Many in the microelectronics field view tunneling field-effect transistors (TFETs) as society's best hope for achieving a >10× power reduction for electronic devices; however, despite a decade of considerable worldwide research, experimental TFET results have significantly underperformed simulations and conventional MOSFETs. To explain the discrepancy between TFET experiments and simulations, we investigate the parasitic leakage current due to Auger generation, an intrinsic mechanism that cannot be mitigated with improved material quality or better device processing. We expose the intrinsic link between the Auger and band-to-band tunneling rates, highlighting the difficulty of increasing one without the other. From this link, we show that Auger generation imposes a fundamental limit on ultimate TFET 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, , 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.
II. BACKGROUND ON AUGER GENERATION
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 ( ∼ 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, (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.
III. DEVICE STRUCTURE AND OPERATION
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 (), 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.
IV. THEORY OF AUGER GENERATION AND RECOMBINATION
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 ().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 () is derived using Fermi's Golden Rule to give21
where is the difference between recombination and generation, and the sum is over all initial and final states including spin. Here, is the in-plane area of the quantum well, is the reduced Planck constant, is the occupation probability function, is the matrix element that couples the initial and final states, and ensures energy conservation. Components of the Auger rate are described in Sections IV A–IV C.
A. Occupation probability function
The driving force for net recombination is the occupation probability function () that gives the difference between the occupation of states required for the forward (recombination) and reverse (generation) processes. Following the diagram of Figure 1, is expressed by
and are the Fermi-Dirac distributions for a state to be occupied in the conduction and valence bands with quasi-Fermi levels and , respectively, and is the thermal energy. Similarly, is the probability for a state to be vacant. The approximation to in Eq. (2) assumes and , appropriate for non-degenerate carrier concentrations (discussed further below Eq. (6)). Equations (2a) and (2b) can be manipulated into their typical forms
For reverse bias, , and Eq. (4) can be written as
the carrier probability at the high-energy state 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 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 is given by . In the non-degenerate limit, and , and the approximation results in little error. As the electron concentration increases towards the degenerate limit, grows exponentially while and decrease slightly. Even at the extreme when the Fermi level is located at and [which assumes that , shown to be the condition for the most probable transition in Appendix C:], , leading to only a 4× error; however, at the same time, has grown exponentially by many orders of magnitude, dominating the characteristics for . It is therefore reasonable to neglect and factors for most cases.
B. Matrix element and wavefunctions
The square of the total matrix element for the Auger process is given by23
[The physics community often uses CGS units such that the denominator of Eq. (8) is missing the -factor. Because our work uses SI units (with energy in eV), we include the -factor.] is the wavefunction for state ⟩, is the elementary charge, and 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 and that causes the Auger transition.
The total matrix element includes coupling due to colliding electrons with the same spins and opposite spins. represents the transition from , whereas indicates . When the particles have opposite spins, they are distinguishable, and no interference occurs. The contribution of opposite spin collisions to Eq. (7) is . When the electrons' spins are the same, the and processes interfere due to the exchange interaction resulting in the 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 -direction (perpendicular to the quantum well)26
Following the procedure detailed in Appendix A, the Auger matrix element for a quantum well is found to be
and is the volume of a unit cell. As shown in Appendix C, the most probable transition occurs when , which gives . Because and 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.
C. Wavefunction overlap
The Auger matrix element depends on the overlap integrals of the cell-periodic function and the envelope function . 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 and are defined to be in different bands.
The overlap of the periodic part of the Bloch functions is not straightforward to calculate. Burt et al.25 show that the use of the effective-mass sum-rule method, often used to calculate , is flawed, and instead compute the cell-periodic Bloch function overlap between the Γ and heavy-hole bands using the 15-band and non-local pseudopotential methods. In a subsequent work,27 the same authors calculate for additional III–V and II–VI direct-gap semiconductors. Both studies show strong agreement between the 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 () and is given by
where equals cm for III–V and 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 ).
The calculation of 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 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 for an InAs quantum well with a uniform electric field. at eigenstate energy alignment () varies from approximately 0.01 to 0.1 as the quantum-well width decreases from 20 to 10 nm.
D. Auger rate equation
Following the procedure in Appendix B, the summation over all states is converted to an integral that evaluates to
The exponential factor in the generation rate results from the Maxwell–Boltzmann probability of a carrier in the high-energy state . For example, in the case of CHCC generation, there is a high probability for an empty state in the conduction band at and and an electron in the valence band at , but a low probability of an electron at that ultimately limits the generation rate (see Figure 1). The minimum energy for that satisfies energy and momentum conservation is
E. Auger generation current
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 is defined to correspond with the sign of the drain current .
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 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 () between 30 and 60 meV/decade, depending on the mass ratio. The change in the envelope wavefunction overlap with (Figure 5) is significantly weaker than the exponential dependence of the Auger rate assuming unity overlap (Figure 6). Therefore, inclusion of a -dependent wavefunction overlap to Figure 6 will lower the calculated rate, but not significantly change the trends with .
To convert from to gate voltage, , the gate efficiency of the device must be known. As described in Ref. 29, the incremental gate-voltage efficiency is poor for materials with a small effective mass. Increasing to decrease (Figure 4(c)) increases the field across the junction leading to increased quantization that increases . 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.
V. COMPARISON OF AUGER AND BTBT PROCESSES
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 () 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). is given by
where the approximation assumes Maxwell–Boltzmann statistics. For BTBT in a sufficiently reverse-biased pn junction, and . [ 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 . This probability factor is crucial to the different dependencies of Auger and BTBT rates.
The matrix element for BTBT tunneling is given by
where is the potential energy in the depletion region of the pn junction, and and are wavefunctions in the valence and conduction band. Following the derivation in Appendix D, the matrix element can be rewritten as
Here, a constant electric field () across the tunneling junction is assumed, ensures perpendicular momentum conservation, is the band gap including quantization energy, is the reduced mass, and . The expression for in Eq. (25) results from Kane's two-band 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, .
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 factor. Using this equation, the intrinsic on/off ratio between the BTBT and Auger rates at the point of ideal TFET turn-on () is found to be
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 (), 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 or ). For many materials, the hole mass is much heavier than the electron mass which leads to a large . 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.
APPENDIX A: DERIVATION OF THE AUGER MATRIX ELEMENT
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.
1. Auger matrix element for a bulk crystal
The conduction and valence band wavefunctions for a bulk three-dimensional crystal can be written as Bloch functions26
Making use of the substitution , the integral can be rewritten as
Integrating over and making use of the identity that
(where is the Kronecker delta) yields
To evaluate the integral over , the integral is transformed into spherical coordinates defined such that the vector lies along . This transformation allows the integral to be expressed as
where . Putting this all together gives
Given the unlikelihood of Umklapp processes and the requirement of momentum conversation,26 terms in which and are neglected. This allows the matrix element to be written as
which can be simplified to
2. Auger matrix element for a quantum well
where refers to the two-dimensional in-plane wavevector (perpendicular to the tunneling direction), , , and other parameters are defined below Eq. (9). Making use of the substitution , the integral can be rewritten as
where . Integrating over and making use of the identity that
To evaluate the integral over , the integral is transformed into circular coordinates defined such that the vector lies along . Most of the contribution to the integral over in Eq. (A14) occurs when is small. Therefore, the square root factor is approximated as . The integral over can now be evaluated as follows:
APPENDIX B: DERIVATION OF THE AUGER TRANSITION RATE FOR A QUANTUM WELL
To derive the overall Auger transition rate, we begin with the summation in Eq. (1) and the foresight that the matrix element will include a Kronecker delta factor enforcing conservation of momentum. The sum over all states can be transformed into an integral by making use of property that there is one -state for a reciprocal-space area of
The initial factor of 2 accounts for the spin-up and spin-down configurations of the initial states. (The matrix element already accounts for the cases in which the initial states have the same or opposite spins as explained in Section IV B.) The Kronecker delta is used to reduce the sum over four states to an integral over three states since .
Rewriting Eq. (1) as an integral produces
Equation (14) has been used to replace the square of the matrix element with . The notation signifies that is a function of the wavevector difference between and . is given by Eq. (10), but the Kronecker delta factor has already been used in the transformation performed in Eq. (B1). For brevity, has been replaced with . Substituting Eq. (6a) for in the above equation gives
and . The above expression is valid for the CHCC process under the assumptions detailed in Section IV A.
Much of subsequent procedure follows the work of Ref. 22. It is reproduced here for completeness. To evaluate the integrals of Eq. (B4), is expressed in terms of polar coordinates (), and and are expressed in Cartesian coordinates [denoted () and ()] such that and lie along . The energy for state can be written as
where .To provide clarity, these expressions use the abuse of notation that the square of a vector implies the dot product with itself, i.e., . The statement of energy conservation can be re-expressed as
where is given by (B6e). In this work, is assumed to be independent of , which makes the integration over trivial. In reality, has a -dependence that should be accounted for in future work. The integral over is straightforward
The integral over is evaluated using the property that
For the integral over to be non-zero, since is real. is now given by
Next, the integral over is evaluated under the condition that . This requires to be within the range of
Using these limits, the integral can now be evaluated
can now be written as
Since must be real, must satisfy
which is found from evaluating the square root term in Eq. (B14). The variable substitution
is used to compute the integral over
where is the complimentary error function. can now be given as
The complimentary error function is peaked at the -value where its argument is a minimum, which occurs when
For , the complimentary error function varies much more quickly than , allowing the matrix element factor to be moved outside the integral to produce
The integration over can be performed using Eq. (4.3.34) from Ref. 32 to find
While the above equation is only strictly true for due to the approximation made in Eq. (B22), numerical calculations suggest that Eq. (B23) overestimates the exact integral of Eq. (B20) by less than a factor of 2, even when . Plugging back into Eq. (B3) gives
Substituting the expression for and using the linear relationship between and given by Eq. (16) produces
In this expression, the -dependence of the matrix element cancels with the -dependence of the cell-periodic Bloch function overlap. This expression can then be simplified to the form given by Eq. (18a) in the paper. A similar procedure can be used to derive the net recombination rate for the HCHH process.
APPENDIX C: DERIVATION OF THE MINIMUM ENERGY FOR STATE
As discussed in Section IV, the probability of a carrier in the high-energy state limits the overall Auger generation rate. The probability is highest when is at its minimum energy (). The following derivation is performed in one dimension for simplicity but can be expanded to multiple dimensions arriving at the same results. To find , it is useful to begin with the equations for energy and momentum conservation
In the above expression, and are the only independent variables because can be rewritten in terms of . The minimum energy can be found when the gradient of Eq. (C4) is zero
Solving this linear system of equations gives
when state is at its minimum energy. Combining the above with Eq. (C2) gives
valid for the CHCC process, and a similar procedure can be used to find the minimum energy for the HCHH process.
APPENDIX D: DERIVATION OF THE BTBT MATRIX ELEMENT
To derive the BTBT matrix element in Eq. (24), we begin with the standard expression in Eq. (23). Assuming a constant electric field () across the tunneling junction, the potential can be given as . The constant term results in zero coupling since and the eigenstates of the system are orthogonal to one another. This allows the BTBT matrix element of Eq. (23) to be written as
with the same definitions as in Eq. (9). Here, is the tunneling direction, indicates directions perpendicular to tunneling, and the integral is over the entire volume of the device.
The confined envelope wavefunction, , can be expressed as a Fourier series of plane waves,
where are the Fourier coefficients for state . This allows Eq. (D1a) to be expressed as
which can be proven by expanding the partial derivative of the first term using the chain rule and cancelling opposing terms. The first term of Eq. (D4) evaluates to zero because the wavefunctions of different bands are orthogonal. This leaves only the second term. The cell-periodic part of the Bloch function varies much quicker than the envelope wavefunctions, allowing the expression to be written as
where is the set of all lattice positions and is the volume a unit cell. Using two-band theory, Kane31 shows that
where is the band gap including the quantization energy and is the reduced mass equal to
APPENDIX E: DERIVATION OF THE BTBT RATE FOR A QUANTUM WELL
To calculate the BTBT transition rate, we begin by converting the sum over all states of Eq. (21) to an integral using the property that
Using the values for and given in Section V, the areal generation rate is found to be
Assuming spherical bands, the integral can be rewritten as
The energy difference in the Dirac delta function can be expressed as
Using the variable substitution that
the integral of Eq. (E5) can be expressed as
Putting this all together, we get