Chapter 1 Introduction
1.1 Mismatch in MOS Transistors
Introduction
Section 1.1 Mismatch in MOS Transistors
This dissertation is a contribution to the discussion on the mismatch properties of MOS transistors based on backscattering theory. Mismatch is the process that causes time-independent random variations in physical quantities of identically designed devices. In MOS transistors, It is well recognized that most MOS parameters variations are mainly caused by the doping variation , such as threshold voltage, drain induced barrier lowering(DIBL), backscattering coefficient, etc. And these variations will directly effect the MOS transistor properties and performance, thus mismatch research plays an important role in the design of accurate analog circuits and digital circuits in the future. What we want is deriving a mismatch model to express the variation from long channel to short channel at different electric field and the mismatch properties of the nanoscale device in the future will also be clearly showed.
Section 1.2 Organization of This Thesis
Chapter 1 is a brief introduction about this dissertation and our work. In Chapter 2 of this dissertation, we have discussed the backscattering theory and some parameters
2
extraction method, including the traditional backscattering coefficient extraction and parabolic potential model [1]. On the other hand, we applied the drain/source series resistance theory in our parameter extraction by using constant mobility method [2].
In Chapter 3, we have derived several mismatch model based on the backscattering theory. The first mismatch model is derived by the parabolic potential theory [1]. On the other hand, it is well-known that the backscattering coefficient can be expressed as:
where A and λ are the critical length in the channel near the source that the conduction band bends down by a thermal energy of KBT (KBT layer’s width) and mean free path. The A variation will be analyzed in this part. Based on this mismatch model, RC mismatch model is also derived and discussed. The third mismatch model is developed by using drain current equation in saturation region:
where Ving, Vtho are the thermal injection velocity at the top of source-channel junction barrier and the quasi- equilibrium threshold voltage. In this model, the drain current mismatch can be expressed as a function of three parameters variation: Vtho, Rc, and DIBL. We have also compared our mismatch models with experiments in chapter 3, whereas the details of data extraction about experiment will be discussed in the chapter 2.
( )
[ ]
C C D
tho G
inj
D
R
V R DIBL V
V V
C W
I +
× −
×
−
−
×
×
×
= 1
'
1
'
r
c= λ + A
A
Chapter 2
Backscattering Theory and Parameters Extraction
Section 2.1 Backscattering Theory
The backscattering theory describes the near-equilibrium scattering process near the source when the electrons travel across the channel. Electrons are injected from the source into the channel and only those over a potential barrier can be collected by the drain. This potential barrier is the conduction band which bends “up” due to the presence of the source-channel junction and then bends down due to the channel electric field caused by drain voltage. This barrier may be linked to drain-induced barrier lowering. As schematically shown in Fig. 1, A is the length across which the conduction band bends down by a thermal energy of KBT, where T is the temperature, and KB is the Boltzmann’s constant. This length is known as the width of KBT layer.
Backscattering is mainly occurring in the KBT layer due to the lattice and carrier. The backscattering coefficient, RC, is determined to describe the scattering in the KBT layer, as shown in Fig. 1, RC can be written as:
) 1 λ (
= + A
A R
Cwhere λ is the mean-free-path and A is the width of KBT layer. We note that RC is determined by the electric field profile very near the source where the electrons have been heated to no more than KBT/q. Thus the λ can be estimated by the near-equilibrium low–field mobility. To derive an equation about drain current based
4
on the backscattering coefficient, we assume that there is a source flux F injected into the channel and part of F will be reflected back to the source. From Fig. 1, the flux entering the drain can be expressed as:
( 1 C) ( 2 )
S
D
F R
F = −
Here we assume the flux reflected back to the source from the channel does not reflect.
Thus the electron density in the source side can be expressed as:
(3)
where Q is the charge density per unit area. In the saturation region, above equation can be also written as:( ) ( 6 )
This is the backscattering theory based current equation in saturation region used in this thesis. We can note that there is no mobility parameter in this formula and the drain current heavily depends on the new parameter: RC. For example, in the ballistic situation RC =0, the drain current is governed by the thermal injection velocity [1].
T
From (6), we can extract RC from several processes in the experiment. In order to get the parameter accurately, the drain-induced-barrier-lowering and drain/source series resistance must be considered. Thus equation (6) should be modified as:
( ) ( ( ) )
The RS, RD and DIBL parameters extraction will be discussed later.
Section 2.2 Parabolic Potential Barrier
If there is no electric field, the backscattering coefficient can be simply written as [4]:
When the electric field is present, the conduction band will bend down and the backscattering coefficient should be expressed as [5]:
)
Because the value of backscattering coefficient highly depends on A which is the width of KBT layer, the quantity of A is an important issue in the backscattering theory. The parabolic potential barrier-oriented compact model is a key to solving this
6
problem. We assume that the conduction band bends down as a parabolic situation and the equation of potential drop can be described as:
)
definition of the width of the KBT layer is that the critical distance that the conduction band potential drops KBT /q. So we substitute x and V(x) in above equation as A and KBT/q. Thus the equation can be rewritten as:)
This is an equation about the relation of L and A based on parabolic potential ~ model, whereas L is expected to be a function of channel length, drain voltage, gate ~ voltage, and temperature. To derive the relation between L and L, thus V~ G0, VD0, and T0 must be defined in order to ensure that L = L in these situations and ~ A can 0 be calculated from this function in these situations (VD-> VD0, VG-> VG0, T->T0). To find the relation of these parameters, the different parameters should be discussed separately. The process of the equation derivation between L and each parameter ~ was discussed in [1] and L can be expressed as: ~
fitting in experiment [1]. In this thesis, we consider that n=4.1(V−0.25). Substituting
the L in equation (12) to equation (11), a compact relation between A and bias, ~ device geometrical parameters can also be obtained:
( )
0.5( ) ( 13 )
25 . 0
q T K V
V
L V
Bth G
D
= η −
−A
From equation (13), We have extracted A for W/L = 0.13um/0.065um, 0.24um/0.065 um, 1um/0.065um, 0.13um/0.1um, 0.24um/0.1um, 1um/0.1um, 0.13um/1um, 0.24um/1um, 1um/1um, 0.13um/10um, 0.24um/10um, and 1um/10um all at VG = 1 V, VG = 0.7 V, and VG = 0.5V for VD = 1 V. On the other hand, A extracted experimentally is also shown for comparison. In Fig. 2, open circle represents the mean value of A extracted by parabolic potential model and open triangle represents the mean value of A extracted experimentally.
Section 2.3 Mean‐Free‐Path
The backscattering coefficient can be written as (1). λ is the low-field mean-free-math and can be linked to μn0 through
( / ) / ( 14 )
2 μ
n0K
BT q V
injλ =
where Ving is the thermal velocity and μn0 is the near-equilibrium mobility because that the backscattering occurs very near the source and it can be regard as a near-equilibrium case even there is a strong electric field in the channel due to the drain voltage [3]. Fig. 3 shows the mobility versus channel length, whereas the mobility is obtained from G-D method. It can be seen that there is a drop of the mobility when the channel length decreases due to the short channel effect. Fig. 4
8
displays the mobility versus gate voltage and the three components, which affect the curve of the mobility, can be clearly expressed by the following scatterings: Coulomb scattering, phonon scattering, and surface roughness scattering. Substituting above mobility data into equation (13), the mean-free-path can be obtained. Fig. 5 and Fig. 6 show the mean-free-path versus the channel length at VG = 1V, and 0.5V, and the mean-free-path versus the intrinsic gate voltage, respectively. Here the thermal velocity is substituted as 107 cm/sec in equation (14). In order to get the backscattering coefficient, we substitute the mean-free-path obtained and the ℓ extracted from parabolic potential model into equation (2). Fig. 7 displays the backscattering coefficient versus channel length at different device widths for VG = 1V. The open circle represents the backscattering coefficient extracted from different mean-free -paths for different channel lengths. The open triangle represents the backscattering coefficient extracted from the same mean free path in the long channel case because we consider the mean free path as a parameter due to the material (independent of L). The backscattering coefficient extracted from experiment is also shown in Fig. 7 and labeled by the open square. It can be seen that the backscattering coefficient decreases as the channel length decreases in these cases and the open triangle are closer to the experimental data compared to those of short channel length devices.
Section 2.4 Drain/Source Series Resistance
Extraction
In order to extract the accurate parameters such as backscattering coefficient in this work, the existence of the series resistance RSD should be considered. The series resistance RSD is the resistance in source and drain side of MOSFET. In the past,
because the channel resistance of the long channel device is huge compared with the series resistance, the series resistance can be ignored. But in today’s MOSFET nanoscale technology, the existence of the RSD may lead to many problems such as the drive current degradation and parameter extraction accuracy. In the traditional method, the RSD can be extracted from different gate voltage and devices of different channel lengths. The conventional method is not accurate at short channel situations. So we use a new method called constant-mobility-method to extract RSD [2]. The constant mobility method is used to extract RSD in the high oxide electric field region where surface roughness scattering becomes the dominant mechanism. In the high electric region, a constant mobility is achieved for a given effective vertical electric field, regardless of impurity scattering and phonon scattering, and the change of VBS will lead to the same mobility in this region. The current equation of MOSFETs operating in the linear region can be expressed as:
( ) ( 15 )
Equation (15) and equation (16) apply to linear operation mode at different VBS and VDS =0.01 V. To extract RSD, we assume that μ(1)= μ(2) and substitute the mobility of equation (15) as equation (16). Then the RSD can be written as:
)
10
ds th
th
gs V V V
V
B 2
) 1 1
( (1) (2)
) 1
( + − − −
=
η η
Fig. 8 shows the RSD extracted by equation (17) for W/L=0.24um/0.1um and W/L=0.24um/0.065um and we focus on VG>1V to ensure a high electric field condition. Fig. 9 shows the RSD extracted at different VBS conditions. Fig. 10 shows the RSD extracted from different devices at the same scale. We have done many experiments in order to ensure that the RSD extracted from different channel length and bias conditions should be a constant. The RSD value extracted from the constant mobility method at different device dimensions will be used in some parameters extraction in this thesis.
Section 2.5 Threshold Voltage and DIBL
The threshold voltage extraction is an important part in the MOS transistors research and a key parameter in this work. We employ a maximum tran-conductance method in the linear region to assess quasi-equilibrium threshold voltage and the constant subthreshold current method in the saturation region to extract the DIBL.
The maximum tran-conductance is a method establishing a tangent line from the point where the tran-conductance is the maximum to show the threshold voltage in the ID-VG gragh. We set the drain voltage VD = 10mV to ensure the quasi- equilibrium case. Fig. 11 shows drain current versus gate voltage and the maximum tran-conductance method.
The Drain Induced Barrier-Lowering is the decreasing of the threshold voltage when the drain voltage increases, caused by conduction band bends lowering. In other words, the channel control from the gate will weaken at high drain electric field,
especially in the short channel devices. To develop the mismatch model accurately, the DIBL must be considered. We use the constant subthreshold current method to calculate the threshold voltage shift due to the VD increase and measure the DIBL in the saturation region. We set the drain voltage VD = 1V to ensure that the device is operated in the saturation region.
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Chapter 3
Mismatch Experiment and Results
Section 3.1 Mismatch Model
The mismatch model is a mathematical equation to express the variation of the parameter. The mismatch properties have two features: the total mismatch of the parameter is composed of many single events of the mismatch-generating process and the effects on the parameter are so small that the contributions to the parameter can be summed [6]. The mismatch properties of threshold voltage are the main topic of mismatch since 1970s. In this thesis, our mismatch model is developed based on the research of threshold voltage in the traditional theory and the backscattering theory.
To reach for our goal, we must develop the mismatch model step by step. First of all, we focus on the width of the KBT layer and analyse its mismatch properties. Then we develop the backscattering coefficient mismatch properties in the next step.
One of the fundamental factors limiting the accuracy of MOS circuits is the current mismatch between identically devices. So, based on above considerations and series resistance extraction, the drain current mismatch model will be presented later. The transistors in the circuits usually operate in the saturation region. Thus, the mismatch models in this thesis are all developed and discussed in saturation region.
Section 3.2 Experiment
The measurement procedure is an important part in the mismatch work. Generally, to obtain the statistical variation, we need to measure a lot of devices. In this work, we measure over than 20 different dimensions of n-channel devices, whereas each scale has more than 30 dies on the wafer. All dies are made from the same process and have the same structure. They were fabricated using 65nm CMOS process. We measure our drain current by sweeping gate voltage from 0 to 1.2 V in a step of 25 mV when we fixed the drain voltage at 0.01 V, 0.1 V, and 1 V in order to cover both the equilibrium case and the saturation region case. In the extraction of the series resistance, we set substrate bias VB at = -0.4 V, -0.8 V, and 0 V. The temperature was fixed at 298 K.
The measurement setup includes the HP4156B and a Faraday box for shielding the wafer.
Section 3.3 Mismatch Properties of K
BT Layer’s Width
The mismatch properties can be expressed as the standard deviation: σ. The standard deviation can be calculated from the parameters extracted in the experiment.
In statistics, there should be more than 20 measured devices to ensure that the standard deviation calculated can be considered as the mismatch properties of all dies on the wafer. In this section, the standard deviations are calculated from over than 20 dies. The extreme variance of the parameter would be lead to high standard deviation.
Fig. 12 shows the standard deviation of KBT layer’s width versus the gate voltage from experiments. It can be seen that the variance would decrease when the gate
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voltage increases. To propose a simple statistical model to explain the phenomenon observed, we pay attentions to the equation of the KBT layer’s width. In Chapter 2, we introduce the parabolic potential barrier model to calculate the width of KBT layer:
(18)
From above equation, we will express the mismatch of A as a function of the variance of threshold voltage. The variance of standard deviation with one random variable x can be expressed as:
(19)
Thus, from equation (18) and equation (19), the standard deviation of KBT layer’s width can be written as:
where . The standard deviation can be expressed as a function of inverse square root of the device area [6]. Thus we obtain a compact model:
where A is a constant depending on channel length and drain voltage and is independent of gate voltage.
Vth
A is the size proportionality constant for the variance
x
xof threshold voltage, as shown in Fig. 13. From the threshold voltage term in above equation, we can observe that the variance of KBT layer’s width would decrease when the gate voltage increases because that the effect of the threshold voltage fluctuation would become smaller. In other words, the variance of KBT layer’s width is sensitive to threshold voltage in the weak inversion region. This is a simple mathematical model to understand the mismatch properties of KBT layer’s width and can calculate it from just three simple parameters: threshold voltage, device size, and drain voltage.
Fig. 14 displays the mismatch calculations compared with the experiments for W/L=0.13um/0.1um, 0.24um/0.1um, and 1um/0.5um at drain voltage VD=1V. We can observe that the differences between the two are small. Fig. 15 shows that the variance of A versus square root of L/W for different devices at fixed gate voltage VG=1V. When the gate voltage is high, the difference of the threshold voltage in equation (21) can be ignored and the standard deviation of KBT layer’s width can be proportional to the square root of channel length divided by device width (L/W). It can be clearly shown in equation (21) by separating the channel length (L) term from constant A and multiplied by the inverse square root of area. Another relation can be observed from dividing each sides of equation (21) by KBT layer’s width, where the channel length term in constant A would be deleted. Then the standard deviation divided by the mean of KBT layer’s width is proportional to the inverse square root of device area when gate voltage is high. Fig. 16 displays this relationship which is a traditional relationship like current factor or mobility [6].
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Section 3.4 Backscattering Coefficient Mismatch
The backscattering coefficient RC can be expressed as (1), whereas RC is a function of two variables: mean free path and KBT layer’s width. To express the mismatch as these two parts, we applied a differential equation below for analysis:
where σx and σy, are the variance of x and y, respectively. COV (x, y) is the correlation coefficient between (x, y). We assume that the mean-free-path and KBT layer’s width are independent of each other, that is, the correlation coefficient COV (x, y) is zero.
Thus the RC mismatch model can be expressed as:
This is a simple mathematical model to estimate the fluctuation of RC in the saturation region. The σA is already discussed in Section 3.3 and the ratio σλ/mean(λ) can be obtained by the experiment. Both of them have a proportional relationship with the inverse square root of area. From these two different mismatch properties, the standard deviation of RC can also be obtained, so does the standard deviation divided by the mean: σRC/mean(RC). The mismatch properties of the mean-free-path should have the same properties as mobility in [7]. Fig. 17 shows the relationship between the σλ/mean(λ) and the inverse square root of area. Fig. 18 displays the standard deviation RC versus gate voltage VG from equation (23), compared with experiments
( ) , ( 22 )
for W/L=0.13um/0.1um, and W/L=0.24/0.1um, all at VD=1V to ensure the saturation region. We can observe that the model and the experiments are quite close to each other.
Section 3.5 Drain Current Mismatch
Drain current mismatch properties is the last part of this work which directly affects the performance of the circuit. To develop the drain current mismatch model accurately, we considered the extracted series resistance:
( ) ( ( ) )
This is a whole drain current equation based on backscattering theory. We define that the current mismatch is the standard deviation divided by the mean: σID/mean(ID).
This is a whole drain current equation based on backscattering theory. We define that the current mismatch is the standard deviation divided by the mean: σID/mean(ID).