One-flux theory of saturated drain current in nanoscale transistors

Solid-State Electronics 78 (2012) 115–120

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One-flux theory of saturated drain current in nanoscale transistors Ting-wei Tang a,⇑, Massimo V. Fischetti b, Seonghoon Jin c, Nobuyuki Sano d a

Department of Electrical and Computer Engineering, University of Massachusetts, Amherst, MA 01003, USA Department of Materials Science and Engineering, University of Texas at Dallas, Richardson, TX 75080, USA c Device Lab, Samsung Semiconductor Inc., 95 West Plumeria Drive, San Jose, CA 95134, USA d Institute of Applied Physics, University of Tsukuba, Tsukuba, Ibaraki 305-8573, Japan b

a r t i c l e

i n f o

Article history: Available online 30 June 2012 Keywords: Saturated drain current Virtual source Multiple reflections Backscattering coefficient MOSFETs SNWTs

a b s t r a c t We present an expression for the saturated drain current in nanoscale transistors based on multiple reflections of carriers at the virtual source from two adjacent scattering ‘‘black boxes’’. Under certain assumptions and simplifications this new expression reduces to the well known Lundstrom’s formula and also to the recent model by Giusi et al. Six macroscopic parameters appear in the ‘exact’ form of this model. We do not discuss how to derive physical expressions for these parameters. Rather, we emphasize the limitations of Lundstrom’s model when applied to nanoscale transistors. Some existing formulae for the carrier backscattering coefficient are examined and compared to our results. We verify our model through a consistency test based on simulation data of a 10 nm gate-length silicon nanowire transistor. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction Since Lunstrom’s seminal publication in 1997 [1], followed by more studies by the same group, a large number of publications have appeared attempting to model in compact form the drain current, ID, in silicon MOSFETs. The published literature is related either to the applications of the model [2–7], or to its verification [8–11], or improvements [12,13]. The original paper has been the subject of controversy [14,15] and some questions have been raised on the concept of the so-called ‘critical layer’ which is required to estimate the backscattering coefficient [16–19]. Lundstrom’s main idea in [1] is based on the analysis of the transfer of carriers from the source contact to the virtual source (VS) at the top of the potential energy barrier and to determine the drain current by considering the backscattering of carriers at the VS. The theory is ‘elementary’, as defined by the author himself, but the idea behind the theory is commendable. In spite of many studies published on the subject of carrier backscattering, very few papers have critically examined the derivation and the explicit expression of ID itself, until a recent analysis by Giusi et al. [13]. Such a critical examination is the goal of this work. Therefore, we emphasize the task of obtaining a correct expression of ID rather than of modeling of the backscattering coefficient appearing in ID. In this work, we derive an alternate expression of saturated drain current ID,sat along the line of the one-flux theory of McKelvey et al. [20]. Although the one-flux theory based on Shockley’s differential equations [21] can be used to calculate ⇑ Corresponding author. E-mail address: [email protected] (T.-w. Tang). 0038-1101/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.sse.2012.05.060

the current flux in integral form, it requires introduction of effective mobility and effective diffusivity which are expressed in terms of backscattering coefficients [22,23]. Since the main purpose of this work is to re-visit Lunstrom’s VS model, we follow closely the derivation of the drain current given in [1]. In relating the carrier injection velocity at the VS to the thermal velocity, however, we utilize a multiple reflection scheme for carriers at the VS. Our derivation of the saturated drain current ID,sat based on the carrier’s multiple reflections from two adjacent (upstream and downstream) scattering regions (which we shall treat as ‘‘black boxes’’ fully described by their transmission and reflection coefficients) reveals more of the underlying scattering physics and clarifies the precise meaning of the backscattering coefficient. The concept of the multiple reflections of carriers at the VS applies equally to both one-dimensional (1-D) and two-dimensional (2D) nanoscale transistors. We consider the case of the 2-D MOSFETs first. 2. Drain current under saturated conditions Lundstrom’s elementary scattering theory assumes that carriers injected at the VS have same injection velocity as if they were injected from the ‘‘ideal’’ source, which is viewed as a reservoir of thermal carriers. A fraction of the flux injected from the source transmitting across the source-channel potential energy barrier is collected by the drain and a fraction of the flux injected into the channel is backscattered from the channel, thus reentering the source. We will re-derive below Lundstrom’s expression for the drain current under saturated conditions, using a slightly different approach.

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2.1. Derivation of ID,sat by Lundstrom and by Giuisi et al. Assuming that the flux injected from the drain can be neglected under the high drain bias condition, VDS  kBT/q, we can express the saturated drain current as:

ID;sat ¼ Iþ  I ¼ Iþ ð1  RÞ ¼ Wqnþ mþ ð1  RÞ;

ð1Þ

where I+(I) represents the positively-(negatively-)directed flux, R is defined as the ratio I/I+, W is the width of the device, q is the magnitude of the electron charge and n+(t+) represents the 2-D electron density (velocity) associated with I+ in MOSFETs, all quantities being evaluated at the VS. Introducing Qinv = q(n+ + n) as the total inversion charge density per unit area at the VS, we can express qn+ = Qinv(1 + ktR)1 where kt  t+/t > 1 represents the ratio between the positively- and negatively-directed carrier velocities. Substitution of this expression into Eq. (1) yields

ID;sat ¼ WQ inv ð1  RÞð1 þ km RÞ1 mþ :

Fig. 1. Schematic of the multiple reflections of carriers at the virtual source from two adjacent ‘‘scattering boxes’’.

ð2Þ

As written, Eq. (2) is exact and does not involve any approximations. In Eq. (2), Lundstrom relates Qinv to Cox(VGS  VT), assumes t+ = t = tth, and obtains:

ID;sat ¼ WC ox ðV GS  V T Þð1  RÞð1 þ RÞ1 mth :

ð3Þ

Assuming also that R = ‘/(‘ + k) where ‘ is the so-called ‘critical length’ and estimating the ‘mean-free-path’ k from the low field mobility at the VS [1], Lundstrom reaches a complete expression for ID,sat. While Eq. (3) seems to work well for long gate-length MOSFETs, as the gate length LG is further reduced toward 10 nm, the accuracy and predicting power of Eq. (3) becomes questionable. Recently, Giusi et al. [13] have attempted to improve upon Lundstrom’s Eq. (3) by replacing it with:

ID;sat ¼ WQ inv ð1  RÞð1 þ RÞ1 ð1 þ km RÞ1 mþS;bal ;

ð4Þ

þ Iþ S;bal =ðWqnS;bal Þ

where mþ ¼ mth ½F 1=2 ðgVS Þ=F 0 ðgVS Þ is the posiS;bal ¼ tively-directed ballistic electron velocity at the source contact, þ Iþ S;bal ¼ WqðN 2D =2Þmth F 1=2 ðgVS Þ; nS;bal ¼ ðN 2D =2ÞF 0 ðgVS Þ; N 2D ¼ ðkB Tm 2 DOSÞ=ðp h Þ is th0e effective 2-D density-of-states, kB is the Boltzmann constant, T is the absolute temperature, mDOS is the densityof-states effective mass, tth = (2kBT/pmC)1/2 is the uni-directional thermal velocity, mC is the conduction effective mass, ⁄ is the reR1 duced Planck constant, F j ðgVS Þ ¼ ð1=j!Þ 0 yj ½expðy  gS Þ þ 11 dy is the Fermi–Dirac integral of order j, gVS = [EF,S  EC(xmax)]/kBT, EF,S is the Fermi energy at the source, and EC(xmax) is the energy of the lowest-energy occupied subbands at the VS, x = xmax [13,24]. For clarity, the summation over all occupied subbands is omitted. A more accurate treatment would make use of the appropriate conductivity and density-of-states effective masses for each of the occupied subbands. The main difference between Eqs. (3) and (4) is, beside difference in tth and mþ , an additional factor (1 + ktR)1 S;bal appearing in Eq. (4) which implies that ID,sat is overestimated in Eq. (3). Consistency tests based on the Monte Carlo (MC) simulation of a LG = 20 nm DG nMOSFET [13] seem to confirm the improved accuracy of Eq. (4) over Eq. (3). 2.2. Multiple reflections of carriers at the VS In the following, we shall derive an alternative expression for ID,sat. Let us consider carriers at the top of the barrier as shown in  Fig. 1 (Fig. 2 of [13] with rDS(rSD) replaced by r þ i ðr j Þ). The positively-directed flux at the source, Iþ , consisting of carriers whose S;bal energy exceeds the potential energy barrier, is transmitted from the source contact across the ‘upstream’ scattering ‘‘black box’’ to reach the VS and is designated by Iþð0Þ ¼ t S Iþ S;bal , where the tS is the transmission coefficient. As carriers I+(0) enter the ‘down-

Fig. 2. Schematic structure of the simulated gate-all-around silicon nanowire transistor (LG = 10 nm, Lext = 10 nm, tox = 1 nm, and rs = 2 nm). Note that the channel and crystal coordinate systems are aligned.

stream’ scattering ‘‘black box’’ from the left for the first time, they are reflected. This results in a negatively-directed flux, Ið0Þ ¼ þð0Þ rþ where r þ 1I 1 is the reflection coefficient of the downstream scattering black box. Subsequently, those carriers I(0) which enter the upstream scattering black box from the right are reflected once more with the reflection coefficient r  1 . As a result, this reflected ð0Þ flux, Iþð1Þ ¼ r  , which is positively-directed, modifies the mag1I nitude of I+(0). Assuming the electrons in the two adjacent black boxes are not interacting, after such back and forth reflections, I+ and I at the VS reach final steady state values given, respectively, by (see Appendix A):

Iþ ¼ tS K þ IþS;bal ;

ð5Þ

and

I ¼ rþ1 tS K  IþS;bal ;

ð6Þ

where

K þ  ½1  r 1 r þ1 ð1  r2 r þ2 ð1  r 3 r þ3 ð1  . . .ÞÞ . . .Þ 6 1;

ð7Þ

K   ½1  r 1 r þ2 ð1  r2 r þ3 ð1  r 3 r þ4 ð1  . . .ÞÞ . . .Þ 6 1;

ð8Þ

 rþ i ðr j Þ

and denotes the reflection coefficient of the reflected flux, entering the downstream (upstream) black box from the left (right) the ith (jth) time. The whole information about the physical scattering processes occurring inside the two black boxes is absorbed into  the reflection coefficients r þ i and r j . We do not wish, and neither would we be able to, express these coefficients explicitly in terms of actual scattering mechanisms, doping concentration, channel þ þ þ length, etc. Here, it suffices to note that r þ 1 > r2 > r3 >    > rn . . .     and r1 > r 2 > r 3 >    > r n > . . ., because after each scattering event in the black box the flux loses its high-energy carrier population via inelastic scattering, thus rendering each subsequent reflection weaker. Both Eqs. (7) and (8) converge since the series has  alternating signs and the magnitude of rþ i and r j is always less than 1. We also note that the sum of the series inside the square bracket

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T.-w. Tang et al. / Solid-State Electronics 78 (2012) 115–120 þ þ is less than 1 and that K P K+ because r þ 1 > r 2 > r 3 > . . .. From Eqs. (7) and (8) we readily obtain:

R  I =Iþ ¼ r þ1 ½K  ðr þi ; rj Þ=K þ ðr þi ; r j Þ P rþ1 :

ð9Þ

Note that in Eq. (9), the backscattering coefficient R is explicitly re lated to the reflection coefficients r þ i and r j which are functions of incident fluxes. Although the magnitudes r þ i and R may be close to each other, their physical meaning is somewhat different. The former represents the first reflection coefficient of the downstream scattering black box while the latter represents a ratio between I and I+ at the VS. Adding Eqs. (5) and (6) together, we readily obtain:

Iþ þ I ¼ t S K þ ð1 þ RÞIþS;bal :

ð10Þ

We may rewrite Eq. (10) as:

nþ mþ ¼ tS K þ ðnþS;bal mþS;bal Þ:

ð11Þ

þ S;bal

Note that m is a function of EC(xmax) which is gate-bias dependent and should not be confused with ht(xmax)bali, the ballistic average velocity at the VS. Substitution of t+ obtained from Eq. (11) into Eq. (2) yields:

ID;sat ¼ WQ inv ½kn t S K þ ð1  RÞð1 þ km RÞ1 mþS;bal ;

ð12Þ

þ where the ratio kn  nþ S;bal =n is introduced. We can also obtain directly from Eqs. (5) and (6):

ID;sat ¼ Iþ  I ¼ tS K þ ð1  RÞIþS;bal :

ð13Þ

Equations (12) and (13) are our main results for 2-D MOSFETs. There are no approximations involved in either expression except that additional variables kn and tSK+ appear in the expression.

hmðxmax Þi=mþS;bal  ð1  RÞð1 þ km RÞ1 ¼ Bconv ðkm –1Þ:

ð19Þ

Since ht(xmax)i = ID,sat/(WQinv), from Eq. (12) we obtain our result:

hmðxmax Þi=mþS;bal ¼ ðt S K þ kn Þð1  RÞð1 þ km RÞ1 :

ð20Þ

+

So, what does the factor tSK (1 + R) represent? For well-designed MOSFETs, we can expect that tS  1. If we look at Eq. (7), for þ þ þ 1  þ 1  r can be approximated by ð1 þ r  1 r1  r2 r2  . . . ; K 1 r 1 Þ . If the entire upstream region can be regarded as a thermal carrier resþ þ 1 ervoir, we can assume r 1  1 and hence K  ð1 þ r 1 Þ . If we fur+  ther assume r þ ¼ r ¼ R as in [13], we obtain t K = (1 + R)1 and S j i + the factor tSK (1 + R) is reduced to 1. However, even under these approximations Eq. (20) does not reduce to Eq. (19) unless þ þ þ nþ S;bal =n ¼ 1 þ R. Note that nS;bal =n  1 is assumed in [13]. When Eq. (15) is compared to Eq. (20), we find that the two expressions are different. Therefore we conclude that the ratio ID;sat =Iþ S;bal and the ratio hmðxmax Þi=mþ S;bal represent two different measures of ballisticity although numerically they may be close. 3. Verification of the expression for ID,sat In this section, we verify the consistency of Eqs. (5),(6),(10), and (13). For this purpose, we use the published data obtained from the simulation of a LG = 10 nm gate-all-around (GAA) silicon nanowire transistor (SNWT) (see Fig. 2) by numerically solving the multisubband Boltzmann equation [25]. Electron scattering mechanisms considered are acoustic and intervalley phonons, surface roughness, and ionized impurities, thus accounting for both intrasubband and intersubband, and for elastic and inelastic transitions. More details on the scattering mechanisms and scattering rates can be found in [26]. We consider the bias condition of VGS = 0.3 V and VDS = 0.5 V to assure the drain current is saturated.

2.3. Comparison of different expressions 3.1. Expression of ID,sat in 1-D In this section, we compare Eq. (12) with similar expressions given by Lundstrom’s [1] and by Giuisi et al. [13]. When Eq. (12) is compared to Eqs. (3) and (4), setting tS kn = 1, K+ = 1 and kt = 1 in Eq. (12) recovers Lundstrom’s Eq. (3), while setting tSkn = 1, K+ = (1 + R)1 in Eq. (12), Eq. (4) of Giusi et al. is recovered. When setting rþ i ¼ r for all reflections from the downstream black box, the backscattering coefficient R becomes identical to the reflection coefficient r. In [13], Eq. (10) is approximated as:

Iþ þ I  IþS;bal ;

ð14Þ

which differs from Eq. (10) by a factor tSK+(1 + R). Note that Eq. (14)  is valid only if rþ i  r j ¼ R as assumed in [13]. To indicate how close the drain current ID,sat is to its ballistic þ limit Iþ S;bal , the definition of the ballistic ratio B  I D;sat =I S;bal is often used. From Eq. (13), we readily obtain:

B ¼ t S K þ ð1  RÞ:

ð15Þ

Conventionally, B is given either as [9]:

Bconv ðkm ¼ 1Þ  ð1  RÞð1 þ RÞ1 ;

ð16Þ

or as [12]:

Bconv ðkm –1Þ  ð1  RÞð1 þ km RÞ1 :

ð17Þ

Comparing Eq. (15) with Eq. (16), the difference is again a factor of tSK+(1 + R) while comparing with Eq. (17), the difference is a factor of tS K+(1 + ktR). Another definition similar to the ballistic ratio is the ratio hmðxmax Þi=mþ S;bal . In [1,6,8] this ratio is given by:

hmðxmax Þi=mþS;bal  ð1  RÞð1 þ RÞ1 ¼ Bconv ðkm ¼ 1Þ; while in [10], it is:

ð18Þ

Since carrier transport in SNWTs is 1-D, in the following we emphasize the main differences between the 1-D SNWTs and 2-D þ þ MOSFETs. First, Eq. (1) is replaced by ID;sat ¼ qnþ 1D m1D ð1  RÞ; qn ¼ 1 1 þ Q inv ð1 þ km RÞ by qn1D ¼ Q 1D ð1 þ km RÞ and Eq. (2) by:

ID;sat ¼ Q 1D ð1  RÞð1 þ km RÞ1 mþ1D :

ð21Þ

The positively-directed 1-D ballistic velocity at the source þ þ contact, mþ is given by mþ 1D S;bal , 1D S;bal ¼ I 1D S;bal =ðqn1D S;bal Þ ¼ mth ½F 0 ðgVS Þ=F 1=2 ðgS ; uB Þ where:

Iþ1D S;bal ¼ qðmC =hÞm2th F 0 ðgVS Þ;

ð22Þ

nþ1D S;bal ¼ ðmC = hÞmth F 1=2 ðgS ; uB Þ;

ð23Þ

and

F 1=2 ðgS ; uB Þ ¼ p1=2

Z

1

uB

y1=2 ½expðy  gS Þ þ 11 dy;

ð24Þ

is the incomplete Fermi–Dirac integral of order ½, gS = (EF,S  EC,S)/ kBT is the difference between the Fermi energy EF,S and EC,S, the energy of the lowest-energy occupied subbands at the source, expressed in thermal units, and uB = [EC(xmax)  EC,S]/kBT is the barrier height, also in thermal units [27]. In Eqs. (22) and (23), we have omitted the summation over all the occupied subbands for clarity. The rest of the equations are similar to those of 2-D MOSFETs except for the fact that WQinv must be replaced by Q 1D; nþ S;bal þ þ þ þ þ þ by nþ 1D S;bal ; mS;bal by m1D S;bal ; n by n1D , and m by m1D . Our main results Eqs. (12) and (13) for 2-D MOSFTs then become, respectively, for 1-D SNWTs:

ID;sat ¼ Q 1D ½kn tS K þ ð1  RÞð1 þ km RÞ1 mþ1D S;bal ;

ð25Þ

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T.-w. Tang et al. / Solid-State Electronics 78 (2012) 115–120

and

ID;sat ¼ Iþ  I ¼ tS K þ ð1  RÞIþ1D S;bal :

ð26Þ

3.2. Simulation data of 1-D SNWT Now let us look at the simulation data. Fig. 3 shows the comparison of ID vs. Iþ S;bal for different VDS for Case A (elastic scattering only), for Case B (both elastic and inelastic collisions including intrasubband intersubband transitions) and for Case C (all scattering mechanisms including surface roughness and ionized impurities). A kink in the output characteristics seen in Case C is caused by scattering-induced intersubband transitions [25]. A more detailed explanation of this kink behavior can be found in [28]. Fig. 4 shows the fluxes I+ and I as a function of position along the channel for Case C (plots for Case A and B are similar). The left vertical line represents the location of the VS and the right vertical line the location of the so-called kBT layer [1]. Fig. 5 shows the positively- and negatively-directed carrier velocities t+ and t along the channel position for Case C (similar plots for Case A and B). Fig. 6 shows the corresponding carrier concentrations n+ and n

 Fig. 5. Plot of velocities mþ 1D and m1D along the channel position (5 nm 6 z 6 5 nm) for Case C.

Fig. 6. Plot of electron density n+ and n along the channel position (5 nm 6 z 6 5 nm) for Case C. Fig. 3. Comparison of ID vs. Iþ IDS;bal for different cases at different VDS.

for Case C. In Figs. 4–6, the spatial coordinate z is used instead of  þ  þ þ   þ  x; J þ s ðJ s Þ represents I ðI Þ; ms ðms Þ represents m1D ðm1D Þ and ns ðns Þ þ  represents n1D ðn1D Þ. From these results, we can directly calculate +  R ¼ I =Iþ ; tS K þ ¼ Iþ =Iþ at the S;bal and the velocity ratio kt = t /t VS. Once R and tSK+ are calculated, we can obtain B = tSK+(1  R). We can also compare Iþ þ I  Iþ S;bal (assumed in [13]) vs. Iþ þ I ¼ tS K þ ð1 þ RÞIþ S;bal . These results and comparisons are summarized in Table 1.

4. Discussion

Fig. 4. Plot of currents I+ and I along the channel position (5 nm 6 z 6 5 nm) for Case C.

As we can see from Table I, the results for three cases, A, B, and C presented above are different. First, the magnitude of tSK+ in each case is always smaller than 1, as predicted by Eq. (6). Second, the current ratio (the backscattering coefficient) R progressively increases and the magnitude of the variable tSK+ decreases when stronger scattering (including inelastic scattering) is included in the simulation. The decrease of tSK+ indicates that the magnitude þ þ  of r  1 r 1 in Eq. (7) increases from Case A to Case C. Different r i ðr j Þ must be used in Eqs. (7) and (8) because the energy spectrum of each entering flux changes. In order words, the black box repre-

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T.-w. Tang et al. / Solid-State Electronics 78 (2012) 115–120 Table 1 Comparison of various parameters and models for Case A, B, and C.

Case A Case B Case C

R

tSK+

kt

B

Bconv (kt = 1)

Bconv (kt – 1)

tSK+ (1 + R)

tSK+ (1 + ktR)

0.156 0.177 0.227

0.956 0.829 0.744

1.705 1.667 1.586

0.803 0.680 0.575

0.730 0.699 0.630

0.667 0.636 0.568

0.976 0.976 0.912

1.074 1.074 1.012

sents a nonlinear scattering medium for the entering flux. Consequently reciprocity of the transmission and reflection coefficients does not hold in this case. Third, in any well-designed MOSFETs, the goal is to obtain tS  1 and r  j  1. However, in real devices, tS and r j could be significantly different from 1. Actually the whole upstream black box consists of two parts: A heavily doped sourceextension region where transistor tries to maintain the charge neutrality and a small transitional region around the source-channel junction area where the carriers encounter a retarding field. This small transition region is not a part of the thermal carrier reservoir but it is always present. Because scattering in this region cannot be  totally neglected, we should expect r  j < 1. A smaller r j in very short gate-length devices plays a more important role in the determination of the value of tSK+ than in longer devices. Fourth, under the same bias condition, the barrier height UB = EC(xmax)  EC,S is approximately 0.01 eV, 0.014 eV, and 0.006 eV (data not shown) for Case A, B, and C, respectively. This indicates that the inclusion of different scattering mechanisms in the device affects the barrier height and therefore the values of Q1D and Iþ 1D at the VS as well [14]. Based on the 1-D SNWT simulation results, the following observations can be made when comparing our expressions with those obtained by others: (i) Even for a SNWT with LG = 10 nm, when all scattering processes are included, we obtain R = 0.227 and B = 0.575, which is still not very close to the ballistic limit. (ii) Within the same device and under the same bias condition, the very different results are obtained when ignoring inelastic processes. Thus, in any device simulation, all pertinent scattering processes must be included. (iii) Bconv(kt = 1) overestimates B by nearly 10% in Case C, while Bconv(kt–1) is very close to B (only about a 1% difference). (iv) The value kt = 1.586 for the simulated SNWT (Case C) is larger than the estimated value 1.35 used in the simulation of a DG nMOS1 FET in [13]. The value of 63% we have obtained for m =mþ ¼ km [19] is in line with other MC simulation results of 60–80% in the range of LG = 20–40 nm [9,10,12,29]. This indicates that kt is by no means constant. (v) The difference between tSK+(1 + R)1 and 1 is about 9% in Case C. According to Eq. (10), I+ + I = tSK+(1 + R)IS,bal, hence the result I+ + I  IS,bal assumed in [13] is off by 9%. Returning to the case of 2-D MOSFETs, it may appear that Eq. (13) is more suitable for modeling ID,sat than Eq. (12) because only estimates for tSK+ and R are required. However, this is misleading because Iþ S;bal requires the knowledge of EC(xmax) in the expression of gVS, knowledge which is not readily available unless MC or numerical simulations are performed. Also, obtaining an accurate determination of ID,sat based on Eq. (12) or Eq. (13) depends not only on the barrier height UB = EC(xmax)  EC,S, but also on the shape (the second spatial derivative) of the potential energy profile at and around VS, since this affects carrier backscattering [30,31]. Judging from the last column in Table 1, the approximation tSK+  (1 + ktR)1 appears to be reasonable in Case C. Using tSK+  (1 + ktR)1 as a possible model for tSK+, Eq. (12) is approximated by

ID;sat  WQ inv ½kn ð1  RÞð1 þ km RÞ2 mþS;bal ;

ð27Þ

while Eq. (13) is simplified to

ID;sat  ð1  RÞð1 þ km RÞ1 IS;bal ;

ð28Þ

yielding

B  ð1  RÞð1 þ km RÞ1 :

ð29Þ

The same approximation also yields

hmðxmax Þi=mþS;bal  ðnþS;bal =nþ Þð1  RÞð1 þ km RÞ2 :

ð30Þ

It should be reminded that Eqs. (27)–(30) obtained as a result of the approximation tSK+  (1 + ktR)1 is based on only one set of data from one 1-D device. Understanding whether this approximation has a universal applicability or not would require further investigation by collecting more simulation data from different devices, gate lengths and gate biases. Even the use of Eq. (29) as a measure of ballistic ratio still requires an accurate estimate of R. Although Eq. (29) has appeared in [12], its origin has not been made clear. Here we show how it is obtained and on which assumptions it is based. Eq. (12) can be considered as a generalization of Lundstrom’s original expression, Eq. (3). In addition to Qinv and R, there are four more variables, kn, tS, K+, and kt in Eq. (12), each of which requires additional assumptions to be accurately estimated. Admittedly some of these variables such as kn, tS, and kt can be ‘guesstimated’, but even if the approximation tSK+  (1 + ktR)1 holds, which reduces the variables from six to four, the estimate of R remains as a very challenging task since the physics of all scattering processes as well as information related to the device structure and operating conditions inside the black box are now lumped into a single parameter R. Viewed from this prospect and from Eq. (9), it is difficult to comprehend how a meaningful critical length can be devised and accurately determined, and how the low-field ‘‘mobility’’ [1] can play an important role in the determination of R in the nanoscale transport regime. Lundstrom’s VS model works well for long channel devices because the carrier transport is primarily determined by the downstream scattering in the long channel devices where the low-field region surrounding the VS is well defined. In our work, as well as in [13], emphasis has been given not to the estimation of the parameters which appear in Eq. (12). There is enough controversy over the estimate of R and we are not readdressing it here. Even the estimate of Qinv is not that trivial [29]. Additionally, to compare the device performance with that of the so-called ‘‘ballistic device’’ is, strictly speaking, not very meaningful for two reasons: First, the ballistic limit may not be attainable [31] and, second, the barrier heights obtained when including or excluding scattering processes are different. Even though expression (12) is exact, without accurate estimates of Qinv, R, K+, etc., in itself is not a very useful tool for predicting device characteristics. That is why we firmly believe that there is no shortcut to the accurate prediction of nanometer-scale device performance other than the self-consistent solution of coupled transport and Poisson’s equations with proper boundary conditions by the MC or other numerical methods. 5. Conclusion We have improved the expression of the saturated drain current as given in [1,13] based on the multiple reflection of carriers at the VS along the line of the one-flux approximation [20]. Our result does not involve any approximations but contains a few more variables. It is microscopically more accurate because it takes into account the energy spectrum of the fluxes entering the black boxes and their effects on the reflection coefficients. By comparing our results with existing formulae available in the literatures, we are

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able to indicate where specific approximations are made and what effects these approximations have on the existing formulae. Our results are verified by simulation results of a 10 nm gate-length SNWT. We have also proposed tSK+  (1 + ktR)1 and B  (1  R) (1 + ktR)1 as possible approximations. Although our verification of the results is based on the 1-D simulation of a SNWT, we believe that the implication of the results is equally applicable to 2-D MOSFETs. From these results, we can also estimate the fractional errors resulting from the use of alternative expressions. Sometimes these approximations still yield reasonable results, mainly thanks to a fortuitous cancelation of errors. Finally we conclude that, because of the difficulty in modeling all variables appearing in the derived expression of the saturated drain current, it is not worthwhile to devise any simple model for the prediction of device performance in the nanometer regime. Acknowledgements T.-w. Tang and M.V. Fischetti acknowledge a research support from Samsung Electronics Co. in Korea. T.-w. Tang also thanks private communications with G. Giusi.

Appendix A Referring to Fig. 1, the positively-directed flux of those electrons injected from the source contact with energy exceeding the barrier height is designated by Iþ S;bal . After being transmitted through the upstream region and reaching the VS, this flux becomes t S Iþ S;bal . This positively-directed flux after entering the downstream region is reflected with the refection coefficient r þ 1 and results in a negaþ tively-directed flux designated by rþ 1 t S IS;bal . This negatively-directed flux, after reflection from the upstream scattering black box with the reflection coefficient r  1 results in a positively-directed flux þ þ r r t I . Such multiple reflections continue until I+ and I reach S 1 1 S;bal þ steady state. Setting t S IS;bal ¼ 1 temporally, we evaluate this multiple reflection process taking place at the VS as follows:

Iþð0Þ ¼ 1: Ið0Þ ¼ r þ1 Iþð0Þ ¼ rþ1 : Iþð1Þ ¼ r 1 Ið0Þ ¼ r1 rþ1 : Ið1Þ ¼ r þ2 Iþð1Þ ¼ rþ2 r1 r þ1 : Iþð2Þ ¼ r 2 Ið1Þ ¼ r2 rþ2 r 1 r þ1 : Ið2Þ ¼ r þ3 Iþð2Þ ¼ rþ3 r2 r þ2 r 1 r þ1 : Iþð3Þ ¼ r 3 Ið2Þ ¼ r3 rþ3 r 2 r þ2 r 1 r þ1 : Ið3Þ ¼ r þ4 Iþð3Þ ¼ rþ4 r3 r þ3 r 2 r þ2 r 1 r þ1 . . . ; where superscripts denote the number of each correction over the preceding flux.

Finally, putting back tSIS,bal, we obtain þ

I ¼ Iþð0Þ  Iþð1Þ þ Iþð2Þ  ¼ tS IþS;bal ½1  r 1 r þ1 ð1  r2 rþ2 ð1  r 3 r þ3 ð1  . . .ÞÞ . . .Þ:

ðA1Þ

I ¼ Ið0Þ  Ið1Þ þ Ið2Þ  . . . ¼ tS IþS;bal r þ1 ½1  r 1 r þ2 ð1  r2 r þ3 ð1  r 3 r þ4 ð1  . . .ÞÞ . . .Þ:

ðA2Þ

The alternate signs appearing in the summation of I+ and I are explained as follows. The initial estimate, I+(0)(I(0) of I+(I) is overestimated and requires a correction by I+(1)(I(1)). But this correction is also overestimated and requires another correction to the correction by I+(2)(I(2)). As this correction process continues with alternate signs, eventually I+(I) reaches a steady state value. The þ alternating series in (A1) converges since 1 > r 1 r1 >  þ  þ r2 r 2 > r3 r 3 > . . . and likewise the series (A2) converges since þ  þ  þ 1 > r 1 r 2 > r 2 r 3 > r 3 r 4 > . . .. References [1] Lundstrom M. IEEE Electron Dev Lett 1997;18(7):361–3. [2] Chen M-J, Huang H-T, Chou Y-C, Chen R-T, Tseng T-T, Chen P-N, et al. IEEE Trans Electron Dev 2004;51(9):1409–15. [3] Lin H-N, Chen H-W, Ko C-H, Ge C-H, Lin H-C, Huang T-Y, et al. IEEE Electron Dev Lett 2005;26(9):676–8. [4] Chen M-J, Yan S-G, Chen R-T, Hsieh C-Y, Huang P-W, Chen H-P. IEEE Electron Dev Lett 2007;28(2):177–9. [5] Liao M-H, Liu C-W, Yeh L, Lee T-L, Liang M-S. Appl Phys Lett 2008;92(063506): 1–3. [6] Khakifirooz A, Antoniadis D. IEEE Trans Electron Dev 2008;55(6):1391–400. [7] Wang R, Liu H, Huang R, Zhuge J, Zhang L, Kim D-W, et al. IEEE Trans Electron Dev 2008;55(11):2960–7. [8] Fuchs E, Dollfus P, Carval G, Barraud S, Villanueva D, Salvetti F, et al. IEEE Trans Electron Dev 2005;52(10):2280–9. [9] Palestri P, Esseni D, Eminente S, Fiegna C, Sangiorgi E, Selmi L. IEEE Trans Electron Dev 2005;52(12):2727–35. [10] Tsuchiya H, Fujii K, Mori T, Miyoshi T. IEEE Trans Electron Dev 2005;53(12): 2965–71. [11] Barral V, Poiroux T, Vinet M, Widiez J, Previtali B, Grosgeorges P, et al. SolidState Electron 2007;51(4):537–42. [12] Barral V, Poiroux T, Saint-Martin J, Munteanu D, Autran J, Deleonibus S. IEEE Trans Electron Dev 2009;56(3):408–19. [13] Giusi G, Iannaccone G, Crupi F. IEEE Trans Electron Dev 2011;58(3):691–7. [14] Svizhenko A, Anantram M. IEEE Trans Electron Dev 2003;50(6):1459–66. [15] Martin J, Bournel A, Dullfus P. IEEE Trans Electron Dev 2004;51(7):1148–55. [16] Natori K. IEEE Electron Dev Lett 2002;23(11):655–7. [17] Gnani E, Gnudi A, Reggiani S, Baccarani G. IEEE Trans Electron Dev 2008;55(11):2918–30. [18] Kim R, Lundstrom M. IEEE Trans Electron Dev 2009;56(1):132–9. [19] Jin S, Tang T-W, Fischetti M. Proc 13-th international workshop on computational electronics, Beijing, China; May 2009. p. 313–6. [20] Mckelvey J, Longini R, Brody T. Phys Rev 1961;123(1):51–7. [21] Shockley W. Phys Rev 1962;125(5):1570–6. [22] Gildenblat G. J Appl Phys 2002;91(12):9883–6. [23] Tang T-W, Fischetti M, Jin S. Proc 2009 international semiconductor device research symposium, College Park, MD, USA. IEEE Explore; December 2009. doi:10.1109/ISDR.2009.5378164. [24] Natori K. J Appl Phys 1994;76(8):4879–90. [25] Jin S, Fischetti M, Tang T-W. IEEE Trans Electron Dev 2008;55(11):2886–97. [26] Jin S, Fischetti M, Tang T-W. J Apll Phys 2007;102(8):083715-1–083715-14. [27] Jin S. Ph.D. dissertation, Seoul National University, Korea; February 2006. [28] Jin S, Fischetti M, Tang T-W. Appl Phys Lett 2008;92(8):082103-1–3-3. [29] Fischetti M, Jin S, Tang T-W, Asbeck P, Taur Y, Laux S, Rodwell M, Sano N. J Comp Electron 2009;8:60–77. [30] Tang T-W, Yoon I-O, Sano N, Jin S, Fischetti M, Park Y-J. Proc 14-th International Workshop on Computational Electronics, Pisa, Italy; October 2010. 61–4. [31] Sano N. Phys Rev Lett 2004;93(246803):1–4.