Design and modeling of tunnel field effect transistors

Introduction

Transistors for Low Power Applications

The primary factor influencing power consumption in digital integrated circuits is the supply voltage, as reducing it leads to a quadratic decrease in dynamic power usage However, this reduction creates a significant tradeoff between speed and power dissipation in MOSFET-based circuits due to the inherent 60 mV/decade subthreshold swing limit To achieve low static power dissipation and minimal off-current, a higher threshold voltage is necessary, which consequently results in a smaller overdrive voltage and reduced drive current, ultimately lowering circuit speed This tradeoff arises because a decrease in threshold voltage negatively impacts both circuit speed and power dissipation.

To address the tradeoff between on- and off-currents, a new transistor design requires a steep on-off switching capability, achieving a subthreshold swing of less than 60 mV/decade This characteristic, depicted in Fig 1.1, is essential for low power applications Traditional high subthreshold swing MOSFETs experience low on-current at minimal applied voltages However, the new transistor's steep subthreshold swing allows for a significant reduction in threshold voltage, ensuring adequate on-current and overdrive voltage while keeping off-leakage current low Consequently, this advancement provides greater flexibility in scaling supply voltage (V dd), further minimizing power consumption.

Subthreshold Swing Limit of Traditional MOSFETs

In standard MOSFET devices, the subthreshold current arises from the thermal diffusion of source carriers as they traverse the source-channel potential barrier to reach the drain This phenomenon can be mathematically expressed as [5].

The equation 0exp (1.1) describes the relationship between the elementary charge (q), Boltzmann constant (k), temperature (T), and threshold voltage (Vt) In this context, the parameter 1/n represents the fraction of the difference between gate voltage (Vg) and threshold voltage (Vt) that influences the source-channel barrier, which is denoted by ox b.

C n=1+ C (1.2) where C b and C ox are bulk and oxide capacitances per unit area, respectively

The subthreshold swing, which is defined as the gate voltage variation per decade of drain current, can be calculated immediately q n kT I d

For the ideal case of n=1 corresponding to zero bulk capacitance or infinite gate capacitance, the achievable minimum subthreshold swing at room temperature (300 K) is mV/decade 60

For a given device structure with an approximately fixed n value, the subthreshold swing of MOSFETs is a constant in the subthreshold region.

TFETs with Sub-60 mV/decade Subthreshold Swing

TFET devices operate on band-to-band tunneling rather than thermal diffusion, leading to distinct subthreshold behavior compared to MOSFETs Unlike MOSFETs, TFETs are not constrained by the Boltzmann limit of 60 mV/decade for subthreshold swing, as demonstrated through theoretical calculations and numerical simulations Experimental evidence supports the existence of sub-60 mV/decade subthreshold swing in TFETs made from various semiconductors, including silicon, silicon-germanium compounds, III-V materials, and carbon nanotubes With the capability of achieving sub-60 mV/decade subthreshold swing at room temperature, TFETs are emerging as promising candidates for low power applications.

Challenges of TFET Devices

Band-to-band tunneling in semiconductors has been known for a long time, but it gained little attention for transistor applications while MOSFETs remained scalable in size and voltage due to advanced fabrication technologies Recently, TFET devices have garnered interest as MOSFETs approach their fundamental limits in physical size and operating voltage Despite their rapid development, TFETs face significant challenges that must be overcome before they can be effectively used in practical applications.

The primary challenge for Tunnel Field-Effect Transistors (TFETs) is to increase the on-state tunneling current to satisfy the drive current needs for future low-power, high-performance logic applications This on-current is heavily influenced by the energy bandgap of semiconductors, as tunneling probability is closely linked to the width and height of the tunnel barrier Despite employing advanced techniques such as high-k gate dielectrics, source/drain doping engineering, and innovative architectures, the on-current in silicon-based TFETs remains limited due to silicon's high bandgap Therefore, the integration of low bandgap semiconductors, alongside advanced structural techniques, is essential for significantly enhancing the on-current of TFETs.

The second significant challenge in transistor technology is the scaling of physical Tunnel Field-Effect Transistor (TFET) length In addition to reducing supply voltage, achieving extremely scaled transistors is crucial for enhancing the packing density of integrated circuits Typical homojunction p-i-n TFETs face scalability limitations down to 30 nm due to their small active tunneling region Recent studies indicate that short-channel p-i-n TFETs demonstrate superior transfer characteristics compared to MOSFETs, but only at longer device lengths Moreover, the scalability of TFET devices deteriorates more rapidly than that of MOSFETs as device lengths are aggressively reduced Consequently, scaling TFETs into sub-20 nm regimes presents significant challenges.

Researchers face significant challenges in creating analytical models for TFETs that effectively describe their input and output characteristics Simple and informative models are crucial for understanding the working principles, device characteristics, and design of TFETs However, unlike conventional MOSFETs, there is currently no favorable model for the tunneling current in TFET devices due to the complex band-to-band tunneling mechanism Although some recent analytical models have been developed, they primarily focus on high bandgap silicon TFETs and tend to be overly complicated, hindering the extraction of fundamental device physics and design insights.

Thesis Objective and Organization

This dissertation addresses critical challenges in advanced TFET devices, focusing on enhancing on-current, improving scalability, and developing analytical models A novel TFET structure is introduced through two-dimensional simulations to boost on-current and physical length scalability Additionally, a new analytical model for line-tunneling is formulated for low-bandgap line-tunneling TFETs The dissertation thoroughly examines the physical properties of band-to-band tunneling and the operational principles of the proposed TFETs, providing clarity on device functionality and design The numerical and analytical findings are validated against experimental data, ensuring their applicability in exploring device physics and informing design discussions.

The dissertation consists of seven chapters, including the Introduction (Chapter 1) and the Conclusions (Chapter 7)

Chapter 2 reviews the band-to-band tunneling theories which are served as the basic knowledge to understand the operating principle of TFET devices The Wentzel-Kramers-Brillouin (WKB) approximation and the most popular Kane models are successively summarized to estimate the tunneling probability of electrons through the forbidden gap of semiconductors The BTBT generation rates are then calculated by applying the Kane models for both direct and indirect bandgap semiconductors TCAD simulation models are briefly introduced to specify the physical models used in simulations

Chapter 3 presents the operation of TFETs and the verification of BTBT models and associated parameters used in simulations and calculations The on-off switching mechanism is explained in term of energy-band diagrams The temperature independent property of subthreshold swing is analytically interpreted to show achievable sub-60 mV/decade subthreshold swing at room temperature The BTBT model parameters are analytically calculated and numerically verified with existing experimental data to ensure their validity

Chapter 4 concentrates on the issue of current enhancement in TFET devices The weakness of abrupt heterojunctions, which have been proposed to resolve the strong tradeoff between on- and off-currents in homojunction TFETs, is clarified to indicate that the abrupt heterojunction technique is limited in solving this unexpected tradeoff in TFETs

A new graded heterojunction approach for TFETs is proposed and demonstrated to further enhance on-current without deteriorating off-leakage

Chapter 5 explores the scalability of TFET devices, focusing on the impact of drain parameters on short-channel effects It highlights the successful demonstration of sub-10-nm TFETs with graded Si/Ge heterojunctions, which exhibit remarkable resistance to short-channel effects and improved subthreshold swing Additionally, the chapter provides design considerations for optimizing key factors in short-channel TFETs utilizing graded Si/SiGe heterojunctions.

Chapter 6 focuses on the analytical and numerical study of line-tunneling tunnel field-effect transistors (TFETs) utilizing low-bandgap semiconductors It investigates the tunneling properties of these materials, highlighting the distinct roles of local and nonlocal electric fields in influencing tunneling current The suitability of low-bandgap semiconductors for low-power applications in line-tunneling TFET structures is examined Additionally, the chapter addresses the preliminary design considerations for key factors in low-bandgap line-tunneling TFETs, emphasizing the importance of minimizing the tunnel path.

Fig 1.1: Schematic current-voltage characteristics of high subthreshold swing MOSFET and new transistor with steep subthreshold swing.

Overview of Band-to-Band Tunneling Theories

WKB Approximation

The Wentzel-Kramers-Brillouin (WKB) approximation is a semi-classical technique that effectively calculates tunneling probabilities through potential barriers This method is valid when the variation of electron potential energy over a wavelength is significantly smaller than its kinetic energy By employing the WKB approach, the electron wave functions can be expressed in both classically allowed and forbidden regions.

(2.1) where the wave vectors are given by

 (2.2) where  is reduced Plank’s constant; m ∗ is the effective mass; E and V(x) are the electron energy and potential, respectively

By applying the connection formulas [48], the tunneling probability (P), which is determined by the ratio of outgoing and ingoing wave-amplitude squares, can be approximated as

To accurately calculate tunneling probability, one must determine the explicit expression of the potential barrier V(x) within the classically forbidden region However, identifying the specific form of this potential barrier for an electron in the forbidden gap is often challenging It is commonly accepted that the potential may take either a triangular or parabolic shape Given that the wave vector is zero at the classical turning points and reaches its maximum near the center of the forbidden region, the parabolic form is generally considered more suitable for representing the potential barrier.

(2.4) where E g is the semiconductor bandgap, E ⊥ is the perpendicular energy For a uniform electric field, E 0 is given by

E x (2.5) where ξ is the electric field of tunnel junction

Using (2.2), (2.4) and (2.5), the tunneling probability (2.3) can easily be calculated as

The analyses focus on the direct band-to-band tunneling (BTBT) of valence electrons into the conduction band However, the WKB approximation may not be suitable for indirect-bandgap semiconductors like silicon and germanium, where indirect BTBT components play a significant role.

Two-Band Kane Models

Kane's two-band model offers an alternative method for estimating tunneling probability by treating the tunnel electric field as a constant perturbation This approach facilitates the transition of tunneling electrons from the initial valence band state to the final conduction band state.

First-order time-dependent perturbation theory is utilized to assess the transition probability, while the Kane models provide a concise overview of both direct and indirect tunneling mechanisms.

In direct tunneling, Kane formulated the time-independent Schrödinger equation within a two-band model, utilizing the “crystal momentum” representation that incorporates Bloch's functions, expressed as ψ nk (r)=e i k r u nk (r), to account for the influence of a uniform electric field.

E (2.8) where the force on electrons F =qx , the interband matrix element dr k u u i

In the context of time-dependent perturbation theory, the transition probability from band n to band n' is approximated using Fermi's golden rule, where the perturbation potential FX nn' is treated as a constant The band energy E n (k) is defined as the difference between the total energy E and the perturbation potential FX nn(k), while a n(k) and a n' (k) denote the states of the valence and conduction bands, respectively This framework allows for a clear understanding of the energy dynamics within the bands.

→ ′  (2.9) where ρ(E) is the density of state given by

( = (2.10) with γ is the width of Brillouin zone in the x-direction

The unperturbed wave function in the crystal momentum space can be easily solved by ignoring the interband terms in (2.8) to give:

By following a technique similar to the method of stationary phase in complex plane [38], Kane’s solution of matrix elements is expressed as:

In direct tunneling, the perpendicular energy at the initial state (E ⊥ i) is equal to the perpendicular energy at the final state (E ⊥ f), indicating that the perpendicular energy (E ⊥) is conserved throughout the process The reduced mass, denoted as m r, plays a crucial role in this context.

1 (2.13) with m + and m − are conduction and valence band effective masses, respectively

To calculate the tunneling probability, it is noted in uniform electric field that electrons cycle repeatedly through the Brillouin zone with the period t 0 given by [52] t 0 = Fγ (2.14)

Using (2.9)-(2.14) the direct tunneling probability of electrons from valence band to conduction band is written as

If the effective masses at conduction and valence band edges are equal:

+ =m =m m , (2.17) then the tunneling probability in the Kane model is in agreement with that in the WKB picture, except a factor

For indirect tunneling, a similar method was carried out by Kane and Keldysh to obtain the indirect tunneling probability: [38], [51]

In the context of semiconductor physics, the effective mass ratio of conduction to valence band density of states is represented as \( m_c/m_v \) The mass density is denoted by \( \rho \), while the deformation potential associated with transverse acoustic phonons is indicated as \( D_{TA} \) The energy of the transverse acoustic phonon is expressed as \( e_{TA} \), and the occupation number for these phonons is calculated using the formula \( N_{TA} = 1/[exp(e_{TA}/kT) - 1] \) Additionally, the perpendicular electric field, \( E_{\perp ind} \), plays a crucial role in this framework.

BTBT Generation Rates

For simplicity in calculating tunneling rates, calculation related parameters are represented in wave vector space (k-space)

From quantum mechanics, there is one allowed state in a k-space of volume:

If we let ρ(k)dk be the number of states per unit volume with a wave vector between k and k+dk, then

The density of states in k-space is defined by the equation p ρ dk x dk y dk z g dk k = (2.21), where g represents the degeneracy factor The term dk x dk y dk z refers to the volume in wave vector space that contains electron energy values between k and k+dk.

By considering the external electric field as a perturbation potential and applying the Bloch’s theorem, one can obtain the so-called “Newton’s law in k-space” as follows: [53] x dt q dk  (2.23)

It is easy to see from (2.23) that the electron velocity in k-space is

In real space, electrons in high electric fields perform Bloch oscillations till they are transferred to higher band by band-to-band tunneling [52]

Considering a ring of perpendicular momentum between k ⊥ and k ⊥ +dk ⊥ , the flux of electrons reaching to the high field junction for possible tunneling transitions can be calculated as follows: [50]

Incident electron flux in a ring = velocity in k-space × density of states in k-space × area of ring × occupancy of states (2.25)

Using (2.22) and (2.24), (2.25) can be formulated as

The Fermi-Dirac distribution functions, denoted as Fv and Fc, represent the occupancy of states in the valence and conduction bands, respectively Equation (2.26) incorporates the conduction band Fermi-Dirac function to eliminate the influence of occupied states within the conduction band.

The relationship of perpendicular energy and wave vector (2.16) can be used to express the incident flux in energy space as:

The transmission factor (F T) quantifies the available conduction states for valence electron transmission In direct tunneling, both energy and momentum are conserved, meaning the final state of the electron must closely match its initial state Consequently, there is only one permitted state for the valence electron to tunnel into, defining the transmission factor for direct tunneling.

In indirect tunneling, the transmission factor must exceed one due to the non-conservation of perpendicular momentum resulting from phonon absorption or emission For an electron with an initial perpendicular energy \( E_{\perp i} \), its final perpendicular energy may vary The difference in perpendicular momenta between the initial and final states corresponds to the perpendicular energy of the phonon that was absorbed or emitted The density of available final states for tunneling electrons in terms of perpendicular energy can be calculated straightforwardly.

The transmission factor can now be written as

To calculate tunneling rates, one must consider the tunneling generation rate per unit volume (G BTBT), which is determined by the product of the incident electron flux, tunneling probability, and transmission factor, applicable for both direct and indirect tunneling processes.

Because the Kane models provide a unique expression for direct and indirect semiconductors with an acceptable accuracy [9], [54]-[56], they are adopted in subsequent investigations to calculate the tunneling rates

Using (2.15), (2.27), and (2.28) the generation rate of direct tunneling can be performed immediately:

When the reverse-biased voltage (V a ) exceeds 6kT/q, the occupancy functions can be simplified to step functions, leading to the conclusion that F v −F c equals 1 Consequently, the integral over the variable E ⊥ i can be computed straightforwardly, yielding a clear result.

 (2.33) where E e is the smaller of E i and E f which are the electron energy measured from the p-type valence and n-type conduction band edges, respectively

Similar procedure can be made by using (2.18), (2.27) and (2.30) to express the indirect tunneling rate as:

The Kane models are specifically designed for uniform electric fields, but in non-uniform electric fields, it is essential to utilize a nonlocal electric field to accurately assess tunneling generation rates This approach has been validated through experimental evidence.

[9], [28], [54], [55], [58], the Kane models provide a simple and unique expression of the tunneling rate in both direct and indirect BTBT processes, which makes it favorable to be integrated in simulation tools.

TCAD Simulation Models

Analytical models of band-to-band tunneling (BTBT) in tunnel field-effect transistors (TFETs) often rely on overly simplified device structures and parameters To effectively investigate the working principles, performance, and design of TFETs, TCAD simulations serve as a valuable approach For this analysis, Taurus Medici is utilized to conduct two-dimensional simulations of the electrical characteristics of TFETs.

The Medici simulator effectively integrates the unified Kane models of direct and indirect tunneling By omitting insignificant terms from the exponentials in equations (2.33) and (2.34), the Kane models utilized in Medici can be succinctly represented.

G = − (2.35) where the inputable factor γ =2 for direct tunneling and γ =2.5 for indirect tunneling Material dependent parameters A and B for direct and indirect tunneling are successively given by: q

Parameters A and B can be input through theoretical calculations or experimental calibrations Medici offers two primary methods for calculating the electric field terms in the Kane formula (2.35) for BTBT simulations: the local field and the nonlocal field The local field refers to the electric field at the specific position where electrons begin tunneling, while the nonlocal field represents the average electric field along the entire electron tunneling path.

Trap-assisted tunneling, which involves the tunneling of valence electrons into the conduction band through intermediate trap levels, significantly influences the low subthreshold current region in high bandgap tunnel field-effect transistors (TFETs) Numerical and experimental studies indicate that while trap-assisted tunneling is crucial for the current-voltage characteristics of high-bandgap silicon TFETs, its relevance diminishes in lower bandgap semiconductors like germanium and indium arsenide, where the increase in band-to-band tunneling (BTBT) current outpaces that of trap-assisted tunneling This thesis investigates TFET devices utilizing low bandgap semiconductors such as SiGe, InGaAs, and InSb, emphasizing the importance of trap density and distribution, which are sensitive to processing conditions To better understand the physical principles, operation, and design of TFET devices, defect-free TFETs are utilized in this research.

In numerical simulations of TFETs, the Fermi-Dirac distribution and Shockley-Read-Hall recombination are utilized, with the nonlocal tunneling model applied due to the presence of non-uniform electric fields at tunnel junctions The significance of quantum confinement effects varies with the semiconductor body thickness; they are considered when the thickness is 10 nm or less, while being negligible for thicker bodies Additionally, the tunneling probability is highly sensitive to the energy bandgap, making bandgap narrowing a crucial factor in accurately estimating tunneling current in heavily-doped tunnel junctions, and is included in simulations unless stated otherwise.

(I) (II) (III) b p-type n-type qV a a

Fig 2.1: Band-to-band tunneling of valence electrons into conduction band in a reverse-biased p-n junction with the uniform electric field model.

TFET Operation and Model Verification

On-Off Switching Mechanism

The working principle and characteristics of TFET devices are exemplified by a typical p-i-n TFET structure, which can function in both n-channel and p-channel modes In this design, the source and drain regions are heavily doped with distinct types of materials, enhancing the device's performance.

The article discusses the doping parameters for semiconductor materials, specifying n-type doping at 10^17 cm^-3 in the intrinsic channel and p-type doping at 10^20 cm^-3 It highlights a realistic doping gradient of 2 nm per decade in the junction regions Additionally, it mentions the use of a high-k gate dielectric, specifically 3 nm thick HfO2, paired with a metal gate that has a work function of 4.4 eV.

Figure 3.2 illustrates the current-voltage characteristics of TFETs in both n- and p-channel modes, highlighting the distinct off-, subthreshold-, and on-states observed in their input characteristics TFETs exhibit very low off-currents due to their operation in a reverse-biased condition typical of p-i-n diodes However, their on-currents remain relatively low, attributed to the high bandgap of silicon, which limits significant tunneling generation Additionally, unlike traditional MOSFETs, the subthreshold swing in TFETs is not constant but varies with gate voltage.

The on-off switching mechanism of Tunnel Field-Effect Transistors (TFETs) is illustrated through energy-band diagrams, showing that in the off-state, a wide tunnel barrier prevents electron tunneling from the source to the drain The off-current primarily arises from the drift current of minority carriers in the reversed p-i-n diode When high gate voltages are applied, the tunnel barriers at the source junctions narrow, enabling electrons to tunnel through the forbidden gap and activate the TFETs This switching is controlled by the gate voltage, which alters the tunnel barrier width by adjusting the channel bands Despite the axial symmetry in current-voltage characteristics between n- and p-channel modes, asymmetric tunneling processes occur, with electrons consistently tunneling from the valence band to the conduction band in either operation mode.

Temperature-Independent Subthreshold Swing

To comprehend the subthreshold swing behavior of Tunnel Field-Effect Transistors (TFETs), it is essential to analytically clarify a novel mechanism of subthreshold swing As demonstrated in the previous chapter, the TFET drain current can be simplified from equation (2.35) [19].

I = − − (3.1) where V j is the effective source-channel tunnel junction bias; ξ is the tunnel junction electric field Generally, both V j and ξ depend on the gate voltage V g

The subthreshold swing can be directly calculated to give

The subthreshold swing of tunneling field-effect transistors (TFETs) presents two key characteristics that differentiate them from MOSFET devices Firstly, TFETs are not constrained by the kT/q limit, allowing for a temperature-independent subthreshold swing that can achieve a favorable value of sub-60 mV/decade at room temperature However, this advantage comes with the drawback that the subthreshold swing is highly dependent on gate voltage.

Figure 3.4 illustrates the relationship between subthreshold swing and applied gate voltage in complementary n-channel and p-channel TFETs, highlighting that both types exhibit significant gate voltage dependence in their subthreshold regions Notably, the subthreshold swing remains below 60 mV/decade at low voltages but increases sharply with rising gate voltage, leading to a detrimental effect on the average subthreshold swing While TFET devices can achieve a minimal subthreshold swing, it is essential to assess the average value across a specific current range to accurately evaluate their on-off switching performance.

Calculation and Verification of Model Parameters

In the Kane formalism, model parameters are crucial for accurately assessing the BTBT generation rates at tunnel junctions This section focuses on calculating the Kane model parameters for key direct and indirect semiconductors, such as SiGe, InGaAs, and InSb The calculated parameters are validated through numerical simulations, ensuring they align with experimental data.

Silicon-germanium compounds (Si1-x Gex) are being explored as a replacement for silicon in tunnel field-effect transistors (TFETs) due to their smaller bandgap and compatibility with silicon technology The bandgap of Si1-x Gex varies with germanium concentration; higher Ge concentrations lead to a reduced bandgap Consequently, increasing the Ge concentration significantly enhances the on-current performance of SiGe-based TFETs.

Si1-xGex grown on silicon substrates can exhibit either relaxed or compressively strained phases, influenced by the structure and purpose of the study Theoretical calculations of the Kane model parameters for relaxed Si1-xGex, with germanium concentrations ranging from 0 to 1, have been conducted across various tunneling directions However, in practical applications, compressively strained Si1-xGex is more significant for SiGe-based TFETs due to its notably smaller bandgap resulting from strain effects This section focuses on the theoretical calculations and experimental validation of the Kane model parameters for indirect band-to-band tunneling (BTBT) in compressively strained Si1-xGex on (100) silicon wafers.

The Kane parameters A and B for indirect band-to-band tunneling (BTBT) are determined for compressively strained Si1-xGex at various germanium mole fractions ranging from 0 to 1 Key factors such as mass density (ρ), the deformation potential of transverse acoustic phonons (DTA), transverse acoustic phonon energy (eTA), and the occupation number of transverse acoustic phonons (NTA) are thoroughly addressed in the literature.

To determine the degeneracy factor g, the reduced tunneling mass m_r, and the effective mass ratio of valence/conduction band density of states m_v/m_c in silicon-germanium compounds, it is essential to consider the effects of compressive strain The conduction-band constant energy ellipsoids and valence-band structures in relaxed and compressively strained Si1-xGex are illustrated in Fig 3.5 Under compressive strain, the heavy hole and light hole valence bands become separated, with the sixfold degenerate Δ6 valleys splitting into fourfold Δ4 and twofold Δ2 valleys The Δ4 valleys are oriented parallel to the SiGe/bulk-Si interface, while Δ2 valleys are perpendicular Consequently, strained Si1-xGex displays a Si-like conduction band minimum at the Δ axis, resulting in a degeneracy factor of 8 The parameters used to calculate Kane parameters A and B are summarized in Table 3.1 As shown in Fig 3.6, both Kane parameters A and B decrease gradually with increasing Ge mole fraction x in compressively strained Si1-xGex.

The composition of Ge is crucial because compressively strained Si1-xGex maintains a Si-like conduction band across all Ge mole fractions As the concentration of Ge rises, the tunneling rate increases significantly due to its sensitivity to the exponential factor B, despite a decrease in the pre-exponential factor A.

This study verifies the validity of indirect Kane model parameters for compressively strained Si1-xGex by conducting numerical simulations that compare calculated results with experimental data from strained-Si0.57Ge0.43 and strained-Si0.43Ge0.57 TFETs The simulated device structure mirrors the experimental TFET structure referenced in previous work Simulation results, illustrated in Fig 3.7, demonstrate that the curves derived from the calculated Kane parameters align closely with the measured data, affirming the accuracy of the indirect Kane model However, a slight deviation in the numerical results for lower Ge mole fractions (x=0.43) may be attributed to trap-assisted tunneling, which plays a more significant role in SiGe-based TFETs with reduced Ge concentrations.

The direct band-to-band tunneling (BTBT) process exhibits a significantly higher generation rate compared to the indirect BTBT process for the same energy bandgap, as it avoids intermediate phonon absorptions and emissions While low bandgap silicon-germanium compounds can enhance tunneling current, the on-current improvement remains limited due to the indirect bandgap of Si1-xGex To fulfill the drive current requirements for future low-power, high-performance logic applications, further enhancement of the on-current in tunnel field-effect transistors (TFETs) is essential A promising approach to achieve this is through the utilization of low direct-bandgap III-V semiconductors.

In y Ga 1-y As and InSb which have recently attracted much attention

To accurately compute the Kane model parameters, including the pre-exponential factor A and the exponential factor B in direct semiconductors, it is essential to determine the reduced mass of carriers The effective masses of electrons and holes play a crucial role in this calculation, directly influencing the tunneling current in direct semiconductor TFETs Material parameters for In y Ga 1-y As and InSb, as outlined in Table 3.2, are vital for these calculations Notably, the key parameters of III-V semiconductors, such as energy bandgap and reduced mass, are significantly smaller compared to other materials.

Si1-x Ge x To clearly see how the Kane parameters change with varying In concentration y,

Fig 3.8 shows parameters A and B as functions of In mole fraction ranging from 0 to 1

As the In fraction increases, both the pre-exponential and exponential factors exhibit a nearly linear decrease In the case of InSb, the Kane model parameters are significantly lower than those of InAs, with values of A at 0.83×10^20 eV^(1/2)/cm·s·V^2 and B at 4.83×10^6 V/cm·eV^(3/2).

Numerical simulations were conducted to validate the calculated parameters for low bandgap InAs (E g = 0.37 eV) and high bandgap In 0.53 Ga 0.47 As (E g = 0.76 eV) tunnel diodes The InAs diode featured a 60 nm n-layer with a doping concentration of 3×10^18 cm^-3, a 3 nm intrinsic layer, and a 300 nm p-layer with a concentration of 1.8×10^19 cm^-3 In contrast, the In 0.53 Ga 0.47 As diode had a 60 nm n-layer doped at 1.6×10^19 cm^-3, a 3 nm intrinsic layer, and a 300 nm p-layer at 5.7×10^18 cm^-3 The results, illustrated in Fig 3.9, show a strong agreement between simulated and experimental tunneling currents for both diodes without fitting factors However, the InAs diode exhibited a more significant overestimation of tunneling current at high applied voltages compared to the In 0.53 Ga 0.47 As diode, indicating that the discrepancies at higher voltages are primarily due to the influence of series resistances.

Figure 3.1 illustrates the schematic structure of typical silicon tunnel field-effect transistors (TFETs), highlighting that in n-channel TFETs, the p++ region serves as the source and the n++ region as the drain, while the roles are reversed for p-channel TFETs, where the n++ region acts as the source and the p++ region as the drain.

Gate-to-Source Voltage (V) p-channel mode n-channel mode

Complimentary TFETs Gate Length: 50 nm

Fig 3.2: Simulated current-voltage characteristics of complimentary n-channel and p-channel silicon TFETs

Electrons n-channel TFET Strong Tunneling (b)

Fig 3.3: (a) Off-state and (b) on-state energy-band diagrams of complimentary n-channel and p-channel silicon TFETs

Subt hr es ho ld Sw ing ( m V /D ec a de )

Gate-to-Source Voltage (V) p-channel mode n-channel mode

Complimentary TFETs Gate Length: 50 nm

Fig 3.4: The subthreshold swing of the n- and p-channel silicon TFETs depends strongly on gate-to-source voltage

Relaxed Si 1-x Ge x Compressively strained Si 1-x Ge x

Fig 3.5: Conduction-band constant energy ellipsoids along ∆ axis (top) and valence-band structures (bottom) of relaxed- and compressive strained-Si 1-x Ge x

Fig 3.6: Calculated Kane model parameters for strained-Si1-x Ge x with various Ge mole fractions

Fig 3.7: Numerical drain currents of strained-Si1-x Ge x TFETs using calculated Kane model parameters are verified with experimental data [55]

Fig 3.8: Calculated Kane model parameters for In y Ga1-y As with various In mole fractions

InAs (E g = 0.37 eV) n-type p-type i -layer

Fig 3.9: Numerical and experimental current-voltage curves of InAs and In0.53Ga0.47As tunnel diodes

Table 3.1 outlines the parameters utilized in the calculations of the Kane model parameters A and B for compressively strained Si1-xGex In this model, the strained Si1-xGex is grown on a (001) Si substrate, with carrier transport occurring along the [100] direction All effective masses are expressed in terms of the free electron mass, and the energy gap is denoted as xEg.

Table 3.2: Parameters of direct semiconductors In y Ga1-y As and InSb used in calculations of Kane model parameters A and B All effective masses are in the unit of free electron mass

Enhancing On-Current of TFETs

Homojunction TFETs

The homojunction structure is the most basic design of TFET devices, utilizing a single semiconductor material for the source, channel, and drain regions This architecture enables the creation of complementary n-channel and p-channel TFETs, akin to NMOS and PMOS transistors By employing CMOS-compatible fabrication processes, complementary silicon TFETs can be efficiently produced on the same silicon substrate To comprehend the physical mechanisms and electrical characteristics of advanced TFETs discussed later, it is essential to highlight the limitations of homojunction TFETs that utilize low bandgap semiconductors, which are necessary for enhancing the TFET on-current.

The schematic structure of homojunction Tunnel Field-Effect Transistors (TFETs) is depicted in Fig 4.1(a), featuring Si1-x Ge x in the source, channel, and drain regions A double-gate design, incorporating a 1 nm SiO2 gate dielectric and a body thickness of 20 nm, was implemented to achieve minimized subthreshold swing and maximized on-current The TFETs operated in n-channel mode, utilizing a heavily-doped p++ source and n++ drain with a concentration of 10^20 cm^-3 to demonstrate ambipolar behaviors Additionally, a 40 nm n-channel with a doping concentration of 10^17 cm^-3 and a realistic doping gradient of 2 nm/decade for the source and drain junctions were utilized in the design.

The input current-voltage characteristics of the uniform Si 1-x Ge x TFETs with various

Ge concentrations significantly affect the performance of Si TFETs, as illustrated in Fig 4.1(b) While the pure Si TFET demonstrates a sharp on-off transition, it suffers from low on-current Increasing Ge concentration enhances the on-current due to a reduced bandgap; however, this also leads to a corresponding rise in ambipolar off-state tunneling currents, resulting in substantial off-leakage currents in low-bandgap homojunction TFETs The energy-band diagrams in Fig 4.2 clarify these characteristics, showing that the Si0.5Ge0.5 TFET has a lower and narrower tunnel barrier during the on-state, facilitating greater tunneling generation at the source-channel junction compared to the high-bandgap Si TFET Conversely, in the off-state, the reduced bandgap narrows the tunnel width and lowers the tunnel height, leading to increased leakage at the drain-channel junction Consequently, the use of low bandgap semiconductors in TFET devices compromises the suitability of the homojunction structure due to excessive ambipolar leakage current.

Weakness of Abrupt Heterojunction

In a typical Tunnel Field-Effect Transistor (TFET), the on-state current primarily arises from tunneling at the source-channel junction, while ambipolar tunneling leakage occurs at the drain-channel junction Utilizing low bandgap semiconductors in single-material TFETs enhances the on-current but significantly increases ambipolar leakage current To address the tradeoff between on- and off-state currents, abrupt heterojunctions are employed, where the source-side region features a low bandgap material to enhance on-current, and the drain-side region uses a high bandgap semiconductor to reduce off-state tunneling current It is crucial for the interface between the high and low bandgap materials to be positioned within the channel, with optimal overlap of the heterojunction extending several nanometers beneath the gate to fine-tune the on- and off-currents.

To examine the role of abrupt heterostructures in suppressing the ambipolar current in low-bandgap TFETs, the electrical characteristics of abrupt heterojunction Si/Si1-x Ge x

The numerical analysis of TFETs in the n-channel operation regime reveals significant insights into their performance The simulated structure, depicted in Fig 4.3(a), showcases abrupt Si/Si1-xGex TFETs with varying Ge mole fractions, while Fig 4.3(b) illustrates their current-voltage characteristics This abrupt TFET design mirrors the homojunction structure shown in Fig 4.1(a), differing only in the use of diverse bandgap semiconductors for the source, channel, and drain A gate-source overlap of 2 nm was implemented to enhance the on-current and reduce off-leakage Notably, the high Ge concentration in TFETs effectively suppresses ambipolar tunneling leakage, all while achieving improved on-current levels However, the ambipolar current remains stable despite increasing Ge concentration at the source, due to the fixed high-bandgap Si employed in the channel and drain regions, where the ambipolar current is generated.

The abrupt heterojunction architecture allows for separate management of the energy bandgap at the source and drain junctions, potentially resolving the tradeoff between on- and off-currents in tunnel field-effect transistors (TFETs) However, the direct contact between materials with differing band structures creates abrupt conduction and valence band offsets at the heterojunction interface These specific abrupt band offsets restrict the application of low bandgap materials that could enhance the on-current in TFETs utilizing abrupt heterojunctions.

As shown in this section, the abrupt heterojunction structure may cause unexpected on-current lowering, depending on band offset properties of heterojunctions as well as TFET operation modes

This article discusses the limitations of abrupt heterostructures in p-i-n structure TFETs, illustrating p-channel and n-channel devices with schematic views using Si/Si1-xGex and In0.17Ga0.83As/InyGa1-yAs heterojunctions Both types of devices utilize a similar structure, as shown in Fig 4.3(a), but differ in their abrupt heterojunctions tailored for specific operational modes For SiGe-based p-channel TFETs, a metal gate workfunction of 5.2 eV is employed, highlighting the distinct parameters involved in these semiconductor devices.

To optimize device performance, a heavily doped n++ source of 10²⁰ cm⁻³, a lightly doped p+ drain of 5×10¹⁸ cm⁻³, and a 40 nm p-channel with a doping concentration of 10¹⁷ cm⁻³ were utilized, focusing on enhancing on-current, subthreshold swing, and ambipolar leakage Compressive strained-Si₁₋ₓGeₓ with minimized bandgaps was assumed to improve tunneling current The Si₁₋ₓGeₓ source featured a 2 nm gate-source overlap for optimal performance For InGaAs-based n-channel TFETs, a gate workfunction of 4.5 eV was specified, with reversed polarity and doping magnitude compared to p-channel TFETs Instead of Si and Si₁₋ₓGeₓ, In₀.₁₇Ga₀.₈₃As and InᵧGa₁₋ᵧAs were defined at the drain and source, respectively, with a 6 nm gate-source overlap.

In0.17Ga0.83As was adopted to give a same bandgap of silicon for comparison of tunneling currents

The current-voltage characteristics of abrupt Si/Si1-xGex p-channel TFETs and abrupt In0.17Ga0.83As/InyGa1-yAs n-channel TFETs reveal that the on-state currents initially rise with increasing Ge and In fractions This increase is linked to the reduction in the bandgaps of Si1-xGex and InyGa1-yAs as x and y values increase However, the on-currents peak at medium Ge and In concentrations before declining at higher fractions.

It seems that this on-current lowering contradicts to the continuous decrease of Si1-x Ge x and

The bandgap characteristics of y Ga 1-y As restrict the use of germanium (Ge) and indium-rich compounds with low bandgaps, which hinders the enhancement of on-currents in abrupt heterostructure-based p-channel and n-channel tunneling field-effect transistor (TFET) devices.

To comprehend the complex behavior of on-state tunneling currents influenced by varying Ge and In compositions, Figure 4.6 illustrates the energy-band diagrams of abrupt TFETs, showcasing two distinct values for each Ge and In fraction.

In p-channel TFETs, the valence band offset at the junction interface creates a thermal emission barrier that hinders hole movement from the channel to the drain The transport process involves electron tunneling from the channel valence band to the source conduction band and the thermal emission of trapped holes to the drain valence band As the Ge concentration increases, the thermal barrier height rises, with a low barrier at a small Ge fraction (x=0.3) allowing for effective hole movement due to band-to-band tunneling However, in high Ge fraction TFETs (x=0.5), the elevated thermal barrier significantly restricts hole movement, causing holes to accumulate in the well, which reduces band bending in the channel and leads to an extended tunnel barrier and narrower tunneling region Consequently, the on-state tunneling current diminishes in x=0.5 TFETs, despite a reduced bandgap, which only benefits performance at lower Ge fractions A similar phenomenon is anticipated for n-channel TFETs, where significant thermal barriers for electrons arise at the conduction band in In0.17Ga0.83As/In y Ga1-y As heterojunctions.

In0.17Ga0.83As/In y Ga1-y As heterojunctions exhibit a unique physical transport mechanism in abrupt n-channel TFETs, where electrons tunnel from the source valence band to the channel conduction band, followed by thermal emission to the drain conduction band The presence of a higher In mole fraction increases the thermal barrier, leading to reduced current in In-rich TFETs In n-channel TFETs, only the conduction band offset creates a thermal barrier for electrons, while the valence band offset does not affect electron or hole flow Conversely, in p-channel TFETs, the valence band offset poses an unwanted thermal barrier for holes It can be concluded that significant conduction and valence band offsets in abrupt heterojunctions lead to on-current reduction in n- and p-channel TFETs, respectively In silicon-germanium heterojunctions, most band offset occurs at the valence band, rendering the conduction band offset negligible, which results in observable on-current lowering only in abrupt Si/Si1-x Ge x p-channel TFETs, with no such effect in n-channel counterparts.

Figure 4.7 illustrates the current-voltage characteristics of abrupt Si/Si0.7Ge0.3 and In0.17Ga0.83As/In0.35Ga0.65As heterojunction TFETs across different temperatures Both p-channel and n-channel TFETs exhibit a notable increase in on-state currents as temperature rises This significant temperature dependence of the on-current indicates the existence of thermal emission barriers for both holes and electrons Furthermore, the Si/Si0.7Ge0.3 TFET demonstrates a stronger temperature dependence in its on-current compared to the In0.17Ga0.83As/In0.35Ga0.65As TFET.

In0.17Ga0.83As/In0.35Ga0.65As TFET because its thermal barrier is higher.

Graded Heterojunction Approach

The formation of thermal emission barriers at junction interfaces is a significant weakness of abrupt heterojunctions in TFET devices, as these barriers hinder the flow of generated electrons and holes, limiting on-current despite a low-bandgap source aimed at increasing tunneling rates To effectively utilize heterojunctions, it is essential to eliminate these thermal barriers The physical mechanisms in abrupt TFETs allow for the application of a graded heterojunction approach in both p-channel and n-channel TFETs, enhancing on-state tunneling current The reduction of on-current in n-channel TFETs, similar to p-channel counterparts, is also linked to these thermal emission barriers Thus, implementing graded band offsets through graded heterojunctions presents a promising solution to address the limitations posed by abrupt heterojunctions in TFET devices.

Instead of using abrupt heterostructures, graded Si/Si1-x Ge x and

In 0.17 Ga 0.83 As/In y Ga 1-y As heterojunctions were adopted in p- and n-channel TFETs, respectively, as shown in Fig 4.8 The source region of p-channel (n-channel) TFETs was defined by strained-Si 1-x Ge x (In y Ga 1-y As) whereas the distribution of Ge (In) composition was gradually graded from x (y) value at the source-channel junction to 0 (0.17) at 20 nm away from the source All other device parameters of the graded TFETs are identical to those of the abrupt counterparts Fig 4.9(a) shows the current-voltage characteristics of graded Si/Si 1-x Ge x p-channel TFETs whereas those of graded In 0.17 Ga 0.83 As/In y Ga 1-y As n-channel TFETs are displayed in Fig 4.9(b) For small Ge and In mole fractions x and y, the on-currents are regularly increased with increasing x and y Similar to the abrupt TFET counterparts, this is due to the decrease of bandgaps with increasing Ge and In fractions

Importantly, the on-currents of the graded TFETs continue to be enhanced at high Ge and

InGe-rich compounds exhibit low bandgaps, which enhance the performance of tunneling field-effect transistors (TFETs) when utilizing graded heterojunctions, addressing the limitations of abrupt heterostructures and significantly improving on-current As illustrated in Fig 4.9, InGaAs-based TFETs demonstrate on-state currents that are approximately two orders of magnitude higher than those of SiGe-based TFETs, despite their similar bandgaps This substantial difference is attributed to the higher probability of direct tunneling compared to indirect tunneling for the same energy bandgap, making InGaAs a promising material for achieving high drive currents in TFET devices.

The weakness of abrupt heterojunctions in graded tunnel field-effect transistors (TFETs) is effectively addressed by utilizing gradually graded band offsets, as illustrated in the energy-band diagrams This design eliminates thermal barriers for both holes and electrons in the channels of p-channel and n-channel TFETs, allowing for immediate movement of charge carriers to the source and drain without accumulation in the channel Consequently, channel band bending is primarily influenced by gate voltage, enabling the narrowing of tunnel widths similar to single-material TFETs Additionally, higher fractions of Ge and In contribute to reduced tunnel widths, significantly enhancing on-state tunneling currents The absence of thermal barriers allows for flexible use of elevated Ge and In fractions to further improve drive currents in graded TFETs Therefore, adopting the graded heterojunction approach is crucial for enhancing on-current in heterojunction-based devices facing thermal emission barriers.

TFET devices regardless of n-channel or p-channel FET

Figure 4.11 illustrates the current-voltage characteristics of graded Si/Si0.7Ge0.3 and In0.17Ga0.83As/In0.35Ga0.65As heterojunction TFETs across various operating temperatures, confirming the absence of thermal emission barriers The data indicates that the on-currents in these graded TFETs remain largely unaffected by temperature fluctuations, with minor variations attributed to the temperature dependence of the bandgap.

Fig 4.1: (a) Schematic structure and (b) current-voltage curves of homojunction Si1-x Ge x n-channel TFETs with various Ge concentrations x

Fig 4.2: (a) On-state and (b) off-state energy-band diagrams of homojunction Si1-x Ge x n-channel TFETs with different Ge mole fractions

Abrupt Heterojunction Si/Si 1-x Ge x TFETs

Fig 4.3: (a) Schematic structure and (b) current-voltage curves of abrupt heterojunction Si/Si 1-x Ge x n-channel TFETs with various x values

Fig 4.4: Schematic structures of (a) SiGe-based p-channel TFETs and (b) InGaAs-based n-channel TFETs with using abrupt heterostructures

Abrupt Si/Si 1-x Ge x p-channel TFETs

Abrupt In 0.17 Ga 0.83 As/In y Ga 1-y As n-channel TFETs (b)

Fig 4.5: Current-voltage characteristics of (a) abrupt Si/Si 1-x Ge x p-channel TFETs and (b) abrupt In0.17Ga0.83As/In y Ga1-y As n-channel TFETs with various Ge and In mole fractions

Abrupt Si/Si 1-x Ge x p-channel TFETs

In 0.17 Ga 0.83 As/In y Ga 1-y As n-channel TFETs

The on-state energy-band diagrams for abrupt Si/Si1-xGex p-channel TFETs and abrupt In0.17Ga0.83As/InyGa1-yAs n-channel TFETs illustrate the electron and hole movement generated by band-to-band tunneling, represented by solid-line arrows.

Abrupt Si/Si 0.7 Ge 0.3 p-channel TFET

Abrupt In 0.17 Ga 0.83 As/In 0.35 Ga 0.65 As n-channel TFET

Fig 4.7: Current-voltage characteristics of (a) p-channel and (b) n-channel TFETs based on abrupt heterojunctions at different operating temperatures

Fig 4.8: Schematic structures of (a) SiGe-based p-channel TFETs and (b) InGaAs-based n-channel TFETs with graded heterojunction approach

Graded Si/Si 1-x Ge x p-channel TFETs

Graded In 0.17 Ga 0.83 As/In y Ga 1-y As n-channel TFETs (b)

Fig 4.9: Current-voltage characteristics of (a) graded Si/Si 1-x Ge x p-channel TFETs and (b) graded In0.17Ga0.83As/In y Ga1-y As n-channel TFETs with some values of x and y

Graded Si/Si 1-x Ge x p-channel TFETs

In 0.17 Ga 0.83 As/In y Ga 1-y As n-channel TFETs

The on-state energy-band diagrams for graded Si/Si1-xGex p-channel TFETs and graded In0.17Ga0.83As/InyGa1-yAs n-channel TFETs illustrate the tunneling processes involved These diagrams highlight the movement directions of electrons and holes, represented by solid-line arrows, as they are generated during tunneling The analysis includes variations in the Ge and In fractions, providing insights into the electronic behavior of these transistors.

Graded Si/Si 0.7 Ge 0.3 p-channel TFET

Graded In 0.17 Ga 0.83 As/In 0.35 Ga 0.65 As n-channel TFET

Fig 4.11: Current-voltage curves of (a) p-channel and (b) n-channel TFETs based on graded heterojunctions at various operating temperatures.

High TFET Scalability with Advanced Techniques

Drain Engineering

In conventional MOSFETs, the threshold voltage and on-off switching are influenced by the channel potential barrier, with short-channel effects arising from drain-induced barrier lowering (DIBL) due to the electric field encroachment from the drain into the channel To mitigate these short-channel effects, lateral channel engineering has been utilized, incorporating a heavy halo pocket around the drain to shield against the lateral electric field Additionally, high drain-induced barrier thinning (DIBT) has been observed in short-channel TFETs, indicating significant short-channel effects Effective drain engineering is therefore anticipated to alleviate these short-channel effects in scaled TFETs.

Figure 5.1 illustrates the schematic of a conventional p-i-n tunnel field-effect transistor (TFET) utilized in numerical analysis Key parameters for investigating highly scaled TFETs include channel length (Lg), drain concentration (Nd), and drain length (Ld) The study employed a silicon-on-insulator structure featuring a 20 nm silicon body to effectively analyze the intrinsic short-channel effects of TFET devices Additionally, a gate work function of 4.2 eV was implemented in the research.

The use of a 3 nm HfO2 gate insulator is essential for achieving the desired threshold voltage and effective gate control in TFETs To optimize the on-state current, a heavily doped p+ source with a concentration of 10^20 cm^-3 was implemented alongside an intrinsic n-body of 5×10^17 cm^-3 The study focused on a 5 nm channel length, characteristic of extremely scaled TFETs, and employed a drain voltage (Vd) of 0.7 V to analyze short-channel effects A drain length of 20 nm was consistently used in the TFETs unless otherwise specified.

5.1.1 Design of drain in extremely-scaled TFETs

Figure 5.2 illustrates the current-voltage characteristics of scaled TFETs with channel lengths ranging from 30 nm to 10 nm, utilizing drain concentrations of (a) 5×10^19 cm^-3 and (b) 5×10^18 cm^-3 to analyze the impact of the drain on short-channel effects The long-channel 30 nm TFETs demonstrate favorable switching performance, while the short-channel 10 nm and 15 nm devices exhibit significant short-channel effects Even with negative gate voltages, these short-channel TFETs struggle to turn off, resulting in notable leakage currents The short-channel effect is particularly pronounced in the TFET with a heavy drain concentration of 5×10^19 cm^-3 Conversely, the 20 nm TFET with a lighter drain concentration of 5×10^18 cm^-3 maintains an acceptable on-off transition without increased off-leakage or degraded subthreshold swing This indicates that the light drain plays a crucial role in mitigating short-channel effects in very short TFETs, suggesting that a light drain design is an effective strategy to reduce these effects in extremely scaled TFETs.

The physical mechanism of drain-induced suppression of off-state currents in 20 nm TFETs is illustrated through energy-band diagrams, highlighting the differences between heavy and light drain regions In heavy drain TFETs, the drain voltage drop is primarily confined to the channel, resulting in a narrow off-state tunnel barrier and increased leakage current Conversely, light drain TFETs exhibit an energy-band bending that extends into the drain, creating a wider off-state tunnel barrier and thereby reducing off-state tunneling The electric field distribution along the channel and drain is more uniform with a light drain, which lowers the maximum electric field in the channel Since BTBT generation is highly sensitive to electric fields, effectively managing the lateral field distribution in both the channel and drain is crucial for the design of extremely scaled TFETs, including the upcoming 5 nm variants.

The current-voltage characteristics of 5 nm tunneling field-effect transistors (TFETs) are significantly influenced by drain concentrations ranging from 1×10^19 cm^-3 to 5×10^17 cm^-3 Higher drain concentrations lead to increased off-current due to the drain electric field's penetration in the channel, making it essential to maintain a concentration as low as 5×10^17 cm^-3 for optimal performance The minimum subthreshold swing varies with channel lengths and drain concentrations; a concentration of 5×10^18 cm^-3 suffices for 20 nm TFETs, while 5×10^17 cm^-3 is crucial for 5 nm TFETs Furthermore, using lighter drain concentrations can enhance the subthreshold swing in highly scaled TFETs Comparative energy band diagrams reveal that the energy band bending at the source-channel junction of 5 nm TFETs is more abrupt than at the drain-channel junction, resulting in a smaller subthreshold swing.

The performance of tunnel field-effect transistors (TFETs) is significantly influenced by both drain concentration and drain length, with a minimum drain length of 20 nm being essential for 5 nm TFETs to achieve optimal on-off switching characteristics Short-channel effects are exacerbated in TFETs with drain lengths under 20 nm, leading to increased off-current and subthreshold swing due to the abrupt energy band bending that fails to form a proper tunnel barrier In contrast, a 20 nm drain length allows for a smoother energy band transition, resulting in a wider tunnel barrier that effectively minimizes off-state tunneling The distribution of the electric field is also affected by the drain length, as insufficient length can create a high electric field and a thin tunnel barrier, further worsening short-channel effects Thus, maintaining an adequate drain length is critical for controlling these effects in highly scaled TFETs.

In scaled Tunnel Field-Effect Transistors (TFETs), a longer drain enhances the short-channel effect, yet it simultaneously reduces the on-current due to increased drain resistance This presents a tradeoff between subthreshold swing and on-current, highlighting the need to optimize drain length in very short TFETs As illustrated in Fig 5.7(a), the minimum subthreshold swing varies with drain lengths across different TFET sizes For 20 nm TFETs, the subthreshold swing shows minimal dependence on drain length due to a mild short-channel effect, while for shorter TFETs, particularly 5 nm, the variations in subthreshold swing become significantly more pronounced.

In 10 nm Tunnel Field-Effect Transistors (TFETs), a significant increase in subthreshold swing occurs when the drain length is less than 20 nm To mitigate short-channel effects, the minimum drain length for sub-20 nm TFETs should be designed to achieve a total length of approximately 25 nm for both the drain and channel This limitation is crucial for maintaining the integrity of the reverse-biased drain junction and preventing excessive tunneling Consequently, reducing the supply voltage is vital for further minimizing these extremely scaled TFET devices.

Si 1-x Ge x heterojunctionis combined with the minimum drain design in 5 nm TFETs to separately optimize the source- and drain-side tunnel regions, generating favorable on-off switching characteristics

The previous section discussed drain engineering to illustrate the scalability of TFETs into sub-10 nm regimes However, the use of high-bandgap silicon results in relatively low on-currents in these highly scaled TFETs To address this issue, asymmetric heterojunction structures utilizing low-bandgap Si1-xGex have been shown to effectively decouple the trade-offs between on- and off-currents, allowing for the optimization of the source-side and drain-side tunnel regions to enhance TFET switching characteristics.

The applicability of drain design in extremely scaled tunnel field-effect transistors (TFETs) with low bandgap sources must be examined Figure 5.8 illustrates the schematic of the asymmetric Si1-xGex heterojunction TFET utilized in numerical analyses In n-channel TFETs, abrupt Si/Si1-xGex heterojunctions can be effectively implemented without causing a reduction in on-current.

Effective drain design allows for the successful scaling of Si1-xGex heterojunction tunneling field-effect transistors (TFETs) into the 5 nm regime, achieving high on-state tunneling currents Current-voltage curves for 5 nm Si1-xGex heterojunction TFETs with varying x values are illustrated in Fig 5.9(a) These devices feature a drain length of 20 nm and a drain concentration of 5×.

The use of a Si1-xGex heterojunction in 5 nm tunnel field-effect transistors (TFETs) significantly enhances on-state currents due to its excellent short-channel effects As the germanium composition (x) increases, both the on-current rises and the subthreshold swing decreases, attributed to the lower energy bandgap A comparison of the on-state energy-band diagrams of 5 nm Si0.5Ge0.5 heterojunction TFETs with their silicon counterparts shows that the on-state tunnel barrier is effectively minimized in the former.

Si 0.5 Ge 0.5 source heterojunction without appreciable influence caused by the proposed drain design

Figure 5.10 illustrates the impact of drain concentration and drain length on the current-voltage characteristics of 5 nm Si0.5Ge0.5 TFETs An optimized drain length of 20 nm combined with a concentration of 5×10^17 cm^-3 achieves the lowest off-state current while facilitating the most abrupt on-off switching Conversely, increasing the drain concentration or reducing the drain length exacerbates the short-channel effects in the 5 nm TFETs To maintain a small subthreshold swing and minimize off-state leakage current, low concentration and adequate length are essential Interestingly, when utilizing the Si1-xGex source heterojunction, the on-state current shows minimal dependence on drain concentration and length, suggesting that the trade-offs between short-channel effects and on-state current can be mitigated in very short TFETs.

Graded Si/Ge Heterojunction

The previous section discussed design criteria for drains to effectively mitigate short-channel effects in homojunction and abrupt heterojunction TFETs By manipulating drain doping and length, the tunnel barrier width can be optimized, requiring a drain concentration as low as 5×10^17 cm^-3 and a minimum total drain and channel length of approximately 25 nm This limitation constrains the scaling of total TFET length, while a lightly-doped, long drain can introduce problematic series resistance that hampers TFET performance This section introduces a novel bandgap engineering approach using a graded Si/Ge heterojunction, enabling precise control over both the height and width of the tunnel barrier By integrating techniques for adjusting barrier width and height, it is possible to achieve a sub-10 nm graded Si/Ge TFET with an exceptionally steep subthreshold swing, even at higher concentrations and shorter drain lengths.

The schematic of a p-body TFET featuring a graded Si/Ge heterojunction is illustrated in Fig 5.11, alongside uniform Ge and abrupt heterojunction TFETs for comparison This design incorporates a double-gate structure with a body thickness of 10 nm, utilizing a gate workfunction of 4.8 eV and a 3 nm HfO2 gate-dielectric to achieve low subthreshold swing and high current density The device employs a heavily doped n+ source at 10^20 cm^-3 and a p+ doped drain at 5×10^18 cm^-3, with a p-body doping concentration of 10^17 cm^-3, enhancing the on-current and subthreshold swing while minimizing ambipolar leakage current A realistic doping gradient of 2 nm/decade is maintained throughout all junction regions of the TFETs.

In numerical simulations of graded Si/Ge heterojunctions, linear assumptions are made alongside physical tunneling models The quantum confinement effect is also taken into account due to the thin body structure Furthermore, an optimized p-channel tunneling field-effect transistor (TFET) is explored, featuring a Si-drain and Si-channel configuration.

A Si0.7Ge0.3 heterojunction device with a gate work function of 5.27 eV was employed to enhance performance The choice of a 0.3 Ge mole fraction and a 2 nm gate-source overlap was optimized to achieve high on-current, low off-current, and a steep subthreshold swing.

In scaled tunnel field-effect transistors (TFETs), the height and width of the tunnel barrier are crucial for effective bandgap engineering For p-body TFETs, minimizing the on-state barrier height and width near the source region enhances on-current using a small bandgap semiconductor, while widening the off-state barrier is essential to reduce leakage current Previous studies have typically addressed the barrier engineering of the source and drain separately, which is ineffective for sub-20 nm devices due to the close proximity of the source to the drain The energy-band diagrams of 8 nm graded Si/Ge TFETs, as illustrated in Fig 5.12, reveal that utilizing a graded Si/Ge heterojunction allows for a narrowed and lowered on-state tunnel barrier at the source region, thereby increasing on-current, while simultaneously raising and extending the off-state tunnel barrier into the drain region to effectively manage off-current.

The steep switching characteristics of tunnel field-effect transistors (TFETs) are significantly influenced by the gate voltage's role in the on-off transition of the tunnel barrier As illustrated in Fig 5.13, the heights and widths of the tunnel barriers are determined at the point where the band-to-band tunneling generation rate is maximized, ensuring an effective tunneling pathway In graded heterojunction TFETs, the barrier height behaves almost like a step function in response to gate voltage changes Furthermore, the barrier width exhibits a strong dependence on gate voltage, narrowing with increasing negative gate voltage and gradually widening when the device is turned off This abrupt transition in barrier dimensions contributes to a steep subthreshold slope, enhanced on-current, and reduced leakage current in sub-10 nm graded Si/Ge TFETs.

Short-channel graded Si/Ge TFETs exhibit a distinct tunneling initiation pattern, as illustrated in Fig 5.14, which compares tunneling generation under on and subthreshold conditions using an 8 nm device In contrast to the long-channel 50 nm TFET, where tunneling begins from the body center towards the channel surface with negative gate voltage, the 8 nm TFET initiates tunneling at the channel surface and propagates inward This variation in tunneling initiation is primarily due to changes in the energy-band diagram within the short-channel graded heterojunction The unique tunneling pattern, characterized by abrupt transitions in tunnel barrier width and height relative to gate voltage, plays a crucial role in controlling the on-off switching of scaled TFETs.

5.2.2 Scaling into sub-10 nm regimes

Scaling conventional Tunnel Field-Effect Transistors (TFETs) into sub-10 nm dimensions presents significant challenges due to severe short-channel effects Current-voltage characteristics of uniform Germanium (Ge) TFETs reveal that a 50 nm Ge TFET demonstrates optimal switching performance with minimal off-current However, as channel lengths shrink below 20 nm, these Ge TFETs experience pronounced short-channel effects, leading to increased leakage currents even in the off-state In contrast, the abrupt Silicon/Silicon-Germanium (Si/SiGe) heterojunction TFET maintains excellent on-off switching characteristics at 20 nm, with a sufficiently wide off-state barrier to mitigate tunneling leakage Despite these advantages, the abrupt Si/SiGe TFET also encounters short-channel effects and subthreshold swing issues in sub-20 nm scenarios.

As for the graded TFET displayed in Fig 5.16(a), sub-10 nm graded Si/Ge devices exhibit excellent immunity against the short-channel effect to be an ideal switching device

In long-channel TFETs, the larger drain-side bandgap has little impact on the active tunneling area at the source, while in short-channel devices, the wide-bandgap drain constrains tunneling under subthreshold conditions As channel length decreases, switching becomes more abrupt, with the 5 nm graded Si/Ge TFET exhibiting a significantly improved subthreshold swing, high on-current, and low off-current The steep on-off switching in scaled sub-10 nm TFETs is attributed to the abruptness of the graded Si/Ge heterojunction and its tunneling generation pattern Additionally, the gradual lowering of the channel valence-band results in slightly lower on-current and a higher threshold voltage Analysis of TFET devices with varying channel lengths reveals a minimum average subthreshold swing within one order of drain current, demonstrating nearly ideal sub-10 mV/decade on-off switching in sub-10 nm graded TFETs Overall, graded Si/Ge heterojunction TFETs effectively minimize subthreshold swing when scaling from 50 nm to 5 nm.

Design of Graded Si/SiGe TFETs

Long-channel Tunnel Field Effect Transistors (TFETs) allow for independent optimization of the drain-channel and source-channel tunnel junctions, enhancing switching characteristics A steep and heavily doped source junction is essential for achieving high on-current and low subthreshold swing, while a mild drain concentration helps reduce tunneling leakage In asymmetric structures, the Ge composition at the heterojunction can be finely tuned to adjust tunnel barriers As TFETs are scaled down to sub-10 nm dimensions with graded silicon-germanium heterojunctions, the complexities of tunneling operations increase due to the direct interaction between drain and source tunneling Therefore, the design of the drain junction in ultra-short-channel TFETs must be aligned with the engineering of graded heterojunction regions, where the Ge composition distribution is vital for controlling tunnel barriers and tunneling current For design purposes, a 50 nm TFET represents a long-channel transistor, while a 10 nm TFET exemplifies a short-channel device.

5.3.1 Ge composition in graded heterojunctions

The Ge composition significantly influences the characteristics of silicon-germanium TFET devices, particularly in graded TFETs where the Ge mole fraction x is defined at the source of the Si1-x Gex heterojunction This x-value is crucial as it affects the bandgap widening distribution from the source to the drain This section explores the impact of the Ge mole fraction on the subthreshold swing and on-current, noting that variations in x also alter the threshold voltage of the graded Si1-x Gex TFETs To maintain a consistent threshold voltage, defined at a drain current of 10^-7 A/µm, the gate workfunction was adjusted accordingly.

The current-voltage curves of 10 nm graded Si/Si1-xGex TFETs reveal that larger Ge mole fractions (x) result in improved on-off switching characteristics, while smaller x values lead to a smoother transition from on-current to off-leakage Specifically, the off-state energy-band diagrams indicate that a small x (0.5) causes abrupt bending in the channel energy-band, preventing the formation of an effective tunnel barrier and resulting in higher drain current and poorer subthreshold swing In contrast, a larger x (0.8) allows for a smoother energy-band transition, creating a wider tunnel barrier that minimizes leakage and enhances the steepness of the subthreshold swing Additionally, short-channel graded Si/Si1-xGex devices with larger x values exhibit a significant bandgap difference from source to drain, as the energy-band offset primarily occurs at the valence-band side, thus ensuring a gradual and high off-state barrier in the graded TFETs.

The calculated subthreshold swing of graded Si/Si1-xGex TFETs varies with different x values, as shown in Fig 5.18(a) The average subthreshold swing reflects the on-off switching characteristics, with drain current ranging from 10^-11 to 10^-12 A/μm For short-channel TFETs, the subthreshold swing decreases as x values increase, benefiting from a smoother channel energy-band diagram that enhances switching performance A sufficiently high Ge mole fraction is crucial for effective on-off switching in short-channel TFETs In contrast, long-channel TFETs exhibit a larger subthreshold swing due to less abrupt graded heterojunctions While the subthreshold swing decreases with increasing Ge mole fraction at lower x values, a significant rise in Ge composition (x > 0.8) leads to a notable increase in the subthreshold swing Fig 5.18(b) illustrates the subthreshold energy-band diagrams at three different x values, highlighting a reversal in trend around x = 0.8 Under subthreshold conditions, the Si0.2Ge0.8 graded TFET features a wider tunnel-barrier compared to the Si0.4Ge0.6 variant, resulting in a higher tunnel-barrier than the x = 1 TFET This optimal tunnel-barrier at x = 0.8 effectively minimizes leakage current, producing the most abrupt swing in long-channel graded TFETs.

The Ge mole fraction x significantly influences the subthreshold swing and on-current in graded TFETs, as illustrated in Fig 5.17(a) The on-state current of graded Si/Si1-xGe x TFETs increases with higher Ge mole fractions, reaching its peak with a pure Ge source, as depicted in Fig 5.19 This rise in on-current is primarily due to a reduced bandgap at the Si1-xGe x source, with a more pronounced effect in short-channel devices due to the abrupt graded heterojunctions Therefore, for optimal performance in minimizing subthreshold swing and maximizing on-state current, a higher Ge mole fraction is advantageous in short-channel graded Si1-xGe x TFETs However, achieving a high Ge mole fraction can complicate the epitaxial growth of Si1-xGe x during fabrication.

5.3.2 Role of drain in graded TFETs

This section analyzes how drain concentration and length influence the current-voltage characteristics of graded silicon-germanium TFETs, assuming a pure Ge source The results, illustrated in Fig 5.20(a), show that while a heavily doped drain with low resistances slightly enhances the on-state current, it has minimal impact on the subthreshold swing Additionally, at off-state gate voltages, the source-side tunneling leakage can be effectively managed regardless of the drain concentration However, a higher drain concentration of 5×10^19 cm^-3 leads to a significant ambipolar current in long-channel TFETs.

The impact of drain concentration on the current-voltage characteristics becomes increasingly significant as channel lengths decrease in short-channel Tunnel Field-Effect Transistors (TFETs) Proper adjustment of the drain concentration is essential to manage both the drain-channel and source-channel tunnel barriers effectively While a heavier drain can induce off-state ambipolar conduction akin to long-channel graded TFETs, it also leads to combined subthreshold and ambipolar conduction The sensitivity of tunnel barriers to drain concentration plays a crucial role in the on-off transition; a heavier drain can create a large voltage drop that narrows the tunnel barrier, resulting in pronounced short-channel effects and increased ambipolar current Conversely, using a lighter drain enhances the tunnel barrier, allowing it to extend into the drain region and form a forbidden barrier For optimal switching characteristics in very short graded TFETs, a mildly doped drain concentration of approximately 5×10^18 to 1×10^19 cm^-3 is essential.

Figure 5.21 illustrates the relationship between drain current and varying drain lengths in long- and short-channel graded TFETs With a drain concentration of 5×10^18 cm^-3, shorter drains exhibit minimal impact on both the subthreshold slope and drain current in Si/Ge TFETs The primary drawback of shorter drains is a slight increase in ambipolar off-current due to the short-channel effect Additionally, the voltage drop in the drain region remains largely unaffected by drain length, allowing for flexible management of drain lengths in short-channel graded TFETs.

Fig 5.1: Schematic view of a p-i-n TFET used in numerical investigation A silicon-on-insulator structure was employed with a 20 nm Si body and 3 nm HfO2 gate insulator

Fig 5.2: Current-voltage curves of scaled TFETs with two drain concentrations of (a) 5×

Fig 5.3: (a) Lateral energy-band diagrams and (b) electric field distribution of 20 nm Si TFETs at off-state condition with two drain concentrations

Subt hr es ho ld Sw ing ( m V /D ec a de )

Figure 5.4 illustrates the current-voltage characteristics of 5 nm tunnel field-effect transistors (TFETs) with varying drain concentrations, as shown in panel (a) Panel (b) depicts the minimum subthreshold swing in relation to channel length for these different drain concentrations Additionally, the inset in panel (b) presents the subthreshold energy-band diagrams for 5 nm and 10 nm TFETs, specifically at a drain concentration of 5×10^17 cm^-3.

Fig 5.5: Current-voltage curves of 5 nm TFETs with various drain lengths

Fig 5.6: (a) Off-state energy-band diagrams and (b) electric field distribution of 5 nm TFETs with two drain lengths of 10 nm and 20 nm

Subt hr es ho ld Sw ing ( m V /D ec )

M ini m um D ra in L eng th ( nm )

Fig 5.7: (a) Minimum subthreshold swing versus drain length with various TFET lengths (b) Design criteria of minimum drain lengths in scaled TFETs with acceptable short-channel effects p ++ n - n

Fig 5.8: Schematic structure of a Si1-x Ge x heterojunction TFET

: Si TFET : Si 0.5 Ge 0.5 TFET

Fig 5.9: (a) Current-voltage curves of 5 nm Si1-x Ge x heterojunction TFETs (b) On-state energy-band diagrams of 5 nm Si and Si0.5Ge0.5 heterojunction TFETs

Fig 5.10: Current-voltage curves of Si0.5Ge0.5 TFETs with various (a) drain concentrations and (b) drain lengths

Fig 5.11: Schematic structures of p-body graded Si/Ge heterojunction TFET and counterpart uniform Ge and abrupt Si/SiGe heterojunction TFETs

: Uniform Ge TFET : Graded Si/Ge TFET

: Abrupt Si/SiGe TFET : Graded Si/Ge TFET

Figure 5.12 illustrates the energy-band diagrams of 8 nm graded Si/Ge tunnel field-effect transistors (TFETs) along the channel surface, contrasting them with uniform germanium (Ge) and abrupt silicon/silicon-germanium (Si/SiGe) TFETs in both the off-state and on-state conditions.

Uniform Ge TFET Abrupt Si/SiGe TFET

Fig 5.13: (a) Tunnel barrier width and (b) barrier height as functions of gate voltage in graded Si/Ge, uniform Ge and abrupt Si/SiGe heterojunction TFETs

Fig 5.14: Images of tunneling generation at subthreshold- and on-states in 50 nm and 8 nm graded Si/Ge TFET devices

Fig 5.15: Current-voltage curves of (a) uniform Ge and (b) abrupt Si/SiGe heterojunction TFET devices with various channel lengths

Subt hr es ho ld Sw ing ( m V /D ec a de )

Fig 5.16: (a) Current-voltage curves of graded Si/Ge heterojunction TFET with various channel lengths, and (b) subthreshold swing of TFET devices against channel length

Graded Si/Si 1-x Ge x TFETs

Graded Si/Si 1-x Ge x TFETs

The current-voltage characteristics of 10 nm graded Si/Si1-xGex tunnel field-effect transistors (TFETs) are analyzed, highlighting the impact of varying germanium mole fractions (x) at the source Additionally, the off-state energy-band diagrams for short-channel graded Si/Si1-xGex TFETs with germanium concentrations of x = 0.5 and x = 0.8 are presented, illustrating the differences in electronic properties based on composition.

Subt hr es ho ld Sw ing ( m V /de c)

Graded Si/Si 1-x Ge x TFETs

Fig 5.18: (a) Subthreshold swing, and (b) subthreshold energy-band diagrams, with various x values of graded Si/ Si1-x Ge x TFETs

Graded Si/Si 1-x Ge x TFETs

Fig 5.19: On-state current of long- and short-channel graded Si/Si1-x Ge x TFETs versus Ge concentration x

Fig 5.20: Drain currents of (a) long-channel and (b) short-channel graded Si/Ge TFETs with various drain concentration

Fig 5.21: Current-voltage curves of (a) long-channel, and (b) short-channel, graded Si/Ge TFETs with various drain lengths.

Modeling and Design of Low-Bandgap Line-Tunneling TFETs