Auger generation as an intrinsic limit to tunneling field-effect transistor performance

Deceleration of Moore's law scaling of complementary metal-oxide-semiconductor (CMOS) field-effect transistors has prompted an urgent search for next-generation logic devices.¹ Of all exploratory devices under consideration, tunneling field-effect transistors (TFETs) offer the most promise due to their potential speed, energy efficiency, and compatibility with existing CMOS infrastructure.^2,3

The key advantage of TFETs over existing technologies is the possibility for significant energy reduction achieved through energy-efficient switching of band-to-band tunneling (BTBT) current. The switching characteristics of transistors are assessed by the subthreshold swing, $dVG/d log10(ID)$⁠, which gives the change in gate voltage to produce a 10× increase in drain current. The quantum-mechanical nature of BTBT allows significantly sharper switching for TFETs compared to conventional transistors. [The subthreshold swing of conventional transistors is governed by the Fermi-Dirac distribution of carriers above a potential barrier whose height is modulated by the gate voltage. The thermionic current resulting from carriers above this barrier produces the often-cited 60-mV/decade subthreshold swing at 300 K. TFETs, on the other hand, can overcome the 60-mV/decade thermal limit because they switch by modulating tunneling through a barrier, instead of over it.] Sharper switching of TFETs, corresponding to smaller subthreshold swings, enables lower voltage operation yielding substantial energy savings over conventional devices.

However, despite many years of considerable world-wide effort, TFETs with switching characteristics superior to MOSFETs (i.e., TFETs with subthreshold swings <60 mV/decade at 300 K) for currents greater than 10 nA/μm have not been demonstrated. (See Lu and Seabaugh's review⁴ for a compilation of the most recent TFET results.) Previous work focused on trap-related phenomena, such as Shockley–Read–Hall generation, degraded gate efficiency, and trap-assisted tunneling, to explain non-ideal device characteristics.^5–8 More recently, Teherani et al.⁹ suggest that intrinsic phenomena, such as Auger, phonon-assisted, and radiative generation, may also limit TFET performance and explain the 60-mV/decade room-temperature swings observed in several experimental papers.^10–12 This work expands on this idea and describes an intrinsic mechanism—Auger generation—that may limit the minimum subthreshold swing, off-current, and on/off ratio of experimental TFETs. Auger generation must be understood to improve TFET switching characteristics so that sizeable energy reduction can be attained.

Auger generation, also called impact ionization, occurs when a high-energy carrier collides with an electron in the valence band, resulting in the high-energy carrier losing much of its momentum and exciting the valence-band electron to the conduction band, creating an electron–hole pair. Auger generation is the reverse of Auger recombination, where the recombination of an electron–hole pair excites a third carrier to a high-kinetic-energy state. For both generation and recombination, the high-energy carrier can be either an electron or a hole (Figure 1).

Auger generation is thought to be significant at very high carrier densities or very large electric fields. While this observation is true for conventional semiconductors, Auger generation can also play a fundamental role in very small band-gap semiconductors, even at moderate carrier densities. As a practical example, consider Auger generation in long-wavelength HgCdTe infrared photodiodes (⁠$EG$ ∼ 0.1 eV). In these devices, it is well understood that, at room temperature, Auger generation dominates the reverse-bias dark-current of high-quality devices and limits the detector sensitivity.^13,14 The sizable Auger-induced leakage current in these devices supports the idea that Auger generation must be considered in small band-gap materials.

Figure 2 plots the reverse-bias dark-current-density for ∼70 photodiodes made from high-quality HgCdTe,¹⁵ InSb,¹⁶ and InGaAs (lattice-matched to InP).^16,17 Reverse-bias currents for experimental devices follow an exponential trend (labeled “Rule 07” after the original work) that increases as the band gap shrinks and the temperature increases. Tennant¹⁵ shows that a simple model of Auger generation [which depends on the probability of a high-energy carrier with enough energy to create an electron-hole pair] explains the exponential behavior and gives excellent agreement over 13 orders of magnitude of dark-current-density. The trend line in Figure 2 suggests a sizable leakage current for a room-temperature device with a small band gap.

The total current for a pn junction is the combination of diffusion current due to the decay of excess carriers in the quasi-neutral regions and depletion current due to generation and recombination in the depletion region. Generation and recombination mechanisms, such as Auger, will create depletion currents for non-zero bias. These same generation and recombination mechanisms also determine the minority carrier lifetime that influences the diffusion current. The generation and recombination rates, however, vary as a function of position due to different carrier concentrations and electric fields.

In a typical reverse-biased photodiode, Auger generation is most dominant in the quasi-neutral regions as compared to the depletion region because few carriers exist in the depletion region. Auger generation reduces the minority carrier lifetime and increases the reverse-bias diffusion current leading to the observed dependence plotted in Figure 2. While other generation and recombination processes are also exponentially dependent on the band gap and temperature, it is well established, through theory and experiment,¹⁴ that Auger generation is the dominant process in high-quality small-band-gap HgCdTe photodetectors.]

Though many TFETs are fabricated from conventional semiconductors with relatively large band gaps, the energy separation between the electron and hole eigenstates of the tunneling region becomes exceedingly small as the device is turned on. For example, the perpendicular n-TFET (Figure 3) is switched on by biasing the structure so that the hole eigenstate of the source aligns in energy with the electron eigenstate of the channel. Ideally, the device remains off and no current flows until the electron and hole eigenstates align in energy (Figure 4(c)). But right before the bands align, a small energy separation exists between the electron and hole eigenstates in the tunneling region. The eigenstate energy separation of the tunneling region resembles a small-band-gap semiconductor whose band gap changes with applied gate voltage.

Consequently, as will be shown, Auger generation, in addition to other generation and recombination processes, will occur at significantly higher rates than traditionally expected due to the small eigenstate energy separation of the tunneling region as compared to the large band gap of the bulk material.

To further underscore the importance of intrinsic generation and recombination in TFETs, consider a device just before ideal band-to-band tunneling occurs where the electron and hole states of the tunneling junction are separated by the thermal energy, $kT$ (e.g., Figure 4(b)). Tunneling occurs only when the states align in energy if there is some spatial overlap of the electron and hole states at the tunneling junction. [A zero spatial overlap signifies that the electrons and holes are completely disconnected from each other. Even with abrupt band-edges, the amplitudes of the electron and hole wavefunctions have an approximately exponential decay in the band gap of the material. Under the WKB framework, this decay is viewed as the exponentially decreasing tunneling probability with barrier thickness.] Given the spatial overlap of electrons and holes and the very small energy separation between the eigenstates, non-defect-mediated generation and recombination processes, often negligible in conventional devices, can couple the electron and hole states with significant transition rates. In this work, we address this problem by first establishing a framework for analyzing Auger generation, and then calculating the Auger rate as a function of the energy separation between the electron and hole eigenstates. The rate is then related to a minimum current, and its impact on subthreshold swing is shown to set intrinsic limits to TFET device performance.

We analyze the Auger transition rate for the perpendicular TFET structures depicted in Figure 3. The ideal turn-on of the device occurs when a sufficient gate bias is applied such that the electron eigenstate of the n-layer aligns in energy with the hole eigenstate of the p-layer (Figure 4(c)).

Two doping versions of the structure exist, the n-TFET and p-TFET (Figure 3). They are named in analogy with n- and p-MOSFETs since they follow the same biasing conventions.

Generally, the potential bias of the p+ source of the n-TFET sets the quasi-Fermi level for holes while the n+ drain sets the quasi-Fermi level for electrons. For an n-TFET, the drain's communication with electrons in the channel is unimpeded, whereas communication with the source is restricted by the tunneling barrier. The drain bias is typically much larger than the thermal voltage (⁠$VD≫kT/q$⁠), and electrons in the channel are depleted by the drain as fast as they are supplied (assuming ballistic transport). If the tunneling probability from the source is low, the electron occupancy of the channel will also be low. The strength of Auger generation in an n-TFET does not depend on the precise electron occupancy, only that the electron concentration is low enough to prevent efficient recombination of the generated carriers. A related situation occurs for p-TFETs, where the hole concentration in the channel is decreased by the negative drain bias.

To achieve reasonable tunneling current and gate control, the thickness of the lightly doped layer must be quite thin (∼10 nm), resulting in carrier-energy quantization. This quantization may or may not occur in the heavily doped layer, depending on its thickness. Previous work¹⁸ suggested beneficial device performance for 2D-to-2D tunneling as compared to 2D-to-3D tunneling. In our work, both electron and hole energies are considered to be quantized in their respective layers, a treatment that can be extended to devices with thick layers by considering carriers to be quantized but with a very small energy separation between eigenstates. This approach allows our methodology to be applied to a wider variety of device structures.

Auger transitions can occur among many different bands. The type of transition is denoted by a four-letter initialism that identifies the bands for the initial and final states of the two particles involved in the process. C, H, S, and L correspond to the Γ-Conduction, Heavy-hole, Split-off hole, and Light-hole bands. We focus on Auger generation for two cases: (1) the CHCC process, when a high-energy electron in the conduction band collides with an electron in the heavy-hole band, knocking it into the conduction band; and (2) the HCHH process, when an energetic hole collides with a valence electron, exciting it into the conduction band. These processes are schematically illustrated in Figure 1. In both processes, energy and momentum must be conserved. While we address CHCC and HCHH processes, other transitions, such as HCHS and HCHL, also occur. HCHS and HCHL processes occur at much higher rates than HCHH in most p-type bulk materials with HCHS dominating in materials with band gaps greater than or equal to the split-off energy (⁠$EG≥Δ$⁠).^19,20 We investigate HCHH in this work due to its similarity with CHCC and as a means to establish the theoretical framework. Additional work is needed to compare HCHH, HCHL, and HCHS rates as a function of eigenstate energy separation.

The relative rates of the different transitions depend on specific parameters, such as the split-off energy, effective-mass, and quantization energy of the different bands. Our work is restricted to transitions between the first heavy-hole subband and the first Γ subband to demonstrate the key dependencies of the Auger transition rate. Transitions not considered in this work increase the transition rate but do not significantly alter our overall conclusion—that Auger transitions play a key role in determining the subthreshold behavior of TFETs.

The dominance of CHCC or HCHH in a specific device depends on the relative electron and hole concentrations in the channel. CHCC generation requires a reasonable electron concentration because multiple electrons are involved in the transition; HCHH generation requires a reasonable hole concentration. Low electron concentration in the n-TFET channel (due to the drain bias, explained in Section III) causes HCHH to dominate, whereas low hole concentration in p-TFETs causes CHCC to dominate. We derive rates for the CHCC and HCHH processes, but typically only one is dominant in a specific TFET structure due to the doping profile and the applied bias.

The expression for the net Auger recombination rate per unit area (⁠$U$⁠) is derived using Fermi's Golden Rule to give²¹ 

where $R−G$ is the difference between recombination and generation, and the sum is over all initial and final states including spin. Here, $A$ is the in-plane area of the quantum well, $ℏ$ is the reduced Planck constant, $P$ is the occupation probability function, $M$ is the matrix element that couples the initial and final states, and $δ(E1+E2−E1′−E2′)$ ensures energy conservation. Components of the Auger rate are described in Sections IV A–IV C.

The driving force for net recombination is the occupation probability function (⁠$P$⁠) that gives the difference between the occupation of states required for the forward (recombination) and reverse (generation) processes. Following the diagram of Figure 1, $P$ is expressed by

where

$fc$ and $fv$ are the Fermi-Dirac distributions for a state to be occupied in the conduction and valence bands with quasi-Fermi levels $EFn$ and $EFp$⁠, respectively, and $kT$ is the thermal energy. Similarly, $f′$ is the probability for a state to be vacant. The approximation to $P$ in Eq. (2) assumes $fc′≈1$ and $fv≈1$⁠, appropriate for non-degenerate carrier concentrations (discussed further below Eq. (6)). Equations (2a) and (2b) can be manipulated into their typical forms

by employing Maxwell–Boltzmann statistics.²² Here, $n$ and $p$ are the 2D densities of electrons and holes, and $no$ and $po$ are their values in quasi-equilibrium. [Quasi-equilibrium means $VDS=$ 0, but $VGS$ can vary.] $Nc$ and $Nv$ are the 2D densities of states of the conduction band and valence band, and $Ec$ and $Ev$ are the subband ground-state energies for the conduction and valence bands. [Equations (4) and (5) are also valid for 3D densities with the appropriate redefinition of variables from 2D to 3D.] The bracketed factor represents the difference between recombination and generation. In equilibrium $n=no$⁠, $p=po$⁠, and the net rate is zero. An $np$-product greater than equilibrium results in net recombination, whereas an $np$-product less than equilibrium results in net generation. When considering the impact of drain and source potentials, it is useful to note that

For reverse bias, $np≪nopo$⁠, and Eq. (4) can be written as

The use of Eq. (6) makes the implicit assumptions that

• the carrier probability at the high-energy state $|2′〉$ is accurately modeled by non-degenerate statistics,

• the reverse bias is sufficient to make recombination negligible, and

• the joint probability of an electron in the valence band along with a vacant state in the conduction band is sufficiently high such that it does not limit the creation of an electron–hole pair.

Assumption (i) is reasonable in most cases because the high-energy state is typically far from the band-edge, making the use of Maxwell–Boltzmann statistics at $E2′$ appropriate.

Assumption (ii) is valid given the conditions explained in Section III.

Assumption (iii) is justified by revisiting the non-degenerate approximation in Eq. (2): the generation term of $PCHCC$ is given by $fc′(E1)fc′(E2)fv(E1′)fc(E2′)≈fc(E2′)$⁠. In the non-degenerate limit, $fc′≈1$ and $fv≈1$⁠, and the approximation results in little error. As the electron concentration increases towards the degenerate limit, $fc(E2′)$ grows exponentially while $fc′(E1)$ and $fc′(E2)$ decrease slightly. Even at the extreme when the Fermi level is located at $E1$ and $E2$ [which assumes that $E1=E2$⁠, shown to be the condition for the most probable transition in  Appendix C:], $fc′(E1)=fc′(E2)=12$⁠, leading to only a 4× error; however, at the same time, $fc(E2′)$ has grown exponentially by many orders of magnitude, dominating the characteristics for $P$⁠. It is therefore reasonable to neglect $fc′$ and $fv$ factors for most cases.

The square of the total matrix element for the Auger process is given by²³ 

where

[The physics community often uses CGS units such that the denominator of Eq. (8) is missing the $4π$-factor. Because our work uses SI units (with energy in eV), we include the $4π$-factor.] $Ψn(r)$ is the wavefunction for state $|n$⟩, $q$ is the elementary charge, and $ϵ$ is the permittivity of the semiconductor. (For the low-energy transitions analyzed in this work, the corresponding frequency is relatively low and use of the static permittivity is reasonable. See Refs. 24 and 25 for a more detailed discussion regarding the use of the static versus optical permittivity.) The bold notation indicates a vector quantity. The middle factor in Eq. (8) emphasizes the electron–electron Coulombic interaction between particles $1$ and $2$ that causes the Auger transition.

The total matrix element includes coupling due to colliding electrons with the same spins and opposite spins. $M12$ represents the transition from $|1, 2⟩→|1′,2′⟩$⁠, whereas $M21$ indicates $|1, 2⟩→|2′,1′⟩$⁠. When the particles have opposite spins, they are distinguishable, and no interference occurs. The contribution of opposite spin collisions to Eq. (7) is $|M12|2+|M21|2$⁠. When the electrons' spins are the same, the $M12$ and $M21$ processes interfere due to the exchange interaction resulting in the $|M12−M21|2$ term in Eq. (7). (See Refs. 23 and 26 for further explanation.)

The wavefunctions for the conduction and valence bands are given by Bloch functions with confinement along the $z$-direction (perpendicular to the quantum well)²⁶ 

where $k$ is the wavevector, $u(r)$ is the cell-periodic function with periodicity of the crystal, $ei(kxx+kyy)$ and $ψ(z)$ are the envelope wavefunctions along the in-plane and confined dimensions, and the summation is a Fourier series expansion of $u$ in terms of $G$⁠, the set of all reciprocal lattice vectors.

Following the procedure detailed in  Appendix A, the Auger matrix element for a quantum well is found to be

where

Here,

and $Ωcell$ is the volume of a unit cell. As shown in  Appendix C, the most probable transition occurs when $k1=k2$⁠, which gives $M12≈M21$⁠. Because $M12$ and $M21$ are nearly equal, the squared matrix element of Eq. (7) can be approximated by

Physically, Eq. (14) suggests that only collisions between carriers of opposite spins contribute significantly to Auger transitions.

The Auger matrix element depends on the overlap integrals of the cell-periodic function $u$ and the envelope function $ψ$⁠. The overlap integrals within the same band are taken to be unity,²¹ so that Eq. (11) can be simplified to

whereby the numbering convention of Figure 1 is followed, and states $|1〉$ and $|1′〉$ are defined to be in different bands.

The overlap of the periodic part of the Bloch functions $〈u1′|u1〉$ is not straightforward to calculate. Burt et al.²⁵ show that the use of the effective-mass sum-rule method, often used to calculate $〈u1′|u1〉$⁠, is flawed, and instead compute the cell-periodic Bloch function overlap between the Γ and heavy-hole bands using the 15-band $k⋅p$ and non-local pseudopotential methods. In a subsequent work,²⁷ the same authors calculate $〈u1′|u1〉$ for additional III–V and II–VI direct-gap semiconductors. Both studies show strong agreement between the $k⋅p$ and pseudopotential methods. Pseudopotential calculations²⁷ give similar results for III–V (InSb, GaSb, GaAs) and II–VI (CdTe, ZnSe) semiconductors, with the overlap for II–VIs approximately half that of III–Vs. For both sets of materials, the cell-periodic Bloch function overlap has a nearly linear dependence on the wavevector difference between the Γ and heavy-hole band (⁠$K=|k1−k1′|$⁠) and is given by

where $cu$ equals $2×10−17$ cm for III–V and $6×10−18$ cm for II–VI semiconductors. Note that Eq. (16) agrees with the work of Verhulst et al.,²⁸ which highlights the zero-coupling between the Γ and heavy-hole band at zone-center (i.e., when $K=0$⁠).

The calculation of $〈ψ1′|ψ1〉$ depends on the details of the structure. Specifically, the envelope wavefunction overlap is affected by

• the effective mass of electrons and holes in the quantization direction: As the effective mass decreases, the particle is less confined to its well and a large fraction of its wavefunction leaks into the band gap.

• the width of the quantum well: As the quantum well widens, a larger separation distance between electron and hole wavefunctions occurs when a perpendicular electric field is applied to the quantum well. The wavefunctions separate in real space and $〈ψ1′|ψ1〉$ decreases considerably. For the same reason, BTBT also decreases considerably as the distance between holes and electrons (i.e., the tunneling distance) increases.

• the doping profile in the tunneling region: Doping modifies the energy-band diagram and thus the wavefunctions. Arbitrary doping profiles require a self-consistent calculation of the coupled Poisson–Schrödinger equations to determine the wavefunctions and the overlap integral.

For a structure with low doping concentration, such as the bilayer TFET²⁹ shown in Figure 3, the energy-band diagram is well-modeled by a uniform electric field across the tunneling junction. The exact wavefunctions for a quantum well with a uniform electric field have been solved analytically and shown by Ref. 30 to be a linear combination of Airy functions. In Figure 5, we calculate the wavefunction overlap $〈ψ1′|ψ1〉$ for an InAs quantum well with a uniform electric field. $〈ψ1′|ψ1〉$ at eigenstate energy alignment (⁠$ΔE=0$⁠) varies from approximately 0.01 to 0.1 as the quantum-well width decreases from 20 to 10 nm.

To arrive at a simplified form of the Auger rate equation, the terms derived in Secs. IV A–IV C are substituted into Eq. (1) to give

Following the procedure in  Appendix B, the summation over all states is converted to an integral that evaluates to

where $G$ is the areal generation rate; $ΔE$ is the energy separation between the first electron and hole subbands; $μ=mc/mv$⁠, and $mc$ and $mv$ are the density-of-states masses for the Γ and heavy-hole bands. [When using energy units of $[eV]$⁠, the $q4$-factor in Eq. (18) is equal to $(1.602…×10−19)2 [e2 C2]$⁠, where $C$ is Coulombs and $e$ is the elementary unit of charge. Using this prescription, the $q2/ϵ$-factor of the Coulombic potential is expressed in units of $[e CF/cm]=[eV cm]$⁠.]

The exponential factor in the generation rate results from the Maxwell–Boltzmann probability of a carrier in the high-energy state $|2′〉$⁠. For example, in the case of CHCC generation, there is a high probability for an empty state in the conduction band at $|1〉$ and $|2〉$ and an electron in the valence band at $|1′〉$⁠, but a low probability of an electron at $|2′〉$ that ultimately limits the generation rate (see Figure 1). The minimum energy for $|2′〉$ that satisfies energy and momentum conservation is

Equation (19) is derived in  Appendix C. By substituting Eq. (19) into (6), the origin of the exponential dependence of the generation rate can be readily understood.

Because Auger generation occurs in the high-field region of a reverse-biased pn junction, nearly all the generated carriers are swept out to the contacts where they contribute to device current. With this assumption, the current-density due to generation can be expressed as

where the sign of $Jgen$ is defined to correspond with the sign of the drain current $ID$⁠.

If the material properties and structural details are known, the Auger generation current can be calculated using Eqs. (18) and (20). We define a generic material to provide an Auger generation current that is relevant to a wide variety of material and doping configurations. The generation current for a specific structure can be determined by multiplying the current for the generic material by relevant scaling factors.

Figure 6 shows a plot of the generation current-density as a function of $ΔE$ for different ratios of the electron and hole mass for CHCC and HCHH processes. The exponential factor of Eq. (18) dominates the current characteristics, giving rise to room-temperature subthreshold swings (⁠$dΔE/d log10(Jgen)$⁠) between 30 and 60 meV/decade, depending on the mass ratio. The change in the envelope wavefunction overlap with $ΔE$ (Figure 5) is significantly weaker than the exponential dependence of the Auger rate assuming unity overlap (Figure 6). Therefore, inclusion of a $ΔE$-dependent wavefunction overlap to Figure 6 will lower the calculated rate, but not significantly change the trends with $ΔE$⁠.

To convert from $ΔE$ to gate voltage, $VG$⁠, the gate efficiency of the device must be known. As described in Ref. 29, the incremental gate-voltage efficiency $(dΔE/d(qVG))$ is poor for materials with a small effective mass. Increasing $VG$ to decrease $ΔE$ (Figure 4(c)) increases the field across the junction leading to increased quantization that increases $ΔE$⁠. For example, the expected gate efficiency for a TFET with a 15-nm InAs body and 1-nm effective oxide thickness is ∼40% (Ref. 29). Such a low gate efficiency will further degrade the subthreshold swing in these devices.

Even though BTBT is a single-particle process while Auger requires two particles, BTBT and Auger transitions are not remarkably different phenomena. Both can be viewed as generation and recombination events, and Fermi's Golden Rule can be used to calculate the transition rate. Changing a device design to increase BTBT likely increases Auger transition rates. We believe that the relationship between Auger and BTBT is instructive for designing devices that minimize Auger and maximize BTBT.

Applying Fermi's Golden Rule to calculate the BTBT rate gives¹⁸ 

The occupation probability function for BTBT (⁠$P$⁠) is the difference between recombination (where an electron in the conduction band tunnels to the valence band) and generation (where an electron in the valence band tunnels to the conduction band). $P$ is given by

where the approximation assumes Maxwell–Boltzmann statistics. For BTBT in a sufficiently reverse-biased pn junction, $np≪nopo$ and $PBTBT≈−1$⁠. [$PBTBT$ is defined for net recombination so a negative value indicates net generation.]

The equations for the occupation probability function for BTBT (22) and Auger (4) are quite similar. The same driving force is seen in both functions, with the Auger function scaled by the probability of a carrier in the high-energy state $|2′⟩$⁠. This probability factor is crucial to the different dependencies of Auger and BTBT rates.

The matrix element for BTBT tunneling is given by

where $qϕ(z)$ is the potential energy in the depletion region of the pn junction, and $Ψ1$ and $Ψ1′$ are wavefunctions in the valence and conduction band. Following the derivation in  Appendix D, the matrix element can be rewritten as

where

Here, a constant electric field (⁠$F$⁠) across the tunneling junction is assumed, $δk⊥1−k⊥1′$ ensures perpendicular momentum conservation, $EG$ is the band gap including quantization energy, $mr$ is the reduced mass, and $zcv=u1′|id/dkz|u1$⁠. The expression for $zcv$ in Eq. (25) results from Kane's two-band $k⋅p$ calculations.³¹ 

The similarity between the matrix element for BTBT (24) and Auger (10) should be noted. Both require perpendicular momentum conservation and are proportional to the overlap of the envelope wavefunctions of the valence and conduction bands, $〈ψ1′|ψ1〉$⁠.

The BTBT rate is derived in  Appendix E, and the areal generation rate is found to be

Equation (26) looks quite different from typical expressions for BTBT because most of the details of the tunneling process are hidden in the $〈ψ1′|ψ1〉$ factor. Using this equation, the intrinsic on/off ratio between the BTBT and Auger rates at the point of ideal TFET turn-on (⁠$ΔE=0$⁠) is found to be

[$Nc$⁠, $Nv$⁠, $n$⁠, and $p$ are given by their 2D values.] This ratio is plotted as a function of the electric field in Figure 7. The ratio depends only on fundamental constants, material parameters, and the electric field and doping of the structure. It is technology-independent, without the effects of band tails, selection rules, defect states, and other non-idealities. While the turn-on of BTBT is often envisioned as a sharp jump from near zero current to its on-state value,¹⁸ Eq. (27) quantifies this jump in current and sets limitations to device performance.

For an ideal TFET, an infinitesimal change in gate voltage creates a huge change in drain current, which translates to a huge on/off ratio. Auger generation, however, increases the off-current, thereby reducing the on/off ratio. Equation (27) provides the maximum possible on/off ratio for an infinitesimal change in a TFET's gate voltage. Because we look only at Auger transitions between the first eigenstates of the heavy-hole and Γ band, the actual on/off ratio will be even smaller than the value calculated from Eq. (27). Overall, Auger generation prevents ideal TFET operation because it decreases the on/off ratio, complicating circuit design and increasing off-state power consumption, which reduce the energy-efficiency and limit the potential applications for TFETs.

The Arrhenius dependence of Auger-generation current on the eigenstate energy separation is a challenge for researchers seeking a steep switching transistor. The turn-on of BTBT is intrinsically linked to the turn-on of Auger generation, which (unfortunately) is thermally dependent. Efforts to improve the steepness of BTBT may be unproductive if Auger generation dominates the subthreshold characteristics of the device. Therefore, Auger generation (along with other non-ideal effects) must be modeled in the subthreshold region of TFET characteristics to capture its negative impact on device performance and resolve the discrepancy between simulation and experimental results.

Auger generation is difficult to mitigate given that it is an intrinsic phenomenon. Decreasing the wavefunction overlap to decrease Auger generation will also, undesirably, decrease the BTBT current. The best approach to decrease Auger generation is to decrease the carrier concentration of the heavily doped source; however, a reduction in the source doping will decrease the field across the junction, decreasing the tunneling rate. These difficulties require careful device design to minimize the impact of Auger generation.

One straightforward method to reduce Auger generation is to use a p-TFET device design with CHCC as the dominant Auger process. As shown in Figure 7, p-TFETs yield higher on/off ratios and lower Auger currents compared to n-TFETs because CHCC generation depends on the Γ-band mass to the third power (⁠$mc3$⁠), which is quite small for many materials.

Our work focuses on perpendicular TFETs (i.e., tunneling perpendicular to the gate), but point TFETs (with tunneling parallel to the gate) are also affected by Auger generation. The area over which Auger generation is prevalent is approximately equal to the tunneling area of the device. For a perpendicular TFET, the tunneling area is equal to the gate area. For a point TFET, the tunneling area is less easily defined but can be considered approximately equal to the width of the device multiplied by the thickness over which tunneling occurs. Our approach provides a rough estimate of the Auger current in point TFETs, but a more detailed analysis is needed to address the dependence of Auger generation on device dimensionality.

Auger generation presents a novel method of achieving sub-thermal subthreshold swings—the minimum room-temperature subthreshold swing for an Auger process is 30 mV/decade (assuming perfect gate efficiency), when the mass of one carrier is many times heavier than the other (i.e., either $μ$ or $μ−1→∞$⁠). For many materials, the hole mass is much heavier than the electron mass which leads to a large $μ−1$⁠. We propose a new device concept—the Auger FET—that uses the steep slope of Auger generation as a switching mechanism. Such a device may look similar to a perpendicular TFET; however, the structure would be optimized to increase Auger rather than BTBT, such as by increasing the doping concentration of the source. More analysis is needed to characterize the Auger FET's potential for achieving high currents.

This work was supported in part by the Center for Energy Efficient Electronics Science (NSF Award No. 0939514). The authors thank Mathieu Luisier, Roger Lake, and Rebecca Murray for their technical comments regarding the manuscript.

The final form of the Auger matrix element for a quantum well given in Eq. (10) can be derived from Eq. (8), which gives the standard expression for a matrix element resulting from a Coulombic potential between two particles. Before deriving the matrix element for a quantum well, it is helpful to first analyze the matrix element for a bulk crystal.

The conduction and valence band wavefunctions for a bulk three-dimensional crystal can be written as Bloch functions²⁶ 

Here, $Ω$ is the volume of crystal, $k$ is a three-dimensional wavevector, and other parameters have the same definitions as in Eq. (9). Substituting the wavefunctions for a bulk crystal into Eq. (8) gives

Making use of the substitution $r1=r12+r2$⁠, the integral can be rewritten as

Integrating over $r2$ and making use of the identity that

(where $δk$ is the Kronecker delta) yields

To evaluate the integral over $r12$⁠, the integral is transformed into spherical coordinates defined such that the vector $k1−k1′ +G1−G1′$ lies along $θ=0$⁠. This transformation allows the integral to be expressed as

where $ρ12=|r12|$⁠. Putting this all together gives

Given the unlikelihood of Umklapp processes and the requirement of momentum conversation,²⁶ terms in which $G1≠G1′$ and $G2≠G2′$ are neglected. This allows the matrix element to be written as

which can be simplified to

where