Band-to-band tunnel field-effect-transistors (TFETs) are considered a possible replacement for the conventional metal-oxide-semiconductor field-effect transistors due to their ability to achieve subthreshold swing (SS) below 60 mV/decade. This letter reports a comprehensive study of the SS of TFETs by examining the effects of electrostatics and material parameters of TFETs on their SS through a physics based analytical model. Based on the analysis, an intrinsic SS degradation effect in TFETs is uncovered. Meanwhile, it is also shown that designing a strong onset condition, quantified by an introduced concept - “onset strength”, for TFETs can effectively overcome this degradation at the onset stage, and thereby achieve ultra-sharp switching characteristics. The uncovered physics provides theoretical support to recent experimental results, and forward looking insight into more advanced TFET design.

Band-to-band tunnel FET is a promising candidate for next generation low-power digital applications, due to its low OFF-current and small subthreshold-swing compared to conventional metal-oxide-semiconductor field-effect transistors (MOSFETs). However, TFETs suffer from low ON-current mainly because of the large band-to-band tunneling (BTBT) barrier, especially for large band gap semiconductors including silicon, the material of choice for mainstream semiconductor technology. To overcome this shortcoming and further improve the subthreshold characteristics, many efforts1–11 have been focused on proposing new structures/materials for TFETs, among which several2–4 exhibit near perfect (step-like) switching characteristics, i.e., ultra-small SS. However, the hidden physics and design rules leading to such ideal subthreshold characteristics are still not apparent. As a result, certain degree of ambiguity prevails over the choice of structures and materials for achieving small SS, such as, which material system among Si, Ge and InGaAs has the greatest potential to reach the smallest SS. This work is aimed at exploring and understanding the physics behind these issues, based on which both theoretical and experimental results could be well explained, and more importantly, a general design rule for TFETs with small SS can be formulated.

Within the Landauer's formalism, the current for one conduction mode in TFETs can be expressed as12 

\begin{equation}I_{ds} = \frac{{2q}}{h}\int_{E_{cc} }^{E_{vs} } {T(E)\left[ {f_s (E) - f_d (E)} \right]} dE\end{equation}
Ids=2qhEccEvsT(E)fs(E)fd(E)dE
(1)

where q is the electron charge, h is the Planck's constant, T(E) is the BTBT probability, and fs/d(E) is the source/drain Fermi-Dirac distribution function. Evs and Ecc (Fig. 1) are the valence band maxima in the source region and conduction band minima in the channel region, respectively. Fig. 1 shows the schematic illustration (top) of a double-gated n-TFET used as the sample device in this paper and the corresponding energy band diagram (bottom) along its channel surfaces. P-type source and n-type drain are highly doped, while the channel is left intrinsic. WT is the minimum tunnel width, Evj is the valence band maxima at the source/channel junction, ΔE is the allowed tunnel window contributing to transport, and Efs/d is the source/drain Fermi level. Assuming T(E) remains constant across ΔE, it can be taken out of the integral in Eq. (1). This assumption is valid for BTBT in the range of small ΔE (within a few kT),13 which is the regime of essence for the SS physics of TFETs, as will be pointed out in the subsequent sections. In circuit applications, the drain voltage of an n-TFET is usually at least 0.1 V larger than the source voltage.5 This way, fd(E) is small enough to be neglected compared to fs(E). Thus, the source to channel tunneling current can be formulated as

\begin{equation}I_{ds} = \frac{{2q}}{h}T(\Delta E)F_{Integral}\end{equation}
Ids=2qhT(ΔE)FIntegral
(2)

where

\begin{equation}F_{Integral} = kT\ln \left( {\frac{{1 + \exp \left[ {(E_{fs} - E_{vs} + \Delta E)/kT} \right]}}{{1 + \exp \left[ {(E_{fs} - E_{vs})/kT} \right]}}} \right)\end{equation}
FIntegral=kTln1+exp(EfsEvs+ΔE)/kT1+exp(EfsEvs)/kT
(3)

FIntegral is the integral of the Fermi function part in Eq. (1). Note that the Fermi-Dirac distribution of electrons (in the source region, which is the relevant region in this work) is truncated by the band gap of the source region at Evs.

To obtain the expression of T(ΔE), we need to solve the 2D Poisson's equation to obtain the channel potential. Simplifying 2D Poisson's equation into a 1D form as expressed in Eq. (4) by using Yan's parabolic approximation14 is an efficient approach in FET compact modeling.

\begin{equation}\frac{{d^2 \phi _s }}{{\partial x^2 }} + \frac{{V_{gs} - V_{fb} - \phi _s }}{{\lambda ^2 }} = \frac{\rho }{\varepsilon }\end{equation}
d2ϕsx2+VgsVfbϕsλ2=ρɛ
(4)

where ϕs is the surface potential; λ is the natural length; Vfb is the flat band voltage; ρ is the charge density; and ɛ is the material permittivity. Following this approach, Zhang15 et al. recently developed an analytical model that can accurately capture the electrostatic potential profile around the tunnel junction in double-gated TFETs. Employing their model, Evj can be expressed as:

\begin{equation}\begin{array}{l}E_{vj} = \sqrt {(E_{vs} - E_{fs} + \delta E)^2 + \gamma ^2 + 2\gamma (\Delta E_{sc} + E_{vs} - E_{fs})} \\[6pt]\quad \quad \;\, - (\Delta E_{sc} + E_{vs} - E_{fs} + \gamma) \\\end{array}\end{equation}
Evj=(EvsEfs+δE)2+γ2+2γ(ΔEsc+EvsEfs)(ΔEsc+EvsEfs+γ)
(5)

where ΔEsc = ΔE + δE + Eg, δE (fixed at 0.1 eV in this work) is the value of Ecc - Evs at Vgs = 0, and

\begin{equation*}\gamma = N_{a,eff} q^2 \lambda ^2 /\varepsilon _{ch},\,N_{a,eff} = N_a - \frac{{\varepsilon _{ox} (\Delta E_{sc} + E_{vs} - E_{fs})}}{{\pi q^2 T_{ox} T_{ch} }}\end{equation*}
γ=Na,effq2λ2/ɛch,Na,eff=Naɛox(ΔEsc+EvsEfs)πq2ToxTch

where Na is the source doping level; ɛch/ox and Tch/ox are the permittivity and thickness of channel material/gate dielectric, respectively.

The minimum tunnel width can be written as15 

\begin{eqnarray}W_T &=& L_1 + L_2 - \lambda \cosh ^{ - 1} \left( {\frac{{\Delta E + \delta E - E_{fs} + E_{v,W_T } }}{{E_{vs} - E_{fs} + \delta E}}} \right) \nonumber\\[-6pt]\\[-6pt]&& -\, \sqrt {2\varepsilon _{ch} (E_{vs} - E_{v,W_T } - E_g)/(q^2 N_{a,eff})} \nonumber\end{eqnarray}
WT=L1+L2λcosh1ΔE+δEEfs+Ev,WTEvsEfs+δE2ɛch(EvsEv,WTEg)/(q2Na,eff)
(6)

where |$\lambda = \sqrt {\varepsilon _{ch} T_{ox} T_{ch} /(2\varepsilon _{ox})}$|λ=ɛchToxTch/(2ɛox)

\begin{equation}L_1 = \sqrt {2\varepsilon _{ch} (E_{vs} - E_{vj})/(q^2 N_{a,eff})}\end{equation}
L1=2ɛch(EvsEvj)/(q2Na,eff)
(7)
\begin{equation}L_2 = \lambda \cosh ^{ - 1} \left( {\frac{{\Delta E_{sc} + E_{vs} - E_{fs} - (E_{vs} - E_{vj})}}{{E_{vs} - E_{fs} + \delta E}}} \right)\end{equation}
L2=λcosh1ΔEsc+EvsEfs(EvsEvj)EvsEfs+δE
(8)
\begin{equation}\begin{array}{l}E_{v,W_T } = \sqrt {(E_{vs} - E_{fs} + \delta E)^2 + \gamma ^2 + 2\gamma (\Delta E + \delta E + E_{vs} - E_{fs})} \\[6pt]\quad \quad \quad \, - (\Delta E + \delta E + E_{vs} - E_{fs} + \gamma) \\\end{array}\end{equation}
Ev,WT=(EvsEfs+δE)2+γ2+2γ(ΔE+δE+EvsEfs)(ΔE+δE+EvsEfs+γ)
(9)

Tunneling probability is then calculated using the Wentzel–Kramers–Brillouin (WKB) approximation and the two-band dispersion relation,16 

\begin{equation}T(\Delta E) \approx \exp \left( { - \frac{{\pi \sqrt {2m_T } E_g^{3/2} }}{{4q\hbar \bar F}}} \right)\end{equation}
T(ΔE)expπ2mTEg3/24qF¯
(10)

where mT is the tunnel effective mass, and |$\bar F$|F¯ is the junction electric field modeled as |$\bar F = E_g /W_T$|F¯=Eg/WT.

According to the definition, SS equals the gate voltage swing needed to change the drain current by one decade. Hence, combining Eq. (10) with Eq. (2), SS can be expressed as

\begin{equation}SS = \left( {\frac{{d\log _{10} (I_{ds})}}{{d(\Delta E/q)}}} \right)^{ - 1} = \frac{{SS_T SS_{FD} }}{{SS_T + SS_{FD} }}\end{equation}
SS=dlog10(Ids)d(ΔE/q)1=SSTSSFDSST+SSFD
(11)

where |$SS_T = \left( {\frac{{d\log _{10} \left( {T(\Delta E)} \right)}}{{d(\Delta E/q)}}} \right)^{ - 1},SS_{FD} = \left( {\frac{{d\log _{10} \left( {F_{{\mathop{\rm int}} egral} } \right)}}{{d(\Delta E/q)}}} \right)^{ - 1}$|SST=dlog10T(ΔE)d(ΔE/q)1,SSFD=dlog10F int egrald(ΔE/q)1

Eq. (11) clearly shows the manner in which the tunneling probability variation and the truncated Fermi-Dirac distribution contribute to the total SS through SST and SSFD, respectively. SSFD can be easily expressed as,

\begin{equation}SS_{FD} = \frac{{\ln (10)kT}}{q}\left( {1 + \exp \left( {\frac{{E_{vs} - E_{fs} - \Delta E}}{{kT}}} \right)} \right)\ln \left( {\frac{{1 + \exp \left( {(E_{fs} - E_{vs} + \Delta E)/kT} \right)}}{{1 + \exp \left( {(E_{fs} - E_{vs})/kT} \right)}}} \right)\end{equation}
SSFD=ln(10)kTq1+expEvsEfsΔEkTln1+exp(EfsEvs+ΔE)/kT1+exp(EfsEvs)/kT
(12)

The expression for SST can be written as,

\begin{equation}SS_T = \frac{{2\sqrt 2 \ln (10)\hbar }}{{ - q\pi \sqrt {m_T E_g } \frac{{dW_T }}{{d\Delta E}}}}\end{equation}
SST=22ln(10)qπmTEgdWTdΔE
(13)

where ħ is the reduced Planck's constant.

Before delving into the SS physics through the developed model, the transfer characteristics and SS behavior of a conventional Si TFET and a high-performance graphene-nanoribbon (GNR) based TFET are comparatively shown in Fig. 2. The purpose of showing this comparison is simply to point out the differences between the I-V curves of these two devices, and thereby serve as the starting point for the discussion on the SS behavior. The width and thickness of the GNR channel are 2.34 nm and 0.34 nm, respectively, while those of the Si channel are 1 μm and 10 nm, respectively. Other features of the two devices are kept identical. An in-house NEGF simulator17 and the Sentaurus device simulator18 are used to simulate the GNR and Si devices, respectively. Fig. 2 displays two main points that will be analyzed thereafter. Firstly, the SS of GNR TFET (red squares) approaches zero at some Vg, while that of the Si TFET (red triangles) does not. Secondly, the SS of GNR TFET increases much faster than that of Si TFET with Vg.

Fig. 3 shows the shapes of SS, SST and SSFD versus ΔE for a double-gated Si TFET, calculated with Eqs. (11)–(13). Relevant parameters are listed in the caption of Fig. 3. When Na is low enough so that Efs is higher than Evs, SSFD expressed by Eq. (12) reduces to ln(10)ΔE/q as pointed out in Ref. 13. However, a high Na required to achieve high ON-current usually pulls Efs well below Evs leading to a different SSFD behavior as displayed in Fig. 3. According to the shape of SSFD and its underlying physics, the ΔE range can be divided into three regions as notated by symbols I, II and III in Fig. 3. In region I (0<ΔE<2kT, the onset stage of TFETs), Eq. (12) reduces to

\begin{equation}SS_{FD,{\mathop{\rm Re}\nolimits} gion\,I} = \ln (10)\frac{{kT}}{q}\left( {1 - \exp \left( {\frac{{ - \Delta E}}{{kT}}} \right)} \right)\end{equation}
SSFD, Re gionI=ln(10)kTq1expΔEkT
(14)

SSFD exponentially approaches zero with decreasing ΔE, which stems from the truncation of the Fermi-Dirac distribution by the band gap of the source region (will be abbreviated as Fermi-Dirac truncation in following text). Starting from ΔE = 2kT to region II (2kT <ΔE<Evs-Efs) and then to region III (ΔE> Evs-Efs), SSFD initially stays at a constant value of 60 mV/dec, and then quickly increases, which mimics the thermionic emission dominated subthreshold behavior of a MOSFET. Note that Eq. (11) can't describe the total SS in region II and III as accurately as in region I (the onset stage), which is the focus of this work. Calculated results in region II and III only provide a qualitatively correct trend of SS, as confirmed by the extracted SS shape in the numerical simulation in Fig. 2.

Based on Eq. (14) and Eq. (11), all TFETs should have a near-zero SS within the onset stage due to their BTBT nature determined Fermi-Dirac truncation. The reason why a near-zero SS in simulated Si TFET does not appear is due to the fact that the OFF-current “immerses” the relatively low but quite sharply changing BTBT current within the onset stage, as schematically denoted (by red dotted curve) on the transfer characteristics curve for Si TFET in Fig. 2.

For a well designed/fabricated TFET, thermionic component of OFF-current is dominant and can be modeled as,

\begin{equation}I_{off} = \frac{{2q}}{h}\int_{E_{cs} }^{ + \infty } {f_s (E)} dE \approx \frac{{2qkT}}{h}\exp \left( { - \frac{{E_g + E_{vs} - E_{fs} }}{{kT}}} \right)\end{equation}
Ioff=2qhEcs+fs(E)dE2qkThexpEg+EvsEfskT
(15)

To bring out the near-zero-SS nature of TFETs, the BTBT current at the onset stage should be higher than the OFF-current. To quantify this effect, Ionset is defined as the BTBT current at ΔE = 0.5kT (∼13 meV). The concept of “onset strength” is proposed as the difference between Ionset and Ioff. It is well known that the source doping level Na, natural length λ, band gap Eg and tunnel effective mass mT are the four critical parameters for TFETs performance.19 The former two can be classified as electrostatic parameters, while the remaining two as material parameters. We make these four parameters as variables in our study to lump the effect of several promising technologies, such as low-dimensional materials/structures, strain technology, III-V compound, etc. Fig. 4 shows the dependence of Ionset and Ioff on these four parameters. As shown in Fig. 4(a), Ionset has a maximum value at a relatively high Na. The initial increase of Ionset with Na is due to the shrinkage of the tunnel width WT leading to higher tunneling probability. However, when Na increases too much, Efs moves far below Evs, leading to severely reduced occupation probability of states around the valence band edge of the source region and thus to the reduction of Ionset. Ioff keeps decreasing with Na because minority carrier density, which contributes to the OFF-current proportionally, decreases with Na. When any of λ, Eg or mT is reduced, Ionset is significantly enhanced as shown in Figs. 4(b)–4(d). The difference between the last three cases is that while Ioff remains constant with λ and mT variation, it keeps decreasing with increasing Eg (similar to Ionset). In other words, Na, λ and mT are the explicit parameters to enhance the “onset strength” and thus “pull” the steepest part of BTBT current above the OFF-current, while Eg is not explicit, since its effect on the “onset strength” depends on the rate of change of both the OFF-current and the BTBT current w.r.t Eg (Fig. 4(c)). There is, however, an exceptional condition in which Eg also becomes explicit. That is the hetero-junction TFET in which Eg, for Ioff calculation, is the band gap of the source material, while for Ionset it is the band gap overlap between the source and the channel material. For this case, the red dashed line for Ioff should be replaced by the blue horizontal dotted line (Fig. 4(c)).

It is worthwhile to note that the OFF-current of a not well designed/fabricated TFET should also include trap assisted tunnel leakage and gate leakage, which are not taken into account in Eq. (15). The trap assisted tunnel leakage may stem from interface states near the tunneling junction, defects at the tunneling junction (especially in hetero-junction TFETs), and high doping induced band edge states. These leakages will increase the OFF-current and somehow degrade the “onset strength” and prevent TFETs from exhibiting ultra-small SS. This issue needs further investigation but is beyond the scope of this article.

In the TFET community, there exists an ambiguity19 about the minimum SS (SSmin) achievable for certain structure or material system. So far, it is clear that SSmin appears at the current level of IBTBT = Ioff (Fig. 2). For devices with strong “onset strength”, SSmin is apparently near-zero. However, strong “onset strength” usually requires stringent fabrication condition and thus is not easy to achieve. In this situation, to clarify the ambiguity, we need to extend the discussion from the contribution of Fermi-Dirac truncation to that of tunneling probability variation, i.e., SST. SST very close to ΔE = 0 does not play an important role, since the near-zero SSFD there results in a near-zero SS, as reflected by Eq. (11). Therefore, a figure of merit−SST@2kT, which equals the value of SST at ΔE = 2kT is defined as an indicator of SST behavior. Fig. 5 shows the dependence of SST@2kT and SSmin on Na, λ, Eg and mT. When Na increases, or λ (or mT, or Eg) decreases, SST@2kT becomes larger. At first glance, these results seem unexpected, since the tunneling efficiency, which is a more familiar performance parameter for TFET, is expected to improve with higher Na, or smaller λ (or mTor Eg), i.e., “Ion improving condition”. Following analysis verifies that these results are physical. The two electrostatic parameters Na and λ, determine the depletion widths L1 (Eq. (7)) in the source region, and L2 (Eq. (8)) in the channel region, respectively. When Na becomes higher and/or λ becomes smaller, L1 and/or L2 shrink leading to the reduction of WT according to Eq. (6). With higher Na and/or smaller λ, WT is smaller as expected. The important point is that the rate of change of WT w.r.t ΔE also becomes smaller. This is because stronger electrostatic screening effect for devices with smaller WT leads to slower rate of change of WT., i.e., smaller dWT/dΔE. Therefore, SST becomes larger for higher Na or smaller λ according to Eq. (13). The manner in which the material parameters Eg and mT affect SST is also clear from Eq. (13), but is governed by the nature of tunneling rather than the junction electrostatics, since Eq. (13) is directly derived from WKB approximation based tunneling probability expressed in Eq. (10). From the above analysis of SST, it can be deduced that these four parameters degrade (through SST in Eq. (11)) the SS in a manner that can be regarded as intrinsic properties of TFETs, and hence, the corresponding degradation in SS can be termed as intrinsic SS degradation. This effect is usually overlooked in the TFET community, but it plays an important role in determining the achievable SSmin.

It can be observed from Fig. 5 that the effects of Na, λ and mT on SSmin are opposite of that on SST@2kT, while the effect of Eg displays the same behavior for both. The reason can be explained in a schematic manner as shown in Figs. 6(a) and 6(b). Compared to the reference condition (grey curves), both IBTBT and SS in “Ion improving condition” (black curves) increase over the entire range of ΔE, due to the reduced tunnel barrier and intrinsic SS degradation, respectively. As analyzed above, the levels of Ioff are critical in determining SSmin, since SSmin appears at IBTBT = Ioff. As shown in Fig. 4, Ioff decreases with Na, and remains nearly constant with λ (or mT), while increases with Eg. Therefore, IBTBT and Ioff meet at much smaller ΔE (labeled as ΔES) than the original ΔE that corresponds to the reference condition (labeled as ΔEO) for Na, λ and mT cases, while at relatively large ΔE (labeled as ΔEL) (may be larger than ΔEO) for the case of Eg. These trends lead to smaller SSmin for higher Na or smaller λ (or mT), while larger SSmin for smaller Eg, as shown in Fig. 6(b). In other words, the enhancement of “onset strength” with higher Na or smaller λ (or mT) overcomes the intrinsic SS degradation, leading to a reduction of SSmin, while that with smaller Eg is not strong enough to reach the same result. Several representative experimental results (Ioff and SSmin) obtained recently on TFETs, are summarized in Fig. 6(c). The Si SOI device is set as the reference device, and improvements on specific parameters are denoted by corresponding arrows. It can be observed that using small-Eg channel materials, such as Ge and III-V, does not improve SS, because increased Ioff degrades the “onset strength”, in agreement with the uncovered SS physics above. It is worthwhile to mention that SS of III-V devices may also suffer from interface traps, which is expected to improve with more advanced interface engineering in the near future. To increase ON-current while retaining small SS, hetero-junction TFET (with large Eg on source side for lowering leakage and small Eg or the band gap overlap of source and channel, for enhanced tunneling) could be a solution, which has been experimentally achieved to some extent in Ref. 6. Excellent electrostatics, arising from using nanowire structure, or using atomically-thin emerging 2D semiconducting crystals20–22 as channel materials, are also promising alternatives.

Thus far, the reasons for faster increase of SS of GNR TFET with ΔE compared to that for Si TFET, and the step-like transfer characteristics in many proposed TFETs2–4 can be well explained. Strong “onset strength” in these devices make their ultra-sharply changing BTBT current visible, i.e., larger than Ioff, at the onset stage, leading to ultra-small SSmin. At the same time, the intrinsic SS degradation of these devices is also strong, i.e., SS increases rapidly with gate bias (i.e., ΔE), leading to quick flattening of the BTBT current, and thus to step-like transfer characteristics. From the application point of view, TFETs with strong “onset strength” should only be used for ultra-low power application in which supply voltages are ultra-low. The reason is that larger supply voltages result in larger regions (flattened part of BTBT current) degraded by intrinsic SS degradation. Thus the average SS over specific Id or Vg (or supply voltage) interval will be increased, with reduced advantage of ultra-small SSmin at the onset stage.

In summary, essential physics of the SS of TFETs is studied in this paper, from the perspectives of Fermi-Dirac distribution and tunneling probability variation. The novelty of this work can be attributed to the much improved insights gained into the SS characteristics of TFETs, wherein it is shown, by examining the effects of the four critical parameters for TFET performance - the source doping level, natural length, band gap and tunnel effective mass, that the truncation of the Fermi-Dirac distribution, OFF-current and the uncovered intrinsic SS degradation compete with each other and determine the minimum achievable SS. Detailed analysis suggests that small natural length, suitably high source doping level, and small tunnel effective mass, are preferred for homo-junction TFET design. On the other hand, for the hetero-junction TFET design, in addition to these three parameters, small band gap only at the tunnel junction (keeping band gap relatively large in the remaining areas to suppress OFF current) is preferred. The uncovered SS physics in this work is consistent with recent representative experimental results and offers the nanoscale device community deeper insight into TFET design.

This work was supported by the National Science Foundation, Grant No. CCF-1162633.

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