

104

1.3

105

106

107

108

−1

Qinv (cm )

Gate voltage (V) Fig. 3. Ballistic ratio and transmission coefficients vs. gate voltage in NW-FETs with different gate lengths at VDS ¼5 mV. As shown in Eq. (10) the ballistic ratio is equal to the transmission coefficients.

Fig. 5. Momentum-relaxation length vs. electron density at the virtual source of NW-FETs with different gate lengths. Upper plot: VDS ¼ 5 mV. Lower plot: VDS ¼ 1 V.

0.4 First subband VDS=1.0 V

Iscat/Ibal/N

1 VDS=1 V

0.8

0.2

0.6

Second sub.

0.0

0.4 0.2

Energy (eV)

0

Rt

0.8 0.6 0.4 0.2 LG=10 nm LG=30 nm LG=50 nm LG=100 nm LG=300 nm

Rt’

0 0.8

1−rsat

0.6

1+rsat

0.4 0.2 0 0.5

VGS=0.5 V VGS=0.6 V VGS=0.7 V VGS=0.8 V VGS=0.9 V VGS=1.0 V VGS=1.1 V VGS=1.2 V

−0.6 −0.8 −1.0

0

0.7

0.9

1.1

QINV=5x106 (cm−1) QINV=9x106 (cm−1)

50

100

0

50

100

Position (nm)

Fig. 4. Ballistic ratio (top) and transmission coefficients (center and bottom) in NW-FETs with different gate lengths vs. gate voltage for VDS ¼1 V. Different quantitative results are found for the three definitions of ballistic ratio.

Finally, the gate voltage dependence of BRlin is very weak for nearly all gate lengths both in the linear and saturation regions, with the only possible exception of the shortest gate length. The knowledge of the backscattering coefficient makes it possible to extract the momentum-relaxation length from Eq. (7). We find Lð1rÞ 2r

First sub.

−0.4

−1.2

Gate voltage (V)

lp ¼

−0.2

ð11Þ

and its plot vs. Qinv is shown in Fig. 5. We notice here a fairly large deviation of lp at the shortest gate length LG ¼10 nm with respect to the other curves. This is due to the increasing importance of the source and drain regions with respect to the channel as the gate length is reduced. Electrons traveling across the channel at the lowest possible energies have in fact a larger offset from the subband edge within the source and drain regions. Due to the decreasing density of states with energy in 1D structures, the

Fig. 6. Subband energy profiles for a NW-FET with Lg ¼ 100 nm. Left: first subband profiles for increasing gate voltages. At low VGS, electrons traveling across the channel experience a higher number of scattering events than at high VGS, where a rapid drop of the subband profile is visible. Right: first and second subband energy profiles for VGS ¼ 0.8 V (circles) and 0.9 V (squares). At VGS ¼ 0.8 V electrons can scatter to the second subband only in the vicinity of the drain, while at VGS ¼ 0.9 V intersubband scattering events can take place from the middle of the channel.

scattering probability is much lower in these regions than in the channel. At low VDS such a difference decreases with increasing carrier densities due to the lowering of the energy barrier with the gate voltage. At high VDS, instead, lp increases with increasing gate voltage since the number of scattering events experienced by electrons traveling across the channel decreases. As can be seen from the lowest subband energy profiles displayed in Fig. 6 (left), at large gate voltages electrons with an energy higher than Ec ðxm Þ rapidly gain kinetic energy as soon as they overcome the barrier. Hence, a crossing of the curves is observed. The wiggles exhibited by the curves at Qinv ¼ 5  106 and 9  106 cm  1 for LG ¼ 100 and 300 nm are due to intersubband scattering mechanisms. In order to clarify this effect, the first and

24

E. Gnani et al. / Microelectronics Journal 44 (2013) 20–25

the second subband energy profiles are displayed in Fig. 6 (right) for the above values of the charge density. For Qinv ¼ 5  106 cm1 , electrons can scatter from the first to the second subband only in the vicinity of the drain while, for Qinv ¼ 9  106 cm1 , intersubband scattering events can occur over a much larger portion of the channel. Hence, the decrease of lp . The final lp roll-off at high Qinv is due instead to the subband repopulation under degenerate conditions, which enhances the scattering probability in the higher subbands due to the larger density of states. The backscattering coefficient r ¼1 Rt is plotted vs. Qinv in Fig. 7 at VDS ¼5 mV (upper plot) and at VDS ¼1 V (lower plot). We notice a small deviation between the lowest-subband (dotted lines) and the global backscattering coefficients (solid lines), which grows bigger at the largest carrier densities. The average electron velocity is plotted in Fig. 8 vs. the electron density for VDS ¼ 5 mV (top) and VDS ¼1 V (bottom). As the electric field scales inversely with the channel length, the

1

Backscattering coefficient

0.8 0.6 LG=10 nm LG=30 nm LG=50 nm LG=100 nm LG=300 nm

0.4 VDS=5 mV 0.2

In this work we examine the ballistic ratio and the backscattering coefficient in NW-FETs operating under quasi-ballistic conditions. We propose three different definitions of the ballistic ratio, namely: for constant terminal voltages, for constant carrier inflow and for constant carrier density at the virtual source, and demonstrate that each of them is characterized by different expressions and numerical values when the device operates in saturation. The three definitions, instead, merge into one at verylow drain voltages, due to the near-equilibrium distribution function. This treatment refers to 1D structures, but the above considerations apply to 2D structures as well, provided the degeneracy factors in Eqs. (1) and (2) are changed to account for the diverse dimensionality of the problem. The quantitative values of the ballistic ratio, backscattering coefficient, electron mean-free path and average velocity are computed with reference to a NW-FET, and are based on a rigorous solution of the BTE in 1D ¨ structures and a numerical solution of the Schrodinger–Poisson equations.

Acknowledgments

0.6

0.2

104

This work has been supported by the EU NoE no. 216171 (NANOSIL) via the IU.NET Consortium.

Solid: total Dashed: first sub.

VDS=1 V 105

106

Qinv

107

108

8

L L L L L

VDS=5 mV

6 4

=10 nm =30 nm =50 nm =100 nm =300 nm

2 0 VDS=1 V

1.5

vT unprimed ladder

1.0 0.5 0.0

104

105

References

(cm−1)

Fig. 7. Backscattering coefficient vs. electron density at the virtual source of NWFETs with different gate lengths. Upper plot: VDS ¼5 mV. Lower plot: VDS ¼ 1 V. Dashed lines: first subband. Solid lines: all subbands.

Velocity (105 cm/s)

5. Conclusions

0.8

0.4

Velocity (107 cm/s)

shorter devices exhibit larger velocities. At VDS ¼5 mV their decrease at large electron densities is mainly due to the degeneracy factor in Eq. (8). At large VDS, instead, the electron velocity increases at large electron densities due to the increased injection velocity under degenerate conditions. This is reflected by the degeneracy factor in Eq. (9), which grows bigger than 1 for increasing gate voltages.

106

107

108

−1

Q inv (cm ) Fig. 8. Average velocity vs. electron density per unit length at the virtual source of NW-FETs with different gate lengths. Upper plot: VDS ¼ 5 mV. Lower plot: VDS ¼ 1 V.

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