Drain Current Model of One-Dimensional Ballistic Reconfigurable Transistors

TL;DR

This paper presents an analytical drain current model for one-dimensional ballistic RFETs with Schottky contacts, incorporating band-to-band tunneling and comparing multi-gate configurations.

Contribution

It introduces a simplified analytical model based on WKB approximation for RFETs, including tunneling effects, and compares two-gate and triple-gate device configurations.

Findings

Analytical solution for the Landauer integral derived.

Model accurately accounts for electron and hole tunneling.

Comparison shows differences between two-gate and triple-gate RFETs.

Abstract

A simple model based on the WKB approximation for one-dimensional ballistic multi--gate reconfigurable field--effect transistors (RFETs) with Schottky-Barrier contacts has been developed for the drain current taking into account electron and hole band-to-band tunneling. By using a proper approximation of both the Fermi-Dirac distribution function and transmission probability, an analytical solution for the Landauer integral can be obtained. A comparative analysis of the two-gate and triple-gate RFETs is performed based on the numerical integration of the current integral.

Equations

$$E_{\mathrm{C}}^{\mathrm{s}}(z)=E_{m,0}-q{\psi }_{cc,1}+E_b^s\left(1-\frac{z}{{\lambda }_s}\right),$$

$$E_{\mathrm{C}}^{\mathrm{d}}(z)=E_{m,0}-q{\psi }_{cc,N}+E_b^d\left[1+\frac{z-L}{{\lambda }_d}\right],$$

$$\begin{gathered} E_{\mathrm{C},\mathrm{n}}^{\mathrm{in}}(z)=E_{m,0}-q{\psi }_{cc,n-1} \\ +q({\psi }_{cc,n}-{\psi }_{cc,n-1})\left(\frac{z_{R,n-1}-z}{z_{L,n}-z_{R,n-1}}\right), \end{gathered}$$

$${f_{\rm{app}}(E)=\left\{\begin{array}[]{l}1-\frac{1}{2}\exp\left(\frac{E-E_{F}}{{c_{1}k_{B}T}}\right),E\leq E_{F}\\ \frac{1}{2}\exp\left(\frac{E_{F}-E}{{c_{1}k_{B}T}}\right),~{}E_{F}<E<E_{F}+c_{2}k_{B}T\\ \exp\left(\frac{E_{F}-E}{{k_{B}T}}\right),~{}E\geq E_{F}+c_{2}k_{B}T\end{array}\right.\ }$$

$$T_b^{s(d)}(E)=\exp \left\{-\alpha \sqrt{|E_b^{s(d)}|}\gamma \left(E/|E_b^{s(d)}|\right)\right\},$$

$$\gamma (x)={\left(1-x\right)}^{3/2},$$

$${\gamma }_{\mathrm{app}}(x)=(1-x)(1-px),$$

$$p=\phi -\sqrt{{\phi }^2-\phi }\approx 0.618,$$

$$T_{\mathrm{BTB}}^{s(d)}=\exp \left\{-\alpha E_g^{3/2}/E_b^{s(d)}\right\}.$$

$$T_b^{s(d)}(E)=\exp \left\{-\beta \zeta \left(\frac{E+E_{m,0}-|E_b^{s(d)}|}{E_{m,0}}\right)/|E_b^{s(d)}|\right\},$$

$$\zeta (x)=\pi /2-x\sqrt{1-x^2}-\arcsin \left(x\right),$$

$$T_{\text{BTB}}^{s(d)}=\exp \left\{-\frac{3\pi }{16}\alpha E_g^{3/2}/E_b^{s(d)}\right\}.$$

$$T_{tun}(E)=T^s(E)T_{BTB}^{in}{T}^d(E).$$

$$I=\frac{4q}{h}\underset{-\infty }{\overset{\infty }{\int }}{T}_{tun}(E)\left[f_{FD}(E-E_{Fs})-f_{FD}(E-E_{Fd})\right]dE,$$

Full text

I Introduction

Some of the recent requirements for CMOS technology listed in the International Roadmap for Devices and Systems (IRDS) [1] include high–mobility channel materials, gate–all–around (nanowire) structures, scaling down supply voltages lower than 0.6 V, controlling source/drain series resistance within tolerable limits, providing lower Schottky–barrier (SB) height, and fabrication of advanced nonplanar multi–gate and nanowire MOSFETs. Along with FETs based on semiconductor nanowires, carbon-nanotube FETs (CNTFETs) satisfy these requirements [2, 3, 4]. Downscaling the transistor dimensions goes along with a transformation of ohmic contacts into Schottky contacts [5, 6]. Due to a possible low channel resistance (or even ballistic conduction), the metal-semiconductor contact resistance can significantly affect or even dominate the performance of SB transistors [7, 8, 9]. In contrast to conventional FETs, multi-gate reconfigurable field–effect transistors (RFET) can be configured between an n– and p–type by applying an electrical signal, which selectively controls charge carrier injections at each Schottky contact, explicitly avoiding the material doping [10, 11]. RFETs have the potential to enable adaptive and reconfigurable electronics, which can lead to the initiation of radically new circuit paradigms and computing schemes based on the reprogrammable logic with the reduced number of required devices. Along with the electron tunneling through SB, the band-to-band tunneling (BTBT) of electrons has significant effect on RFET characteristics. This leads to an increase of current and decrease of a subthreshold swing, which can be less than its limit value of 60 mV/dec typical for MOSFETs at room temperature [12]. For both tunnel- and multi-gate RFETs, it has been experimentally demonstrated, that the subthreshold limit value can be decreased down to 30 and 40 mV/dec, respectively [13, 14, 15, 16].

For circuit design, the description of the device behavior based on the nonequilibrium Green’s function (NEGF) method, Wigner transport equation, and Boltzmann equation formalism is unsuitable in terms of memory and time [2, 17, 18, 19]. To reduce the computation time, TCAD simulation tools have been used to analyze the ${I-V}$ characteristics of RFETs with SB contacts solving the current integral involved in the transport calculations numerically  [20, 21]. For practical circuit design based on simulations in a SPICE-like environment, compact models are required. In the framework of the constant effective SB approximation using an energy independent transmission probability, different simple analytical expressions for the drain current have been reported in the literature for RFETs [21, 14, 22, 23]. In these models, the simulated ${I-V}$ characteristics agree with experimental data in a limited bias range [19]. The reason is that the analytical expression for the drain current corresponding to the thermionic emission with a shifted Fermi level and including energy-independent transmission can be used at small bias, when the contribution of thermally excited electrons in the total current is large enough [24]. The analytical current calculations on the basis of drift-diffusion model do not properly take into account the effect of SB tunneling and BTBT on the electron transport [25, 26]. The empirical continuous compact dc model based on a set of empirical fitting parameters is reliable in the framework of experimental data [27], but it cannot be used for predictions.

In this paper, we demonstrate the drain current model, which allows to simplify solving of the current integral. It potentially enables to simulate ${I-V}$ characteristics of one-dimensional reconfigurable multil–gate transistors with SB contacts with reduced computation time. We adopt the pseudo-bulk approximation [22] to self–consistently estimate the channel potential variation under applied bias with respect to channel charge. The drain–current model captures a number of features such as ballistic transport, transmission through the SB contacts, band-to-band tunneling and ambipolar conduction. It can be applied to quasi-1D RFETs based on both nanowires and nanotubes at large bias voltages.

II Transport Model

II-A Energy Band Model

We consider $N$ gates with left- and right-end coordinates ${\left[z_{L,n},z_{R,n}\right]}$ ($n=1,2,\dots,N$) placed along the channel. The given band model was adopted from the evanescent mode analysis approach [28, 29, 30, 26]. The electrostatic potential, ${\psi(r)}$, inside a transistor contains a transverse potential ${\psi_{t}(r)}$, which describes the electrostatics perpendicular to the channel and represents a partial solution of Poisson’s equation, as well as a longitudinal potential ${\psi_{l}(r)}$ called evanescent mode, responsible for the potential variation along the channel. The transverse potential inside the channel is reduced to ${\psi_{t}(r)\approx\psi_{cc}}$, where ${\psi_{cc}}$ is the channel (surface) potential at the current control point [31, 32]. The longitudinal solution ${\psi_{l}(r)}$ is obtained solving the Laplace equation along the transport direction. Therefore, near the source and drain contacts, the conduction subband edge is given by exponentially decaying functions. Since electrons with high energy mainly tunnel through the Schottky barrier, the conduction subband edge $E_{\rm{C}}^{\rm{s}}$ ($E_{\rm{C}}^{\rm{d}}$) in the vicinity of the source (drain) contact can be approximated by a linear decaying function

[TABLE]

where $L$ is the total length of the channel, $\lambda_{s(d)}$ is a characteristic length of the decaying electrostatic potential that can be interpreted as an effective SB width and ${E^{s(d)}_{b}=\phi_{b}+q\psi_{cc,1(N)}-E_{m,0}-qV_{s(d)}}$ is the bias dependent potential barrier height with respect to the bottom of the $m$th conduction subband ${E_{m,0}-q\psi_{cc,1(N)}}$ at the source and drain contacts, correspondingly. For cylindrical gate-all-around FETs, the asymptotic value of $\lambda$ is approximately given by ${(2\kappa t_{\rm{ox}}+d_{\rm{ch}})/4.81}$, where ${\kappa=\epsilon_{\rm{ch}}/\epsilon_{\rm{ox}}}$ can be obtained if the oxide thickness, $t_{\rm{ox}}$, is significantly smaller than the channel diameter, ${d_{\rm{ch}}}$ [28]. For gate-all-around CNTFETs, the CNT diameter, ${d_{\rm{CNT}}}$, is often smaller than the oxide thickness, therefore, the asymptotic value of ${\lambda}$ is slightly modified [33]. In the case of double-gate FETs, the similar approximation of the characteristic length reads ${\lambda\approx(2\kappa t_{\rm{ox}}+t_{\rm{ch}})/\pi}$, where ${t_{\rm{ch}}}$ is the thickness of the channel [28].

Between two adjacent gates with bias voltages ${V_{g,n-1}}$ and ${V_{g,n}}$, the electrostatic potential is supposed to be linearly dependent on space variable in the inner part of the channel. Hence, the conduction band edge ${E_{\rm{C,n}}^{\rm{in}}}$ in the nth adjacent interval ${(z_{R,n-1},z_{L,n})}$ is defined as

[TABLE]

where index $n=2,3,\dots,N$. In the case of mirror-symmetric band structure, the valence subband edge ${E_{\rm{V}}}$ is described as ${E_{\rm{V}}(z)=E_{\rm{C}}(z)-2E_{m,0}}$.

Fig. 1 shows the conduction band profile along the channel. The gate length ${L_{g}}$ of the device coincides with the channel length ${L}$. The metal-semiconductor barrier height referenced to source Fermi level $E_{Fs}$ is described by a bias independent parameter, ${\phi_{b}}$, which is commonly defined by the difference between the metal work function, $\phi_{M}$, and semiconductor electron affinity, $\chi_{SC}$, i.e., $\phi_{b}\approx\phi_{M}-\chi_{SC}$ [34, 35, 36]. For holes, the similar parameter ${\phi^{h}_{b}}$ is given by ${\phi^{h}_{b}=E_{g}-\phi_{b}}$, where ${E_{g}=2E_{m,0}}$ is the band gap. The source and drain Fermi levels $E_{Fs}$ and $E_{Fd}$, respectively, are related as $E_{Fd}=E_{Fs}-qV_{ds}$, where $V_{ds}=V_{d}-V_{s}$ is the drain–source voltage.

The contribution of electrons injected from the source and drain to the total current depends on both the energy dependent transmission through the channel and electron distribution in the contacts.

II-B Piece-Wise Approximation of Fermi-Dirac Distribution Function

The electron distribution in the source/drain contacts is given by the equilibrium Fermi-Dirac distribution function ${f_{FD}(E-E_{F})=1/\left\{\exp\left[\left(E-E_{F}\right)/{k_{B}T}\right]+1\right\}}$. To find an analytical expression for the current, we use a piece-wise approximation for $f_{FD}(E)$ given by [24]

[TABLE]

where ${c_{1}=2\ln(2)}$ and ${c_{2}=2\ln^{2}(2)/(2\ln(2)-1)\approx 2.49}$.

The approximation $f_{\rm{app}}(E)$ provides accurate values of the electron distribution function at different temperatures in the whole energy range with a maximum relative error of about 6-9 percent in the vicinity of Fermi level $E_{F}$.

II-C Transmission Probability

In order to estimate the transparency of the source/drain contacts, we use the transmission probability across each SB obtained in the framework of the Wentzel–Kramers–Brillouin (WKB) approximation. Using the effective mass (parabolic one–band) approach, the probability $T^{s(d)}_{b}(E)$ for electrons to tunnel through a linear decaying potential barrier of the kind ${E^{s}_{b}\left(1-z/\lambda\right)}$ or ${E^{d}_{b}\left[1+(z-L)/\lambda\right]}$ is given by the following expression [34]

[TABLE]

where ${\alpha=4\lambda_{s(d)}\sqrt{2m^{*}}/(3\hbar)}$. For CNTs, the electron effective mass is ${m^{*}=4E_{m,0}\hbar^{2}/(3a^{2}V_{\pi}^{2})}$ with $a=$2.49\text{,}\mathrm{\SIUnitSymbolAngstrom} - carbon–carbon atom distance and $V_{\pi}=$3.033\text{\,}\mathrm{eV} – carbon $\pi-\pi$ bond energy in the tight binding model [37].

To obtain an analytical expression for the current, we use the following approximation for $\gamma(x)$ in (5)

[TABLE]

where the quantity $\varphi=(1+\sqrt{5})/2$ represents the golden ratio and $x=E/E^{s(d)}_{b}$ is a dimensionless variable. The absolute error of $\gamma_{\rm{app}}(x)$ is less than 0.016 for all ${x\in[0,1]}$. Nevertheless, the implementation of $\gamma_{\rm{app}}(x)$ in (5) leads to an increase of relative error of the approximate transmission probability $T^{s(d)}_{\rm{app}}(E)$ with gate voltage due to term ${E^{s(d)}_{b}}$. To reduce the relative error, we introduce a correction factor ${\exp\left[\alpha\Delta(E^{s(d)}_{b})^{1/2}\right]}$ with the constant $\Delta<\text{max}\left|\gamma(x)-\gamma_{\rm{app}}(x)\right|$ in the final expression of current. The approximate transmission probability $T^{s}_{\rm{app}}(E)$ based on (5) and (7) is used in region 2 if there is only one potential barrier.

If ${E^{s(d)}_{b}>E_{g}}$, electrons can tunnel through the band gap from valence band to conduction band and vice versa. The probability of such band-to-band (BTB) tunneling for electrons and holes with equal masses is given in the parabolic one-band approximation by

[TABLE]

In the non-parabolic two-band approximation, the energy dispersion for electrons and holes in CNT is ${E_{m,l}=\pm\sqrt{E_{m,0}^{2}+(\hbar v_{F}k_{l})^{2}}}$, where ${v_{F}\approx 10^{8}}$ cm/s is the Fermi velocity. The electron effective mass $m^{*}$ and ${v_{F}}$ are related by ${m^{*}=E_{m,0}/v^{2}_{F}}$. In this case, the probability $T^{s(d)}_{b}(E)$ for electrons to tunnel through a linear decaying potential barrier is obtained in the WKB approximation as

[TABLE]

where ${\beta=\lambda_{s(d)}E^{2}_{m,0}/(\hbar v_{F})}$ and $x=E/E^{s(d)}_{b}$ is a dimensionless variable. In the non-parabolic two-band approximation, the probability of BTB tunneling reads[38]

[TABLE]

A comparison of (9) and (12) shows that the probability of BTB tunneling obtained in the non–parabolic two–band approximation is greater than that obtained in the parabolic one–band approximation. In the inner part of the channel, BTB tunneling ${T^{\rm{in},n}_{\rm{BTB}}}$ of electrons between two adjacent gates is given by (9) or (12), where ${E^{s(d)}_{b}}$ is replaced by the difference ${q|\psi_{cc,n}-\psi_{cc,n-1}|}$ and the characteristic length $\lambda_{s(d)}$ is replaced by the distance ${z_{L,n}-z_{R,n-1}}$ between two adjacent gates $n-1$ and $n$ ($n=2,3,\dots,N$).

If ${q\psi_{cc,1(N)}=E_{m,0}-\phi_{b}+V_{s(d)}}$, there is no SB located at the source (drain), then the transmission probability of electrons or holes to inject from the source (drain) into the channel is equal to 0 if the electron energy belongs to the band gap (${\phi_{b}-V_{s(d)}-2E_{m,0}<E<\phi_{b}-V_{s(d)}}$) and it is 1 otherwise.

The probability of electron transmission through the potential barrier increases with energy. At a large gate voltage, electrons with high energy or close to the Fermi level tunnel through the thin potential barrier with a rather small reflection probability ${1-T^{s(d)}_{b}(E)}$ and mainly contribute to the current, whereas the contribution of electrons with low energy is not essential due to a small transmission probability ${T^{s(d)}_{b}(E)}$. Hence, the multiple reflections between two potential barriers can be neglected. In this case, the total transmission probability reads

[TABLE]

The approximate total transmission probability ${T^{tun}_{\rm{app}}(E)}$ can be obtained by using (5)–(13).

The presented approach is valid if electron-phonon scattering is relatively small, i.e., the channel length $L$ is of the order of an electron mean free path $L_{\rm{mfp}}$, such that ${L/L_{\rm{mfp}}<1/\overline{T}_{b}}$, where ${\overline{T}_{b}}$ is an average value of the SB transmission probability characterizing a source/drain contact transparency [39]. Depending on the applied bias, the mean free path $L_{\rm{mfp}}$ can vary from 60 to 200 nm [40, 41, 42, 43, 44] at room temperature in CNTFETs. Also, the model does not take into account direct source-to-drain tunneling and short–channel effects (e.g., SS degradation and Drain-Induced Barrier Lowering), which are determined purely by electrostatics and essentially affect the current at ${L\approx\lambda}$ [39].

II-D Total Current

To calculate the total electron current, we use the Landauer-Buttiker approximation for a one-dimensional system [45]

[TABLE]

where the product of the spin and electron subband degeneracies gives a factor of 4 in front of the integral (14) for CNTFETs.

III Results

Fig. 2 shows the total current $I$ calculated numerically in the framework of two-band approximation (10)-(12) and (14) as a function of tube potential ${\psi_{cc,1}}$ at different values of drain–source voltage ${V_{ds}}$ and similar values of the program gate for the two–gate RFET with equal gate lengths of 45 nm. The program gate voltage ${V_{pd}}$ is supposed to be equal to the corresponding tube potential ${\psi_{cc,2}}$.

At a larger drain–source voltage (${V_{ds}=2}$ V), the total current strongly depends on the gate voltage in the whole interval of ${\psi_{cc}}$, because the contribution of electrons injected from the drain into the channel is negligibly small due to the large reflection of such electrons from the potential barrier in the channel. The On/Off ratio is equal to $2.12\cdot 10^{2}$, ${1.08\cdot 10^{6}}$, and ${2.11\cdot 10^{2}}$ at the drain–source and program gate voltages ${V_{ds}=V_{pd}}$ equal to 0.1, 0.5, and 1 V, correspondingly. The subthreshold swing ${SS=\left({d\log_{10}I}/{dV_{gs}}\right)^{-1}}$ equals 63, 31, and 118 mV/dec, respectively. Therefore, it can be considerably less than 60 mV/dec at ${V_{ds}=V_{pd}=0.5}$ V, when the corresponding On/Off ratio is ${1.08\cdot 10^{6}}$.

Fig. 3 depicts the total current $I$ calculated numerically in the framework of two-band approximation (10)-(12) and (14) as a function of tube potential ${\psi_{cc,2}}$ at different values of drain–source voltage ${V_{ds}}$ and similar values of the program gates for the triple–gate RFET. The program gate voltages ${V_{ps}}$ and ${V_{pd}}$ are supposed to be equal to the values of corresponding tube potentials ${\psi_{cc,1}}$ and ${\psi_{cc,3}}$. The On/Off ratio is equal to $2.53\cdot 10^{2}$, ${7.17\cdot 10^{5}}$, and ${4.77\cdot 10^{3}}$ at the drain–source and program gate voltages ${V_{ds}=V_{pd}=V_{ps}}$ equal to 0.1, 0.5, and 1 V, correspondingly. The subthreshold swing ${SS}$ equals 62, 59, and 63 mV/dec, respectively. A comparison of transfer characteristics shown in Fig. 2 and Fig. 3 indicates that the contribution of holes to the current of the n–type triple-gate RFET is diminished by an order of magnitude compared to that of the n–type two-gate RFET, i.e.  the ambipolarity is greatly reduced. As a result, the On/Off ratio for the triple-gate RFET is greater by an order of magnitude in comparison to that for the two-gate CNTFET at large ${V_{ds}}$. But, the $SS$ is similar to the thermionic limit value of 60 mV/dec, which is about twice greater than ${SS=31}$ mV/dec for the double-gate RFET at ${V_{ds}=V_{pd}=0.5}$ V.

IV Conclusion

A simple model for ballistic one-dimensional multi-gate transistors with SB contacts taking into account band-to-band tunneling has been developed. The model allows to find an analytical solution of the current integral, therefore, it can significantly decrease the evaluation times and eases the implementation of the model in Verilog-A. We have introduced a piece-wise approximation for Fermi–Dirac distribution function and modified the transmission probability using simple elementary functions, which allow to simplify the current calculations. Our model can be used for the analysis of experimental data as well as for performance predictions for different SB heights, characteristic lengths, gate lengths, and either electron effective mass or band gap of channel material for quasi-1D multi–gate RFETs based on both semiconductor nanowires and nanotubes. A comparative analysis showed, that the ambipolarity in the triple-gate RFETs is strongly suppressed compared to that in the two-gate RFETs. In contrast, the subthreshold swing for two-gate RFETs can reach a minimum value of 31 mV/dec, which is about twice less than 60 mV/dec typical for the triple-gate RFETs.

The author would like to thank Prof. Michael Schröter and Dr. Martin Claus for valuable discussions.