1. Introduction

Size-shrinkage as a main trend of the electronics development has already brought not only cutoff frequency but also energy consumption increase. In addition, a current leakage of the fieldeffect transistor (FET) has also increased. The leakage current consists of a current from the drain to the source (Isd) due to overlapping of the p-n transition regions in the contacts and a tunneling current from the gate to the channel (Ig). Moreover, in the FET size-shrinkage, the I<sup>g</sup> part becomes more important; for example, at 130 nm technology, it takes less than 5% of whole leakage, at 90 nm, it takes 40%, and at 65 nm, it takes 90%, respectively [1]. To decrease the Isd, SOI substrates and vertical orientations of the FET are used. This effectively decreases the width of the cross section for the Isd. Note that this accompanies the two-dimensional character of the FET channel. To diminish the Ig, high-k dielectrics are used as the gate insulators. This increases the capacity between the gate and the channel and decreases the pinch-off voltage or energy of

© 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

the tunneling carriers. Another way is to use quantum well for the FET channel, for example, InSb layer [2]. This also diminishes gate voltage because the band inversion is not required. As a result, the cut-off frequency can be increased up to 300 GHz. One can see that two-dimensional systems of carriers (2DSC) are inherent to the modern nanoFET.

In this chapter, an application of the 2DSC in a FET gate is considered for further leakage reducing.

### 2. Resonant tunneling of carriers

Tunneling has been revealed by Esaki [3] and studied mainly in semiconductor diodes since 1958. Several years before, Shriffer had proposed size-quantization of the carriers in semiconductor films [4] that was observed by Tsui in InAs tunneling diode [5]. Then Esaki [6] and Kazarinov and Suris [7] proposed carriers resonant tunneling (CRT) in semiconductor heterostructures. In 1974, this effect was observed [6]. On the base CRT, a resonant-tunneling diode (RTD) [8] and resonant-tunneling transistor (RTT) [9] are realized as highest-frequency solid-state devices up to date. Carriers tunneling is well-known to play a negative role in modern c CMOS transistors made on the base technology of 45 nm or less. However, the instances of the RTD and RTT give us a hope that a proper application of the CRT can improve the situation in the FET. To clarify this, let us consider the CRT in detail.

Usually, the CRT is observed in a double-barrier heterostructure, the conduction band profile of which is shown in Figure 1. In a thin layer of a narrow band gap semiconductors, the localized states are forming and called subband states or levels. The ground subband state has energy Ez0. As a result, the barriers transparency has sharp peaks up to 1 in its energy dependency when the incident-electron energy along z direction E<sup>z</sup> approaches to Ez0 and, what is more important to this chapter, it decreases down to <sup>T</sup><sup>z</sup> <sup>¼</sup> <sup>10</sup>�<sup>4</sup> at intermediate energy [6].

To calculate current-voltage characteristics, one can consider model of sequential tunneling [10]. In this model, tunneling of the electron can be described as sequential quantum transition perturbed by tunnel Hamiltonian T [11]. In first term of the perturbation theory, one can expect to find the probability of the transition as follows:

$$\mathcal{W}\_{\vec{\eta}} = \frac{2\pi}{\hbar} |\Psi\_i| T |\Psi\_f|^2 \delta(E\_f - E\_i) \tag{1}$$

where ћ is Planck constant, Ψi,f are electron wave functions, Ei,f are energy of initial and final states. Since the potential is in the one z direction, the electron wave functions are as follows:

$$
\Psi\_{i,f} = \chi\_{i,f}(\mathbf{z}) \exp\left(\frac{i}{\hbar} \left[p\_{xi,f}\mathbf{x} + p\_{yi,f}y\right]\right) \tag{2}
$$

Then the matrix element of T is as follows:

$$
\Psi\_i|T|\Psi\_f = T\_{\sharp}\delta(p\_{x\sharp} - p\_{xi})\delta(p\_{y\sharp} - p\_{yi})\tag{3}
$$

Figure 1. Energy profile of the conduction band bottom of the two barrier heterostructure.

the tunneling carriers. Another way is to use quantum well for the FET channel, for example, InSb layer [2]. This also diminishes gate voltage because the band inversion is not required. As a result, the cut-off frequency can be increased up to 300 GHz. One can see that two-dimensional

In this chapter, an application of the 2DSC in a FET gate is considered for further leakage

Tunneling has been revealed by Esaki [3] and studied mainly in semiconductor diodes since 1958. Several years before, Shriffer had proposed size-quantization of the carriers in semiconductor films [4] that was observed by Tsui in InAs tunneling diode [5]. Then Esaki [6] and Kazarinov and Suris [7] proposed carriers resonant tunneling (CRT) in semiconductor heterostructures. In 1974, this effect was observed [6]. On the base CRT, a resonant-tunneling diode (RTD) [8] and resonant-tunneling transistor (RTT) [9] are realized as highest-frequency solid-state devices up to date. Carriers tunneling is well-known to play a negative role in modern c CMOS transistors made on the base technology of 45 nm or less. However, the instances of the RTD and RTT give us a hope that a proper application of the CRT can improve the situation in the FET. To clarify

Usually, the CRT is observed in a double-barrier heterostructure, the conduction band profile of which is shown in Figure 1. In a thin layer of a narrow band gap semiconductors, the localized states are forming and called subband states or levels. The ground subband state has energy Ez0. As a result, the barriers transparency has sharp peaks up to 1 in its energy dependency when the incident-electron energy along z direction E<sup>z</sup> approaches to Ez0 and, what is more important to

To calculate current-voltage characteristics, one can consider model of sequential tunneling [10]. In this model, tunneling of the electron can be described as sequential quantum transition perturbed by tunnel Hamiltonian T [11]. In first term of the perturbation theory,

<sup>ℏ</sup> <sup>j</sup>ΨijTjΨ<sup>f</sup> <sup>j</sup>

where ћ is Planck constant, Ψi,f are electron wave functions, Ei,f are energy of initial and final states. Since the potential is in the one z direction, the electron wave functions are as follows:

i

2

<sup>ℏ</sup> <sup>½</sup>pxi,f <sup>x</sup> <sup>þ</sup> pyi,f <sup>y</sup>� 

ΨijTjΨ<sup>f</sup> ¼ Tif δðpxf � pxiÞδðpyf � pyiÞ ð3Þ

δðEf � EiÞ ð1Þ

ð2Þ

this chapter, it decreases down to <sup>T</sup><sup>z</sup> <sup>¼</sup> <sup>10</sup>�<sup>4</sup> at intermediate energy [6].

one can expect to find the probability of the transition as follows:

Wif <sup>¼</sup> <sup>2</sup><sup>π</sup>

Ψi,f ¼ χi,fðzÞexp

systems of carriers (2DSC) are inherent to the modern nanoFET.

28 Different Types of Field-Effect Transistors - Theory and Applications

2. Resonant tunneling of carriers

this, let us consider the CRT in detail.

Then the matrix element of T is as follows:

reducing.

where Tif ¼ χijTjχ<sup>f</sup> . As one can see from Eqs. (1) and (3), the tunneling electrons save its energy and planar components of the momentum. Since the electron effective mass is equal on both sides of the barrier, the tunneling electron also saves E<sup>z</sup> energy. To calculate current, one should sum transition probabilities timed on electron charge from given equation:

$$I = \sum\_{i,f} \varepsilon \mathcal{W}\_{\circ}(f\_i - f\_f) \tag{4}$$

where fi,f, Fermi-Dirac distribution functions of electrons in initial and final states. Let us suppose that <sup>T</sup>if is a constant, then <sup>T</sup>if <sup>¼</sup> <sup>τ</sup>�<sup>1</sup> , where τ is a tunneling rate of the electrons. According to Eqs. (1) and (3), one can get the following:

$$I = \frac{e}{\pi} \sum\_{i \in \mathcal{E} \cup \{\mathcal{E}\_0\}} (f\_i - f\_f) = \frac{e}{\pi} (N\_i(\mathcal{E}\_{\mathfrak{z}0}) - N\_f(\mathcal{E}\_{\mathfrak{z}0})) \tag{5}$$

where i(Ez0) and f(Ez0) are the initial and final states which have the same energy Ez0 of motion in z direction; Ni(Ez0) and Nf(Ez0) are the number of electrons populating the initial and final states. Using low-temperature limit that is kT << E<sup>F</sup> and kT << Ez1�Ez0 and also supposing final states as empty that is Nf(Ez0) ¼ 0 as usual, one can calculate Ni(Ez0) as a number of filled states on a Fermi-hemisphere intersection disk taking at a momentum pz0 in the phase space (see gray disc in Figure 2) where Ez0 ¼ (pz0) 2 /2m\* and m\* is an electron effective mass. Thus, the tunnel current can be found as follows:

$$I = \frac{e(p\_F^2 - p\_{z0}^2)S}{(2\pi\tau\hbar^2)} = e\mathfrak{g}\_{2D}\mathcal{S}(E\_{Fe} - E\_{z0})/\tau\tag{6}$$

where e is an electron charge, p<sup>F</sup> is a Fermi momentum of electrons, g2D is a density of twodimensional states of electrons, and S is a sample area.

Let us suppose the emitter grounded, i. e., μfe ¼ const, then the voltage dependence of Ez0(V) determines the I-V curve. If the barriers width D is greater than the quantum well (QW) width d, then Ez0(V) can be found from linear Stark effect:

$$E\_{z0}(V) = E\_{z0}(0) - eV/\alpha \tag{7}$$

where α is a leverage factor, i.e., α ¼ D/(d<sup>1</sup> þ d2). Since usually Ez0(0) > μfe, there is a threshold voltage Vth higher than a resonant current I that has appeared when Ez0(Vth) ¼ μfe. Then combining Eqs. (6) and (7), one can get the following expression for the current I:

$$I = e^2 \mathcal{g}\_{2D} \mathcal{S} (V - V\_{th}) / \alpha \pi \tag{8}$$

Eq. (8) is justified when μfe > Ez0(V) > Ece. At the current peak voltage Vp, the subband energy Ez0 approaches to Ece, i.e., Ez0(Vp) ¼ Ece, and after that the resonant current drops down to zero.

As a result, the I-V curve of the RTD is shown in Figure 3 as solid line. It is worth noting that Eq. (8) describes only resonant part of the current. Nonresonant current usually is monotonic

Figure 2. Momentum space of the emitter and states available for resonant tunneling.

on a Fermi-hemisphere intersection disk taking at a momentum pz0 in the phase space (see

where e is an electron charge, p<sup>F</sup> is a Fermi momentum of electrons, g2D is a density of two-

Let us suppose the emitter grounded, i. e., μfe ¼ const, then the voltage dependence of Ez0(V) determines the I-V curve. If the barriers width D is greater than the quantum well (QW) width

where α is a leverage factor, i.e., α ¼ D/(d<sup>1</sup> þ d2). Since usually Ez0(0) > μfe, there is a threshold voltage Vth higher than a resonant current I that has appeared when Ez0(Vth) ¼ μfe. Then

Eq. (8) is justified when μfe > Ez0(V) > Ece. At the current peak voltage Vp, the subband energy Ez0 approaches to Ece, i.e., Ez0(Vp) ¼ Ece, and after that the resonant current drops down to

As a result, the I-V curve of the RTD is shown in Figure 3 as solid line. It is worth noting that Eq. (8) describes only resonant part of the current. Nonresonant current usually is monotonic

combining Eqs. (6) and (7), one can get the following expression for the current I:

I ¼ e 2

Figure 2. Momentum space of the emitter and states available for resonant tunneling.

/2m\* and m\* is an electron effective mass. Thus, the

<sup>ð</sup>2πτℏ<sup>2</sup><sup>Þ</sup> <sup>¼</sup> eg2DSðEFe � Ez0Þ=<sup>τ</sup> <sup>ð</sup>6<sup>Þ</sup>

Ez0ðVÞ ¼ Ez0ð0Þ � eV=α ð7Þ

g2DS Vð Þ � Vth =ατ ð8Þ

2

<sup>F</sup> � <sup>p</sup><sup>2</sup> <sup>z</sup>0ÞS

<sup>I</sup> <sup>¼</sup> <sup>e</sup>ðp<sup>2</sup>

dimensional states of electrons, and S is a sample area.

d, then Ez0(V) can be found from linear Stark effect:

gray disc in Figure 2) where Ez0 ¼ (pz0)

30 Different Types of Field-Effect Transistors - Theory and Applications

tunnel current can be found as follows:

zero.

Figure 3. Current-voltage characteristics of RTD with a single quantum well between barriers (solid line) and a double quantum wells (dashed line).

function of the voltage and includes scattering tunneling and tunneling across all barriers. This provides nonzero current at any nonzero voltage. Thus, one can see that two-dimensional state in the QW produces the resonant tunneling in a finite resonant voltage range from Vth to V<sup>p</sup> and depresses the resonant tunneling at other voltages. This resonant voltage range can be further shrunk if another QW will be used (see Figure 4). In this case, the resonant tunneling is possible only at resonant voltage Vp1 when E01(Vp1) ¼ E02(Vp1). This decreases significantly the width of the current peak in the I-V curve (see Figure 3 dashed line).

Thus, the application of 2DSCs could significantly decrease the carriers tunneling in a wide range of the applied voltage. This means there is a new way to decrease carriers tunneling between a

Figure 4. Energy profile of the conduction band bottom of the heterostructure with two quantum wells.

gate and a channel that is application 2DSCs in them. Semiconductor heterostructures with two 2DSCs separated by a tunnel barrier have been studied and demonstrated their properties [11].
