Organization

This thesis is organized as follows. In Chapter2, we present a parabolic channel

potential model of UTB MOSFETs with thin BOX under subthreshold region. In Chapter 3, the quantum-confinement model based on the parabolic channel potential model is derived and we discuss the impact of different channel materials and surface orientations on the degree of quantum confinement. In Chapter 4, we investigate the impact of quantum-mechanical effects on the V roll-off for GeOI devices. Chapter _T 5 is the conclusion of this thesis.

Chapter 2

Channel Potential Model for

Ultra-Thin-Body MOSFETs with Thin BOX Under Subthreshold Region

2.1 Introduction

Ultra-thin-body (UTB) MOSFET with thin buried oxide (BOX) is a promising candidate to extend CMOS scaling because of its superior electrostatic integrity than bulk devices [10]-[12]. In addition, due to its better control of short-channel effects, lower subthreshold swing, and reduced leakage current, UTB MOSFET is also an ideal structure for the subthreshold circuit applications [13]. A channel potential model with series form for UTB MOSFETs under subthreshold region has been reported in [14]-[15]. Using the potential model, the electrostatic characteristics such as threshold-voltage, subthreshold swing, and subthreshold current can be estimated for various UTB devices.

To further simplify the series solution, in this chapter we present a compact form of model to be used in Chapter 3. The model verification is shown in Section 2.3. For various UTB structures with long- and short-channel length, thin- and thick-channel thickness, low and high channel doping concentration, and different buried oxide thickness, comparisons between the model and TCAD simulations have been carried out. Besides, we also examine the model at high and low drain biases and different

2.2 Series Solution of Channel Potential

Our theoretical 2-D potential model for UTB MOSFETs is derived from the Poisson’s equation. Fig. 2.1 shows a schematic sketch of a UTB device with thin buried oxide and silicon substrate. In the subthreshold region, the channel is fully depleted with negligible mobile carriers. Therefore, the channel potential distribution



^{x y}



ch ,

 satisfies the Poisson’s equation

   

potential distribution _BOX



^x,y



satisfies the Laplace equation

   

     

the flat-band voltages of gate and back-gate, respectively.  is the built-in potential _ms of the source/drain to the channel. E_i,ch and E_i,sub are the intrinsic Fermi level of channel and substrate, respectively.

The corresponding 2-D boundary value problem can be divided into two sub-problems, a 1-D Poisson’s equation and a 2-D Laplace equation. Using the superposition principle, the complete channel potential solution is



^{x y}



ch

 

^x ch



^{x y}



ch ,  _,1  _,2 ,

   , where ch 1,

 

^x and ch_,2



^x,^y



are the solutions of 1-D and 2-D sub-problems in the channel, respectively. The 1-D solution ch 1,

 

^x can be expressed as

 

^{x qN x A x B} channel-material region. The electric field discontinuity across the gate oxide and channel interface can thus be eliminated. For the channel/buried oxide interface, both the potential distribution in the channel (ch_,2



^x,^y



) and that in the buried oxide

The coefficients c , _n c_n^', and e in (2.5a) can be expressed as _n

2.3 Compact Form and Verification

2.3.1 Compact Form

From equations (2.4) and (2.5), we can obtain the classical potential in whole channel [2.5]-[2.6]. However, the series solution is too complicated to be used in solving the Schrödinger equation in Chapter 3. To simplify the solution, the potential in the channel _ch



^x,y



is further reduced to a parabolic form. Since the direction of quantum confinement is perpendicular to the interface between oxide and channel, we express the channel potential along the x-direction for each slice of y-direction as

   

corresponding location. In fact, equation (2.6) is the Lagrange interpolating

polynomial of degree 2 that passes three points. If the three points are determined, the equation (2.6) can be expressed as the parabolic form

_ch

 

^{x a}₂^{x2 a}₁^xa₀ (2.7) , where a , ₂ a , and ₁ a can be calculated from equation (2.6). ₀

So the next issue is to choose the three points that can make the parabolic form faithfully represent the series solution. We propose a methodology to determine the three points, and the methodology obeys the following principles:

1. The three points contain the highest potential and another two boundaries along the x-direction.

2. If the highest potential is also the boundary, we choose the midpoint and the other boundary along the x-direction.

The reason why we choose the highest potential is that the carrier flow (electron flow in NMOS) may be the largest through this point. In the words, the point is critical to the determination of electron density and subthreshold current. Using the methodology, we can reconstruct the channel potential by the compact form instead of series solution. Notice that the sets of a , ₂ a , and ₁ a will be different if we choose ₀ different cross-sections of y-direction.

Comparisons between the compact parabolic form and series solution are shown in Fig 2.2. It shows that the reduced parabolic form is fairly accurate with L=30nm and 15nm and V_DS=0.05V and 1V for Ge-channel devices. The curves of compact model in the y-direction are slightly discontinuous since we only reduce the channel

form instead of series solution.

2.3.2 Verification

We use Ge-channel devices in the model verification. The channel length (L) is 30nm and 15nm, the gate oxide thickness (T ) is 1nm, the channel thickness (_ox T ) is _ch 5nm and 10nm, the buried oxide thickness (T_BOX ) is 1nm, 5nm and 10nm. The drain bias (V_DS) is 0.05V and 1V and the back-gate bias (V ) is 0V, 0.2V, and 0.5V. The _BS channel doping concentration (N ) is 1x10_A ¹⁵ cm^-3 and 1x10¹⁸ cm^-3. Besides, we use the heavily-doped (1x10²⁰ cm^-3) silicon substrate and treat it as ground plane.

Fig. 2.3 shows the potential distribution across half T and half _ch L in the long- and short-channel devices and in the thin- and thick-channel thickness devices for model and TCAD results. It shows that our model is fairly accurate for various channel sizes. In Fig 2.4, our model is suitable for different channel doping concentrations (N ). In Fig 2.5, our model is also satisfactory for _A T_BOX =1nm and 5nm and implies that the model may be applied for double-gate (DG) devices when

BOX

ox T

T  . Fig 2.6 shows the potential distribution at high and low drain biases (V_DS) and at different back-gate biases (V ). The model is also accurate compared with _BS TCAD simulations. The compact parabolic potential model shows excellent agreement with TCAD simulations for UTB devices.

2.4 Summary

We have developed a channel potential model for UTB MOSFETs under subthreshold region. Specifically, we propose a compact form of model instead of

series solution. To examine the accuracy of the compact parabolic model, we have carried out extensive verification for various L, T , _ch N , _A T_BOX, V_DS, and V . All _BS verification results show sufficient accuracy compared with TCAD simulations. The compact form not only makes the expression of potential model clear but also simplify the derivation of eigenenergy (E ) and eigenfuction (_n  ). That will be _n discussed in Chapter 3.

Fig. 2.1 Schematic sketch of the UTB structure investigated in this study.

Fig. 2.2 Potential distribution for UTB devices with (a) L=30nm and 15nm and

Fig. 2.3 Potential distribution for UTB devices with (a) L=30nm and 15nm, and (b) T =5nm and 10nm using model and TCAD simulation. _ch

Fig. 2.4 Potential distribution for UTB devices with N =1x10_A ¹⁵ cm^-3, and N =1x10A ¹⁸ cm^-3 using model and TCAD simulation.

Fig. 2.5 Potential distribution for UTB devices with (a) T_BOX=5nm, and (b) TBOX=1nm using model and TCAD simulation.

Fig. 2.6 Potential distribution for UTB devices with (a) V_DS=0.05V and 1V, and

Chapter 3

Quantum-Confinement Model for

Ultra-Thin-Body MOSFETs with Thin BOX Under Subthreshold Region

3.1 Introduction

With decreasing channel thickness, quantum-confinement effect may be significant [17]-[18] and may affect the electrostatic characteristics of ultra-thin-body (UTB) MOSFETs [19]. In this chapter, an analytical solution of the Schrödinger equation for UTB MOSFETs under subthreshold region is presented. Based on the parabolic channel potential model developed in Chapter 2, we derive the eigenenergy and eigenfunction of UTB devices. Therefore, the electron density in the channel considering quantum mechanism can be obtained by using the calculated eigenenergy and eigenfunction. Besides, we also discuss the impacts of channel material and surface orientation [20]-[21]. Quantum-mechanical effects on Si-, Ge-, and In0.53Ga0.47As-channel UTB devices will be assessed.

3.2 Model Derivation

The eigenenergy and eigenfunction of channel carriers are crucial to the quantum-mechanical effects, and they can be determined by solving the Schrödinger equation [3.6]. The schematic sketch of a UTB device has been shown in Fig 2.1.

Because the direction of quantum confinement (x ) is perpendicular to the interface

between oxide and channel, for each cross-sections of y-direction, the Schrödinger equation for the UTB devices can be expressed as

) interface between oxide and channel, and ^E_C

 

^x is the conduction band edge.

states in the conduction band, and N is the effective density of states in the valence _C band, respectively. Therefore, from equation (2.7) and equation (3.2), the conduction band edge can be expressed as

^E_C

 

^{x a}₂^{'x2 a}₁^'xa₀^' (3.3)

, where a , ^'₂ a , and ₁^' a₀^' are known values and can be obtained from the parabolic channel potential model presented in Chapter 2.

Using equation (3.3), equation (3.1) can be expressed as

If we use the power series method and assume



, the coefficients c can be determined by the following recurrence relationship [22] _i

 

0

The required boundary conditions can be described as

_n



^{x0 }



⁰ (3.6a) the eigenfunction _n(x) for UTB devices under subthreshold region can be derived.

Generally, 60 terms in the summation of (3.5a) are needed to give sufficiently accurate results.

Using Fermi-Dirac statistics, the discrete nature of the quantized density of states reduces the integral over energy to a sum over bound state energies [23]. Besides, since we consider the quantum-confinement effect and possibly different types of valleys, the expression for the electron density then becomes [24]

 

the channel along the y-direction.

In equation (3.5), the ^exp

 

^E_F

 

^{y E}_n.



^kT



term is usually much smaller than 1 under subthreshold region. So we can further approximate the equation (3.5) as

 



3.3 Verification and Discussion

In Fig 3.1, we compare the conduction band edge and corresponding eigenenergies between long- and short-channel devices as T =10nm. It shows that _ch the shape of the conduction band edge depends on the channel length. The

short-channel one. That is, the degree of quantization is affected by the conduction band edge with different channel length when T =10nm. _ch

Fig 3.2 shows the calculated quantized n th eigenenergy (E ) in Ge-channel _n devices with various T . We can find that as _ch T =5nm, _ch



^E₂ ^E₁



^{0.35eV is}

much larger than kT 0.026eV . From equation (3.6), it can be expected that the first eigenenergy (E ) will be dominant in the determination of the electron density when ₁ quantum-confinement effect is strong.

In Fig. 3.3, we show the square of the first eigenfunction (₁ ²) with T =5nm _ch and 10nm. Fig 3.4 shows the electron density across half L for the UTB devices.

Although the square of the first eigenfunction with T =5nm is larger than the one _ch with T =10nm, the electron density with _ch T =5nm is smaller than the one with _ch T =10nm as shown in Fig 3.4. From equation (3.6), the ch T =5nm _ch UTB device is expected to have smaller electron density since it possess large



E 1 EC,min



as shown in Fig 3.2. Our quantum-mechanical model is fairly accurate compared with TCAD simulations.

3.4 Various Channel Materials and Surface Orientations

Changing channel materials may improve device performance through the enhancement of carrier mobility. In this chapter, we will use Si, Ge, and In0.53Ga0.47As as channel materials to examine our quantum-mechanical model. In addition, since Si and Ge have three common surface orientations (100), (110), and (111), we will discuss the impact of surface orientation considering quantum mechanism.

To consider quantum mechanism, the first thing is that different channel material or surface orientation may have different corresponding effective mass (m^*_x). The effective mass will determine the degree of quantization. The channel with lighter effective mass will have stronger quantum-mechanical effects. For a given surface orientation, Si or Ge may have two distinct effective masses with corresponding degeneracy (d ) and effective density-of-state mass (_ m_d,). Table 3.1 shows m^*_x, d , _ and m_d, of the three channel materials considering various surface orientations [24]-[26].

In this work, we use our parabolic channel potential model for different channel materials. Then based on our quantum-mechanical model, we can use the parameters in Table 3.1 to solve the Schrödinger equation and to calculate the electron density considering the impact of surface orientation.

Fig 3.5 shows the channel potentials of Si and In_0.53Ga_0.47As, respectively. It can be seen that our parabolic channel potential model is accurate. Fig. 3.6 shows the



E 1 EC,min



for Si with (100) and (110) surface orientations. Because these

orientations have two types of valley (two effective mass m^*_x ), there exists two



E 1 EC,min



. We can find that the difference between the two



E 1 EC,min



becomes large (>>kT 0.026eV) when T about 3nm. It means that the valley with smaller _ch



E 1 EC,min



will determine the electron density in ultrathin T devices. Similarly, _ch

Fig 3.7 shows the



^{E E}



for Ge with (110) and (111) surface orientations.

When T about 5nm, the difference becomes obvious and hence we can treat the _ch valley with smaller



E 1 EC,min



as the dominant type for calculating the electron density. From Fig 3.6 and Fig 3.7, we can find that in ultrathin T devices _ch (T_ch 3nm for Si, T_ch 5nm for Ge), the type of valley with heavier effective mass will determine the electron density.

Table 3.2 collects the critical effective masses (m^*_x,crit) which dominate the electron density. The table can help us understand what critical effective mass may affect the degree of quantization and corresponding electrostatic characteristics in ultrathin T devices. Fig 3.8 shows the dominant _ch



E 1 EC,min



of Si-channel with (100) orientation, Ge-channel with (100) orientation, and In0.53Ga0.47As-channel UTB devices, respectively. We can find that the In_0.53Ga_0.47As-channel UTB device has the largest



E 1 EC,min



and thus experiences the strongest quantum-mechanical effects.

Fig. 3.9 shows the square of the first eigenfunction (₁² ) for Si- and In_0.53Ga_0.47As-channel UTB devices. Note that Si-channel UTB device with (100) orientation has two types of ₁ ². In Fig 3.10, we show the electron density for different channel materials in the UTB devices. It can be seen that for Si-channel UTB device with (100) orientation, the electron density of the dominant type of valley (2-fold valley) essentially determines the total electron density for the T =5nm _ch device. This supports our use of the critical effective mass to determine the degree of quantization.

3.5 Summary

In this chapter, we present the UTB MOSFET model considering quantum mechanism based on our previous parabolic channel potential model. Using the model, we can obtain eigenenergy, eigenfuction, and quantum electron density. Besides, we have demonstrated that



E 1 EC,min



will be the dominant term to determine the electron density, especially for ultrathin T devices. _ch

We have also discussed the impacts of different channel materials and surface orientations considering quantum mechanism. We find that the channel material which experiences the strongest quantum-mechanical effects has the lightest effective mass (m^*_x). We have constructed the table (Table 3.2) of the critical effective masses (m^*_x,crit) which determine the degree of quantum confinement.

Fig. 3.1 The conduction band edge and corresponding eigenenergies for T =10nm devices with (a) ch L=60nm and (b) L=30nm.

Fig. 3.2 The calculated quantized n th eigenenergy (E ) for the devices with _n various T . _ch

Fig. 3.3 The square of the first eigenfunction (₁ ²) for T =5nm and 10nm in _ch the Ge-channel UTB devices.

Fig. 3.4 The electron density for T =5nm and 10nm in the Ge-channel UTB _ch devices.

Table 3.1 m^*_x, d , and _ m_d, of three channel materials considering surface

Fig. 3.5 The potential distribution of (a) Si-channel UTB device, and (b) In0.53Ga0.47As-channel UTB device for model and TCAD results.

Fig. 3.6 The



E 1 EC,min



of Si-channel UTB devices with (a) (100) and (b) (110) surface orientations with model and TCAD results.

Fig. 3.7 The



E 1 EC,min



of Ge-channel UTB devices with (a) (110) and (b) (111) surface orientations with model and TCAD results.

Table 3.2 The critical m^*_x,crit of three channel materials considering surface orientations when quantum-mechanical effect is strong in ultrathin T _ch devices.

(100) (110) (111)

Si

* ,crit

mx

0.916m ₀ 0.316m ₀ 0.259m ₀

Ge

* ,crit

mx

0.117m ₀ 0.218m ₀ 1.57m ₀

In_0.53Ga_0.47As

* ,crit

mx

0.041m ₀

Fig. 3.8 The dominant



E 1 EC,min



for UTB devices with various channel

Fig. 3.9 The square of the first eigenfunction (₁ ²) for (a) Si-channel with (100) surface orientation and (b) In_0.53Ga_0.47As-channel UTB devices.

Fig. 3.10 The electron density of Si-channel with (100) surface orientation and In0.53Ga0.47As-channel UTB devices.

Chapter 4

Impact of Quantum-Mechanical Effects on Threshold-Voltage Roll-Off

in UTB GeOI MOSFETs

4.1 Introduction

Germanium as a channel material has been proposed to enable mobility scaling.

However, its higher permittivity makes it very susceptible to Short Channel Effects (SCEs). To improve the electrostatic integrity, ultra-thin-Body (UTB) Germanium-On-Insulator (GeOI) MOSFET has been proposed as a promising device architecture and shows better control of SCEs than the bulk counterpart [27]-[28]. By scaling down the channel thickness, UTB GeOI MOSFETs can show comparable subthreshold swing as compared with the UTB SOI counterparts [29]. As the channel thickness scales down, the quantum-mechanical effect becomes more significant and its impact on the threshold-voltage (V ) roll-off in UTB SOI MOSFETs has been _T reported in [30]. However, the impact of quantum-mechanical effect on the V _T roll-off in UTB GeOI MOSFETs has rarely been examined.

In this chapter, we investigate the impact of quantum-mechanical effect on the V roll-off characteristics for UTB GeOI MOSFETs by the developed model in T

Chapter 3 and numerical solution of coupled Poisson and Schrödinger equations.

4.2 UTB GeOI Devices and Simulation

The schematic cross section of UTB structure was shown in Fig 2.1. In this study, the gate oxide thickness (T ) is 1nm, the channel thickness (_ox T ) ranges from 4nm to _ch 10nm, the channel length (L) ranges from 2.4 to 10 times the T (proportional to _ch T ), and the buried oxide thickness (ch T_BOX) ranges from 10nm to 20nm. The channel doping concentration ( N ) is 1x10_A ¹⁵cm^-3. Besides, we use the heavily-doped (1x10²⁰cm^-3) silicon substrate and treat it as ground plane.

For UTB devices, our TCAD simulations self-consistently solves the Poisson’s equation (for channel potential) and 1-D Schrödinger equation (for eigenenergy and eigenfunction) at each slice perpendicular to the interface between oxide and channel.

The process of the numerical calculations doesn’t have approximations. Therefore, the TCAD simulations can be exactly to assess quantum-mechanical effects for UTB GeOI devices.

In this study, the V is defined as the _T V_GS at which the average electron density of the cross-section at y  y_crit exceeds the channel doping concentration.

The y_crit stands for the position from the source of highest potential barrier for carrier flow. The y_crit is about L 2 for V_DS=0.05V, and about L 3 for V_DS=1V.

4.3 Results and Discussion

4.3.1 Channel Thickness

Fig. 4.1(a) and Fig 4.1(b) show the V roll-off of Ge- and Si-channel UTB _T devices with quantum-mechanical (QM) and classical (CL) considerations for T =10nm and 5nm, respectively. In Fig 4.1(a), both the Ge- and Si-channel devices ch

with QM consideration show larger V roll-off than that with the CL one. However, _T in Fig 4.1(b), the Si-channel UTB MOSFET with QM consideration shows comparable V roll-off as compared with that using the CL one. The Ge-channel _T UTB MOSFET with QM consideration even shows smaller V roll-off than that _T with the CL one.

In [30], Y. Omura reported that the V roll-off would be increased by QM effect _T in UTB SOI MOSFETs. Their study used the simulator with density-gradient model (DGM) [31]-[32]. In our study, we can see the same trend of increased V roll-off in _T UTB SOI MOSFET. However, the V roll-off is suppressed by QM effect for UTB _T GeOI MOSFET with T =5nm. _ch

Fig 4.1 can be explained by [33]-[34]

QM QM

T m

V  

 (4.1) , where m is the subthreshold slope factor, ^QM is the surface potential shift, and

QM

VT

 is the V shift due to QM effects. In this work, we choose the peak of the _T channel potential at y  y_crit cross-section as the reference potential. Therefore, the



E 1 EC,min



at y  y_crit can stand for the q^QM when considering QM effects.

Fig 4.2 shows the



E 1 EC,min



for GeOI devices with T =10nm and 5nm. Fig _ch 4.2(a) shows that for GeOI devices with T =10nm, the _ch



E 1 EC,min



of the long-channel (L6T_ch) GeOI device is about 2.5



E 1 EC,min



the short-channel

(L2.4T_ch) one. In other words, the short-channel GeOI device with T =10nm _ch

(L2.4T_ch) one. In other words, the short-channel GeOI device with T =10nm _ch

在文檔中 超薄絕緣鍺金氧半場效電晶體在量子侷限下的短通道效應模型與分析 (頁 13-0)