In this thesis, we explore the phenomenon of mobility degradation and the possibility of applying the channel backscattering theory to the result simulated by traditional transport model with channel scaling down at three different temperatures. The following Chapter 2 will explain the channel backscattering theory and the temperature coefficient method proposed elsewhere [3]. And in Chapter 3, we will show the simulation result and extract the temperature coefficients. Furthermore, we will give the reasons of mobility degradation and compare the errors between the channel backscattering theory and temperature coefficient method. We also will bring up a possible solution to make us the extracted ballistic ratio proper. Finally, we make a conclusion in Chapter 4.
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Chapter 2
Channel Backscattering Theory
Section 2.1 Introduction
Channel backscattering theory is a simple one-flux scattering theory used in MOSFETs [2], [10], [11], [12]. The theory gives a new expression of current-voltage characteristics used to analyze short-channel devices.
The new formula is expressed in terms of scattering parameters rather than a mobility. For long-channel transistors, the results reduce to conventional drift-diffusion (DD) theory. But DD theory also applies to short channel device even as the channel length is shorter than the mean-free-path. In backscattering theory, the transconductance is limited by carrier injection from the source for ultra short channel. Another channel backscattering concept to analyze short channel effect has been proposed by [3] in terms of the Temperature Coefficient Method. In this concept, temperature is used to analyze the variation of channel status for different temperatures. By extracting the mean free path, critical scattering length, the temperature coefficients of mobility, injection velocity and the power law coefficients to define the ballistic ratio.
Section 2.2 Channel Backscattering Theory
With channel scaling down, traditional current model would touch a limit caused by the saturation velocity. The key issue of channel backscattering theory is to maximize the saturation drain current for
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short-channel devices. Electrons are injected from the source into the channel across a potential barrier varying with the gate voltage and the drain voltage. Traditionally, carriers drift and diffuse across the channel and finally are collected at the drain. Fortunately, TCAD can work well on the 2D simulation. That is we could simply grasp the status on the virtual source point. And drain-induced barrier lowering (DIBL) has been taken into account by using TCAD simulator. In the theory, the virtual source at the barrier height is considered as a carrier reservoir; which injects a flux from the source side across the barrier into the channel.
There are not all of the carriers that can transmit across the barrier. A fraction of the flux, rc, goes back to the source. The backscattering region, k-T layer, is the key region which dominates the backscatter flux. As shown in Fig. 1-2, k-T layer is the length from the barrier height to the point with a kT potential drop rather than the barrier point. The length is called critical scattering length, LkT. Physically, the channel length is equal to the critical scattering length at very small drain bias. Therefore, whenever the critical length is larger than the channel length, LkT is assumed to be the same as LG. We can find the corresponding drain current from the backscattering coefficient rc as below. In Fig. 1-2, F+ is the injection flux from source to channel, and Fb
- is the injection flux from drain to source. T+ is the transmission coefficient for F+ flux, and T -is the transm-ission coefficient for Fb
- flux. At equilibrium, T -approximates T+=T,
(2-1) (2-2)
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where is the equilibrium unidirectional thermal velocity (i.e., the average velocity of carrier across the barrier in the positive direction).
And position 0 is defined at the Xvs point. From Eq. (2-1) and (2-2), the drain current can be expressed as:
Under ballistic condition, the negative flux caused by thermal emission from the drain is given by thermionic emission as:
(2-4) (2-5) Substituting Eq. (2-4) and (2-5) into (2-3), we have
. (2-6) And taking the series resistance effect into Eq. (2-6), we can get the drain current at linear and saturation regions:
the degenerate case, we use the injection velocity to replace the thermal
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velocity. The injection velocity can be expressed as
. Furthermore, rc can be expressed as:
(2-9) is the mean free path and is the mobility extracted at thermal equilibrium. In simulation, DIBL has been included. Therefore, Qinv has DIBL effect in itself.
Section 2.3 Mean Free Path and Critical Scattering Length
The mean free path is a controlling factor to determine the performance of device. It is associated with the mobility and the thermal injection velocity. Besides, the critical scattering length is widely evaluated by each group’s study [5], [6], [14]; that is, how to calculate LkT
would really decide the characteristics of device. Therefore, we will slightly correct the critical scattering length by using the direct extraction of LkT from the simulation result.
Section 2.4 Temperature Coefficient Method (TC Model)
In section 2.2, we explained the component of the backscattering coefficient, rc. It can be clearly found that it has some relationship with the temperature. This means there is another way to extract the scattering coefficient rc. This is TC model. Replacing rc with and LkT, leading to
(2-10)
8 Differentiating Eq. (2-5) with respect to temperature:
Substituting Eq. (2-12) into (2-14),
Again substituting Eq. (2-15) into (2-11),
This is the temperature coefficient method to extract the ballistic ratio RB
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and the scattering coefficient rc.
Section 2.5 Conclusion
From Eq. (2-7), we can get BR directly and compare it with the result of TC model by using Eq. (2-16). Generally, the two methods may not get the same BR value for any conditions, but it would suggest that there is another factor needed to add into the simulation or the theory should be corrected. In Chapter 3, we will show the calculated result between the two methods.
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Chapter 3
Analysis of TCAD simulation result
Section 3.1 Introduction
In this chapter, we focus on: (i) the extraction of five power-law parameters ( , , , , ); (ii) the extraction of BR; (iii) the comparison of BR between the direct extraction and the calculation result by using TC model; and (iv) the difference between the key point, X=Xvs, and the balance point, X=0. In the study, the whole channel status is taken into consideration by using the TCAD simulator. We used the quantum mechanics to accurately deal with the sub-band energy in the y direction along the channel, and drift and diffusion transport model to get a self-consistent solution along the channel.
Section 3.2 Extraction of Five Temperature Coefficients
Section 3.2.1 Schrödinger–Poisson–Drift–Diffusion (SPDD) model
The SPDD model was proposed in [1], [2]. From Schrödinger equation (3-1), a set of quantum energy states {Eij}, , , can be solved:
(3-1) With device parameters and the operation temperature and applied bias as input, TCAD can accurately calculate the sub-band energy and the
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Fermi-level, thus producing the inversion layer carrier density (2DEG) per sub-band as:
, (3-2) where i=1, 2 (valley), j=1, 2, 3 (sub-band)…; nvi is the degeneracy factor
of ith valley; mdi is the density of states effective mass of ith valley. Ef is the quasi-Fermi level.
And by using the Poisson equation as Eq. (3-3), we can get a self-consistent result of the Schrödinger–Poisson as shown in Fig. 3-1:
(3-3) We get the 1-D Schrödinger–Poisson solution in the y-direction, and we can apply it to 2-D case by appending the DD model to constitute a self-consistent solution along the two directions as shown in Fig. 3-2. The electron and hole continuity equations are written as:
(3-4)
, (3-5) where Rnet is the net electron/ hole recombination rate, is the electron current density, and is the hole current density.
The DD model is widely used as a carrier transport model in semiconductors and is defined by the following equations for the current densities of electrons and holes:
(3-6) (3-7)
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where μ n and μ p are the electron and hole mobilities, and Φn and Φp
are the electron and hole quasi-Fermi potentials, respectively.
Section 3.2.2 Channel Status at Equilibrium
Usually we thought the highest energy of conduction band would locate at the mid of channel, but it would vary from the mid of channel toward the source and drain with increasing gate voltage. As shown in Fig.
3-3 is a profile of the lowest sub-band, E11, along the channel. We found the highest energy is located 2~5nm away from the junction of source/channel and channel/drain in channel. Fig. 3-4 show the sheet charge. In basic Poisson equation, we considered only forward or backward direction. In reality, we should take both into consideration.
This leads to the result of Fig. 3-3. Apparently, the highest point of E11 is not located at the mid-channel. The difference between VD=0V and 1mV is due to the variation of Fermi-Level.
The local higher potential causes a relative lower carrier density at that region. As shown in Fig. 3-5 is the inversion carrier density Ninv
along the channel. We get almost the same Ninv at VD=0V and 1mV. This shows channel is at equilibrium condition for VD=1mV. Furthermore, we find that the inversion carrier density can be gradually affected as the channel length is scaled down. The effect of drain induced barrier lowering, DIBL, can cause the carrier density increasing for LG=15nm, except at LG=90nm. It shows a similar characteristic as [13]. The short channel effect would appear for channel length is smaller than 40nm or the temperature of 150K.
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Finally, we want to compare how much difference between each channel length different gate voltages can produce. Here, we used a quite correct simulator, Schred [7], evaluated by Prof. Lundstrom, et al. at Purdue University, as the standard to check and analyze the TCAD’s simulation result. As shown in Fig. 3-6-1, we can get a perfect match between TCAD and Schred at long channel, but source and drain affect the channel as channel scaling to 15nm. Usually, we think DIBL is a constant for sub-threshold and above threshold region. Indeed, the channel resistance is high below the sub-threshold region; and above threshold region, the source and drain resistance are relative higher.
Therefore, DIBL is not a constant for any gate voltage as shown in Fig.
3-6-2.
Section 3.2.3 Mobility Extraction at Equilibrium
After analyzing the status of channel at equilibrium, it provided two ways to calculate the mobility for us. First, according to the experimental concept, using the results of ID against VG and the inversion carrier density calculated by C-V measurement, we should assume that the drain current uniformly flows through the channel and the carrier uniformly distributes in channel. The results are shown in Eq. (3-8). Second, according to the ballistic theory, carriers come from the virtual source and may be reflected with a critical scattering length Lkt and a mean free path λ. Whatever the methods used, it should follow the DD model as Eq. (3-6) and (3-7). As shown in equation (3-8), we can get a relationship between ID per unit width and current density of electrons and holes. We can even reduce Eq. (3-8) to Eq. (3-9), because the majority carriers are electrons:
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(3-8)
(3-9) Although the drain current is conserved along the channel, we can clearly find that from Fig. 3-5 the carrier density is not the same along the channel. From electron quasi Fermi-level along the channel as shown in Fig. 3-7, we can extract the series resistances of source and drain as shown in Fig. 3-8-1 and Fig. 3-8-2:
. (3-10) Fig. 3-8-1, 3-8-2 and 3-8-3 show Rtot, RSD and RChannel, respectively, for four different channel lengths from 90nm to 15nm and three temperatures. There is a slight difference between two applied VD, which is caused by the mobility degradation at VD=0.025V. That suggests that it is not appropriate to extract mobility at drain bias of 0.025V for nanoscale device. We could predict the trend of mobility by using the characteristic of the channel resistance that is proportional to 1/LG as revealed by Eq. (3-11). However, is not zero at LG=0nm at all for VD=0.025V:
; (3-11)
, (3-12) where is the apparent mobility, is the low field mobility, and is a correction term due to the mobility degradation.
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In Fig. 3-9-1, 3-9-2 and 3-9-3, we sight on the mid-channel (X=0) and X=Xvs. When VD is 1mV, there is no decrease on mobility with the channel length, but we can find a strong degradation on mobility at VD=25mV. This suggests the decrease of mobility with scaling down is caused by the saturation velocity. According to the Canali model evaluated by Canali et al. [4], it is used to explain the high field saturation phenomenon. The Caughey-Thomas formula is shown in below:
(3-13)
where is the saturation velocity and is a fitting parameter for Caughey-Thomas formula. Parameters’ setting is shown in Table 1. We confirmed the mobility degradation by using the Caughey-Thomas formula. As shown in Fig. 3-9-4, we set the low field mobility the same as the mobility extracted from DD model at VD=1mV and LG=90nm. The result agrees with our prediction. That is the mobility degradation caused by the limit of saturation velocity with channel length shrinkage.
Section 3.2.4 Critical Scattering Length L
kTExtraction in Saturation Region (V
G=0.8V and V
D=1V)
Within the framework of the channel backscattering theory, carrier will be scattered in the k-T layer. As the drain voltage is smaller than the thermal energy, the critical scattering length must be equal to the channel length. That explains why we should extract LkT in the saturation region.
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LkT is expressed as:
(3-14) where and are taken as parameters that can fit the simulation result. Generally, we extract LkT from the conduction band energy profile . But would vary with x and y direction. Therefore, we used the lowest sub-band energy profile along the channel to substitute . As shown in Fig. 3-10-1, LkT increases with increasing channel length and increasing temperature. Fig. 3-10-2 shows the extracted LkT and the temperature coefficient for each channel length and temperature. It has the similar trend with Zilli, et al. [9]
Section 3.2.5 Injection velocity v
injExtraction in Saturation Region (V
G=0.8V and V
D=1V)
Within the framework of the channel backscattering theory, carriers come from the virtual source. Therefore, the effective thermal injection velocity should be extracted at the source-channel barrier position [8].
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Fermi-Dirac integral of order one-half. For two-fold valley, and for four-fold valley,
. Here the longitudinal mass =0.916 and the transverse mass =0.19 . The result has been shown in Fig. 3-11-1 and Fig.
3-11-2. It shows an increasing trend with temperature increasing and the temperature coefficient will increase with the channel length increasing.
Section 3.2.6 Inversion Carrier Density N
invExtraction in Above Threshold Region (V
G=0.8 and V
D=1mV/ 1V)
With channel length scaling down to nanoscale, an important question we faced is the short channel effects. The short channel effects are attributed to two physical phenomena: (i) the limitation of the drift characteristic; and (ii) the variation of the threshold voltage. Here, we focus on the phenomenon (ii). This is the variation of the carrier density with channel length scaling down for each temperature. As shown in Fig.
3-12-1 and Fig. 3-12-2, the applied drain voltage will cause the channel carrier density decreasing at LG=90nm and 45nm; and for LG=20nm and 15nm, the channel carrier density increases due to the DIBL effect.
Therefore, the temperature coefficient of the inversion carrier density will vary with the channel length change. In the extraction of beta, there are two methods to do:
; (3-17)
(3-18) Furthermore, DIBL is not a constant for any gate voltage as shown
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in Fig. 3-6-2. It complicates the status of channel with the varying applied gate voltage. Therefore, we focus on the above threshold region and use the maximum transconductance method to extract now. Result is shown in Fig. 3-12-4. Beta appears to slightly increase with channel length decreasing.
Section 3.2.7 Temperature Coefficient of Drain Current in Saturation Region (V
G=0.8V and V
D=1V)
The temperature coefficient of drain current α is the most important factor. It represents the combination of all the effects. Therefore, it should be the same as the result of the combination of each factor. Fig. 3-13-1 shows the same characteristic as Fig. 3-12-2. Here α can be expressed as:
to extract the real variation of drain current between three temperatures.
Owing to the same reason as α , is used to calculate the BR.
Section 3.2.8 Verification of Temperature Coefficient Method
Like the derivation shown in section 2.3, differentiating Eq. (3-9) with respect to temperature is performed:
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The result of each differential term has been shown in section 3-3, 3-6 and 3-7. At the saturation region, we assume the third of the right-hand side of Eq. (3-11) term is zero and
. Therefore, we can derive Eq. (3-22) as:
. (3-22) Result is shown in Fig. 3-14. It suggests that using the temperature coefficient method is feasible and the accuracy is better than 50%.
Section 3.3 Extraction of Ballistic Ratio (BR)
In this section, we will directly extract BR from the drain current and compare it with the temperature coefficient method. As shown in Eq.
(3-23), we calculate BR at saturation region and there are two terms,
and , that should be calculated first:
(3-23) In the experiment part, we can measure C-V characteristics to get the corresponding inversion carrier density . Traditionally, the measured result is the macroscopic average. But in the channel backscattering theory it is not. It represents the carrier density at the barrier high in channel. Therefore, in the following we will extract BR at X=Xvs. As shown in Fig. 3-15-1, BR is extracted at the virtual source; it seems to increase with channel length shrinkage. This result shows the reflection rate decreasing with channel length decreasing. Considering the
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physical limit on saturation velocity, it must saturate at : BR is about 0.75 and rc is about 0.15 at room temperature. Replacing BR with and Lkt, we have higher temperature and BR increases with channel length shrinkage for the two methods. Although they have the same characteristics, the difference between each temperature is not the same. Therefore, we can predict BR extracted from temperature coefficient method. As shown in Eq. (2-16), we can use the five temperature coefficients to extract . Result is shown in Fig. 3-15-3 and it shows a nearly constant characteristic for each channel length. Furthermore, using the temperature coefficient set A assumed in [3] even shows a negative trend at LG=15nm/
20nm. The reason is caused by the error in the five coefficients. In the five coefficients, we could simply assume the error would occur on critical scattering length. Other coefficients have been defined clearly.
Therefore, substituting Eq. (3-24) into (3-23), we have
. (3-25) The calculated critical scattering length is shown in Fig. 3-16-1. And according to the new , we can extract the new BR as shown in Fig.
3-16-2. As our anticipation, it shows similar characteristics as the result of direct calculation.
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Chapter 4
Conclusion
In the study, we used TCAD simulator to predict the characteristic of DG nMOSFETs with channel length scaling down to nanoscale. Here, the most important issue is the high field velocity saturation. It causes the carrier mobility degradation for the even general applied measurement drain bias of 0.025V, and generates DIBL effect for channel length is smaller than 45nm. Therefore, we should measure the mobility at small enough drain bias like 1mV. But the mobility degradation effect still exists for the operation voltage. For this reason, Prof. Lundstrom, et al. at Purdue University brings up the channel backscattering theory. They used another concept to explain the velocity saturation effect. In the theory, they think the critical scattering length stems from the virtual source point to the point with kT potential drop, rather than the virtual source point.
The result between the DD model and channel backscattering model has not too much difference. But as we used the temperature coefficient method to calculate BR, the slight difference in the critical scattering length causes a strong variation on the temperature coefficient LkT. We could infer carriers would be scattered in a range larger than kT-layer for higher temperature and in smaller than kT-layer for lower temperature.
The result between the DD model and channel backscattering model has not too much difference. But as we used the temperature coefficient method to calculate BR, the slight difference in the critical scattering length causes a strong variation on the temperature coefficient LkT. We could infer carriers would be scattered in a range larger than kT-layer for higher temperature and in smaller than kT-layer for lower temperature.