Mean-Free-Path for Backscattering

If we want to use (10) to evaluate the simulation

r

c data, we need information for two important parameters: mean-free-path for backscattering λ₀ and corresponding k_BT-layer’s width A . However, according to the backscattering framework [1-2], A can be explicitly expressed as a function of the thermal energy kBT, the conductor length L, and the applied voltage Va. For this reason, we have

B a

Lk T

= qV

A for the linear potential profile and ^B

a

L k T

= qV

A for the parabolic one.

Since the k_BT-layer’s width is clearly defined, in order for calculations from (10) to have good agreement with simulated

r

c, mean-free-path for backscattering λ₀

should be obtained accurately. We can extract λ₀ via (11) on the longest conductor (L = 100 nm) in the low electric filed, V_a <<k_BT/q, that is to say, L>>l . Figure 2-4 shows the extracted results for three different temperatures of 150, 200 and 300 K. In the figure, the bar represents the range of simulated

r

c values. The average

r

c of a linear potential profile is close to that of the parabolic one, which indicates λ₀ is essentially independent of the potential profile we adopted in the low field regime.

The average

r

c over these two potential profiles leads to the average λ₀ of 56, 105,

and 155 nm for 300, 200 and 150 K, respectively. The extracted average value of λ₀ at 300 K is identical to that of our group previous work [5].

Section 2.3 Result and Verification

Since the mean-free-path for backscattering is extracted via (11) in the low electric field or near-equilibrium condition. The corresponding A is ^B

a

Lk T

= qV

A for

the linear potential profile and ^B

a

L k T

= qV

A for the parabolic one. Figure 2-5 and 2-6

show the simulated

r

c in symbols and (10) in dotted lines. Comparison between linear potential profile and parabolic one, it is easy to find out that there is discrepancy in the linear potential profile between calculations and simulation values. This deviation increases with decreasing conductor length and decreasing lattice temperature.

However, Figure 2-6 shows good agreement between the simulated

r

c and that of (10).

In order to discuss this discrepancy, we have to investigate two important parameters, A and λ₀, in (10). First, according to the open literature [1-2], when Va is much larger than kBT/q, the definitions of the kBT-layer’s width in both potential profiles have been defined. Second, the other parameter λ₀ is called near-equilibrium

mean-free-path for backscattering. We extracted λ₀ via (11) in the low electric field or near-equilibrium condition, and it is the only mean-free-path used for all operating conditions. In [10], the similar expression of (10) is also derived via Boltzmann transport equation in the framework of a “relaxation length” approximation, and within that approximation, mean-free-path λ₀ must be independent of carrier energy.

In other words, λ₀ has to be constant no matter how large the applied voltage is.

However, Monte Carlo simulation results at room temperature in linear potential profile show that λ₀ should change. Certain relationship,λ'=λ γ₀/ with γ =1.5 to 2.0, is discovered [15]. We will discuss this issue later.

Chapter 3

A Parabolic Barrier Oriented Compact Model for the k

_B

T Layer’s Width

The channel backscattering theory [1-2] establishes a connection between the kBT-layer’s width, which locates in the beginning of the conductive channel near the source, and the drive performance of the device. However, the ability to quantitatively determine the width of this critical zone is necessary. We have discussed the backscattering coefficient through scattering matrix approach and performed Monte Carlo simulation in Chapter 2. Although l can be explicitly expressed as a function of the conductor length L, the thermal energy k_BT, and the applied voltage V_a, this expression can not elucidate the effect of gate voltage on l . Moreover, the influence of temperature on l should be discussed more clearly. In this chapter, a parabolic potential barrier oriented compact model for k_BT-layer’s width in nano-MOSFETs is derived, and will be verified by experiment and simulation.

Section 3.1 Discrepancy in Temperature Dependency

Mostly adopted model of l can be quoted in the literature [6],

(

^B

)

a

L k T qV

=

α

l

(13)

where l is the k^BT-layer’s width and L is the conductor channel length. In the previous chapter, we can easily define l in both linear and parabolic potential profile with α= and 1 α=0.5, respectively. However, some division among the magnitude of the temperature exponent α should be clarified. In the first place, the I-V characteristics of a simulation double-gate MOSFET at room temperature can be

best fitted by α=0.57 [6]. In addition, comparable α=0.5 has been produced on experimental bulk n-MOSFETs in a temperature range of 233 to 298 K [5], [7].

Besides, α=0.75, which covers the same range as the former case, has been experimentally determined [3]. Still, α= has already been adopted in a 1 temperature dependent backscattering coefficient extraction method [4]. Another open literature [8] also shows that l is approximately proportional to the temperature form 100 to 500 K, which means α= . Apparently, the issue of widely-ranged 1 α values must be addressed. Furthermore, it is difficult for (13) to elucidate the effect of gate voltage. We will make a simple approach by assuming a parabolic potential near the source-channel junction. Then a compact model with the channel, gate overdrive, drain voltage, and temperature as input parameters will be physically derived.

Section 3.2 Parabolic Barrier Profile

A parabolic potential profile close to the source is schematically plotted in Figure 3-1. This potential profile that extends to the remaining channel can be described as

( )

D

( / )

2

V x = V x L %

₍₁₄₎

The origin x=0 is the peak of the barrier. L% indicates the imaginary channel length corresponding to a certain position where the parabolic potential drop from the top of the barrier is equal to V . At the other side, we neglect the barrier height with respect _D to the source side because of the large drain voltage. Then, by substituting x= l and V x( )=k T q_B / into (14), which means a local potential drop being equal to thermal energy, the l expressed in terms of imaginary L%, applied voltage V , and _D thermal energy k T_B can be obtained:

1

Obviously, once L% can be expressed as a function of channel length, gate voltage, drain voltage and temperature, the expression of l in terms of these parameters can be obtained.

That is to say, this imaginary L% just locates on the real channel length. Then, we will discuss each component in later sections.

Section 3.2.1 Temperature Effect

First of all, the thermal energy layer width, corresponding to L% →L, T → , T₀

Then only the temperature changes individually from T₀ to T . By dividing (15) by (17), a power-law relation can be found:

1 2

( / 0)

L% =L T T (18) Here, we have assumed that the potential profile does not change with temperature. In other words, the local electric field from the peak of the barrier to the end of kBT-layer is approximately the same in these two temperatures. According to the backscattering

theory [1], the local electric fields across this section are equal to k T_{B 0}/ql and ₀

B /

k T ql at T0 and T, respectively.

Section 3.2.2 Gate Voltage Effect

Once the potential profile is assumed, we can easily differentiate (14) twice with respect to position x, which leads to

2

Under the same V , and according to Poisson’s equation, (19) relates to the charge _D density. From [6], we can know that (19) also can be linearly related to the underlying inversion-layer density, which is proportional to the gate overdrive. It turns out that

1

Section 3.2.3 Drain Voltage Effect

The local electric field can be obtained by differentiating (14) once. Thus, if the drain voltage increases from V_D0 to V , local electric field, _D 2V x_D2

L% , must be larger that 2V x_D2⁰

L . If the local electric fields were equal, we could know that

0.5

( _D/ _D0)

L%= L V V . Because V_D >V_D0, the electric field is increased. Thus the power exponent must be no more than 0.5. As a result, one obtains

0.25

( _D/ _D0)

L%= L V V ^ν= (21)

The power exponent ν, which is equal to 0.25, has been experimentally determined in the previous work of our group [7]. For V_D <V_D0, the same expression as (21) can be obtained.

Section 3.2.4 Combination of Temperature and Bias Effects

Through the combination of (18), (20), and (21), these power-law relationships establish a unique expression for L% :

0.25 0.5

Combining (15) and (22) can further lead to

0.5 0.5 fitting parameter in this expression. Indeed, it is supposed that η is a constant value over the channel length, gate and drain voltage, and temperature. Thus, the capability of this parabolic potential barrier oriented compact model will be more general.

Section 3.3 Model Verification

On the one hand, the experimental l in this section was obtained from 55-nm bulk n-MOSFETs by means of a parameter extraction process proposed by our group, and the details can be found in [3], [5], [7]. On the other hand, we quoted the rich literature [8],[19], in which Monte Carlo simulations including quantum corrections to the potential and calibrated scattering models are used to study carrier transport in bulk and double-gate silicon-on-insulator MOSFETs. Also cited are those of the open

literature [2], [6], [16], [17], [18].

Section 3.3.1 Experimental Validation

Figure 3-2 shows the experimental l , which is obtained from 55-nm bulk n-MOSFETs with tox=1.65nm, Npoly=1x10²⁰cm^-3, and NA=7x10¹⁷ cm^-3, versus gate voltage for two drain voltages of 0.5 and 1.0 V and three different temperatures of 233, 263, and 298 K. With these experimental l values, temperature T, and applied drain voltage V_D, we can obtain the corresponding L~

using (15). These results are shown in Figure 3-3 for two drain voltages versus gate voltage. The corresponding near-equilibrium threshold voltage, denoted V_tho for 233, 263, and 298K are 0.360, 0.345, and 0.328 V, respectively. The drain-induced barrier lowering (DIBL) magnitude for 233, 263, and 298K are 120, 123, and 130 mV/V, respectively.

Throughout this work, the threshold voltage Vth at higher drain voltages is equal to

tho D

V −DIBL V× .

Section 3.3.2 Simulation Validation

First, in Ref. [8], the extracted l at V^D = V_G = 1 V is available in a wide range of the channel length from 14 to 37 nm and also a wide range of the temperature from 100 to 500 K. The underlying threshold voltage V_tho and DIBL are reasonably 0.3 V and 110 mV/V, respectively. Second, the citation [19] can further provide the relevant information at room temperature: l from 2.0 to 7.0 nm, L from 14 to 65 nm, V^D (=

V_G) from 1.0 to 1.2 V, and DIBL from 11 to 230mV/V. The above information is listed in Table 3-1. Moreover, we have also extracted l from the published literature [6], [8], [16-18], on double-gate device simulation and l can be extracted directly from the channel potential profiles. The corresponding key parameters are: (i) L = 10 nm, V ≈ 0.33 V, DIBL ≈140 mV/V, V = 0.6 V, V = 0.6 V, and T = 300 K [2]; (ii) L = 20

nm, Vtho ≈ 0.33 V, DIBL ≈ 25 mV/V, VD = 0.2 V, VG = 0.55 V, and T = 300 K [6]; (iii) L = 25 nm, Vtho ≈ 0.3 V, DIBL ≈100 mV/V, VD = 0.8 V, VG = 0.5, 0.8, and 1.0 V, and T

= 300 K [16]; (iv) L = 15 nm, V_tho ≈ 0.2 V, DIBL ≈120 mV/V, VD = 0.7 V, V_G = 0.7 V, and T = 300 K [17]; and (v) L = 15 nm, Vtho ≈ 0.3 V, DIBL ≈ 77 mV/V, VD = 0.7 V, VG

= 0.7 V, and T = 300 K [18]. The above key parameters are also listed in Table 3-1.

With all these key parameters, a scatter plot can be created as shown in Figure 3-4 in terms of the experimental and simulated l versus the quantity of the functional expression LV_D^0.25(V -V ) (k T/q) (k T/qV ) . Apparently, all data seem to _{G th} ^-0.5 _B ^0.5 _{B D} ^0.5 fall on or around a straight line. The slope of this line furnishes η with a value of

4.1V⁻0.25. Still, η remains constant, regardless of the channel length, gate and drain voltage, and temperature.

Section 3.4 Temperature Power Exponent Clarification

Some remarks can now be made to clarify the confusing α values in the open literature [3]-[8]. First of all, it is noticed that in case of bulk n-MOSFET two different values of α were produced: one of 0.5 [5], [7] and one of 0.75 [3]. This difference can be attributed to the different subband treatments during the parameter extraction process: Schrödinger-Poisson equations are numerically solved in [5], [7]

whereas in [3] this was done by a triangular potential approximation [20]. Therefore, different subband levels can lead to different average thermal injection velocities, which in turn give rise to different l values. Secondly, according to (22) a temperature range of 233 to 298 K in case of 55-nm bulk device [5], [7] is not large enough to affect the calculated L~

. In other words, L~

is considerably insensitive to such a narrow temperature range. Consequently, the resulting apparent temperature power exponent was limited to 0.5 as reported in the previous work [4], [5]. Indeed, with the known η as input, fairly good reproduction can be achieved as depicted in

the α ≈ 0.57 case [6]: since the room temperature of operation was involved alone, the temperature effect of L~

can no longer be examined. Only in a wide temperature range as done in the comprehensive study of [8], [19] can the linear relationship of

∝T

l as shown in Figure 3-4 actually occur. Thus, from the aspect of temperature dependencies, excellent coincidence with the data as shown in Figure 3-4 stresses that the existing backscattering coefficient extraction method [4] is valid.

Section 3.5 Results

This new compact kBT-layer’s width model, which links the width of thermal energy k_BT layer to the geometrical and bias parameters of the devices, is physically derived on the basis of a parabolic potential profile around the source-channel junction barrier of nanoscale-MOSFETs. Moreover, this proposed model is supported not only by experimental data, but also by various simulation works presented in the open literature. Only one fitting parameter remains constant in a wide range of channel length (10 to 65 nm), gate voltage (0.4 to 1.2 V), drain voltage (0.2 to 1.2 V), and temperature (100 to 500 K), which means that the capability of this compact model is universal.

Chapter 4

Re-examination of Mean-Free-path for Backscattering

When applying (1) to nanoscale MOSFETs to predict relevant current-voltage characteristic, we need to quantify the key backscattering coefficient rc. However, there are two important parameters needed in rc calculation. We have discussed the l on the basis of a parabolic potential profile around the source-channel junction barrier and obtained a compact model of l with the channel length, gate overdrive, drain voltage and temperature as input parameters. On the other hand, another parameter λ0 has not been discussed yet. In Section 2.2.3, we extracted this information under near-equilibrium low field condition. When comparing (10) to compare with Monte Carlo results, it seems to have some differences between calculated and simulated rc

in the linear potential profile. In parabolic case, Figure 2-6 shows good agreement between calculation and simulation results. In this chapter, we will discuss the near-equilibrium mean-free-path for backscattering.

Section 4.1 Apparent Mean-Free-Path

At the end of Chapter 2, we pointed out that in the citation [12], the Monte Carlo simulations at room temperature in case of non-degenerate statistics on a linear channel potential profile have exhibited certain relationship: λ'=λ γ₀/ with γ =1.5 to 2.0. Here λ is the apparent mean-free-path that constitutes the ' following expression in the high field case:

c

'

r ⁼ + l λ

l

⁽²⁴⁾

Thus, some clarifications are demanded.

In order to achieve this goal, we perform Monte Carlo particle-based simulation on a silicon bulk conductor like what we have done in Chapter 2, but pay more attention on the velocity distribution on the top of the barrier and flux ration at the end of the kBT layer.

Section 4.1.1 Mean-Free-Path Extraction in Linear Potential

First of all, we extract the mean free path from the simulated rc using (25), proven in Chapter 2. By substituting the simulated rc into (25), the underlying λ ' can be obtained as depicted in Figure 4-1 versus applied voltage. From Figure 4-1, we can see that: (i) λ falls below ' λ₀, and decreases with increasing applied voltage, leading to λ'=λ γ₀/ with γ = 1.5 to 2.5; and (ii) on average, λ decreases with '

decreasing conductor length. However, it is supposed that the upper limit of λ0 can be recovered only with increasing conductor length or decreasing applied voltage. On the other hand, the ratio of λ to ' λ₀ appears to be a weak function of the lattice temperature. The same argument has been produced by the recent Monte Carlo simulations [12] devoted to a linear potential profile at lattice temperature of 300 K as we mentioned at the end of Chapter 2.

Section 4.1.2 Mean-Free-Path Extraction in Parabolic Potential

For a parabolic potential profile, the corresponding simulated

r

c is displayed in Figure. 2-6. Here, the lines are calculated from (25) with λ of 56, ' 105, and 155 nm for 300, 200, and 150 K, respectively. Good reproduction of the data for different conductor lengths, different temperatures, and, especially, different applied voltages can be gotten. Moreover, the reproduction can all be achieved with simply λ'=λ₀, without adjusting any parameters. This means no matter how the quasi-ballistic transport prevails in the kBT layer (λ'> A), the quasi-equilibrium conditions still govern the backscattered carriers.

Section 4.2 Evidence for Carrier Heating

In Section 4.1, we see that the discrepancy in the mean-free-path in linear potential profile. This inconsistency can be contributed to the presence of the carrier heating. In citation [10], all the derivations are carried on the basis of the “relaxation length” approximation, which means that the mean-free-path is a constant, and is independent of the carrier energy or carrier temperature. However, as carriers transport under applied electric field, their energy should be changed, especially in the linear potential profile. Because of the lack of weak field regime near the injection point, the carriers experience a larger electric field in the linear one, and thus the deviation between the lattice temperature and carrier temperature would be possible.

On the other hand, for a parabolic potential profile, as showing Figure 2-2, there is significant fraction of the kBT layer, which can be determined as the weak field regime. Take this into consideration, although the deviations from the lattice temperature would be possible as entering into the remainder, the overall carrier heating in the kBT layer should be weakened. Consequently, when the carriers are injected into the channel at the beginning of the k_BT layer, they immediately undergo the strong acceleration. Also, owing to the quasi-ballistic transport, the carrier

mean-free-path is therefore no longer independent of the carrier energy.

Section 4.2.1 Velocity Distribution at the Injection point

Concerning the difference between the lattice temperature and carrier temperature, the carrier velocity at the injection point is helpful as demonstrated in Figure 4-2 and 4-3. Figure 4-2 shows the velocity distribution of L=25nm at Va=0.8 V and 300 K for two potential profiles. Figure 4-2 clearly shows that two significant differences between the potential profiles. First, the injected single hemi-Maxwellian velocity distribution is retained in the positively-directed carriers with the parabolic one, but it is split into two distinct components in the linear one: One of the longitudinal effective mass and one of the transverse effective mass. Second, the distribution of the negatively-directed or backscattered carriers appears to be wider in the linear potential profile than the parabolic one. The same result can be seen in Figure 4-3 for L=50nm at V_a=1.0 V and 150 K. This is evidence of carrier heating.

Section 4.2.2 Flux Ratio at the End of k_BT Layer also can be cited in the literature [10], which was derived on the basis of a Maxwillien shape distribution in both the forward and backward directions with in the context of the relaxation length approximation. In order to get the ratio of negatively-directed flux to positively-directed flux, we substitute (10) into (26) for r_c=a(0) / (0)b . Then,

accounting for the conservation of the current, the ratio ( ) / ( )b l a l can be straightforwardly calculated. Monte Carlo simulation program can provide the velocity distribution at the end of the kBT layer. For example, Figure 4-4 shows velocity distribution of L=50 nm, T=200 K, and V_a=0.2 V at x= A. The velocity distribution also is split into two components, and velocity distribution of backscattered carrier is wider. The ( ) / ( )b l a l is the ratio of deep grey area to light grey area, like what we did to determine

r

c at the injection point in Section 2.2.2. The calculated lines from (26) are shown in Figure 4-4 for linear potential profile. Also plotted in the figure are those of the Monte Carlo simulation. However, the calculated values are seen to fall below the simulation ones. This reveals the fact that the reflected flux at the end of k_BT layer can be enhanced in the presence of carrier heating. Only with decreasing applied voltage can the deviation between the simulation and model calculation be shortened.

Section 4.3 Results

We executed Monte Carlo simulations on a silicon bulk conductor to re-examine the channel backscattering theory in bulk nano-MOSFETs. Through these simulations, some important points can be addressed again:

(i) The near-equilibrium mean-free-path for backscattering λ₀, is independent of the potential profile.

(ii) λ in a linear potential profile is lower than ' λ₀ of parabolic one due to the presence of the carrier heating. Evidence is highlighted by both the velocity distribution at the injection point and the flux ratio at the end of the k_BT layer.

Chapter 5 Conclusion

The backscattering coefficient is derived through the scattering matrix approach and verified by Monte Carlo simulations. Two important parameters, λ₀ and A ,

The backscattering coefficient is derived through the scattering matrix approach and verified by Monte Carlo simulations. Two important parameters, λ₀ and A ,

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