direct and indirect band-to-band tunneling in germanium-based tfets

292 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 59, NO. 2, FEBRUARY 2012

Direct and Indirect Band-to-Band Tunnelingin Germanium-Based TFETs

Kuo-Hsing Kao, Student Member, IEEE, Anne S. Verhulst, William G. Vandenberghe, Student Member, IEEE,Bart Sorée, Member, IEEE, Guido Groeseneken, Fellow, IEEE, and Kristin De Meyer, Fellow, IEEE

Abstract—Germanium is a widely used material for tunnelFETs because of its small band gap and compatibility with silicon.Typically, only the indirect band gap of Ge at 0.66 eV is considered.However, direct band-to-band tunneling (BTBT) in Ge should beincluded in tunnel FET modeling and simulations since the energydifference between the Ge conduction band edges at the L andΓ valleys is only 0.14 eV at room temperature. In this paper, wetheoretically calculate the parameters A and B of Kane’s directand indirect BTBT models at different tunneling directions ([100],[110], and [111]) for Si, Ge and unstrained Si1−xGex . We highlighthow the direct BTBT component becomes more important as theGe mole fraction increases. The calculation of the band-to-bandgeneration rate in the uniform electric field limit reveals that directtunneling always dominates over indirect tunneling in Ge. Theimpact of the direct transition in Ge on the performance of tworealistic tunnel field-effect transistor configurations is illustratedwith TCAD simulations. The influence of field-induced quantumconfinement is included in the analysis based on a back-of-the-envelope calculation.

Index Terms—Direct tunneling, field-induced quantum confine-ment (FIQC), germanium (Ge), silicon–germanium (SiGe), tunnelfield-effect transistor (TFET).

I. INTRODUCTION

THE TUNNEL field-effect transistor (TFET) is one ofthe devices that can reach sub-60 mV/dec subthreshold

swing (SS) since the injection mechanism of carriers from thesource is based on band-to-band tunneling (BTBT) [1]–[7].However, poor SS and low ON-current have been seen in all-SiTFETs due to the large indirect energy band gap (∼1.12 eV).Therefore, a Si-based heterojunction TFET has been proposedto reduce the effective tunneling barrier height by replacing the

Manuscript received August 5, 2011; revised October 5, 2011 andOctober 28, 2011; accepted October 30, 2011. Date of publicationDecember 7, 2011; date of current version January 25, 2012. This work wassupported by the Interuniversity Microelectronics Center’s (IMEC) IndustrialAffiliation Program. The work of W. G. Vandenberghe was supported bya Ph.D. stipend from the Institute for the Promotion of Innovation throughScience and Technology in Flanders (IWT-Vlaanderen). The review of thispaper was arranged by Editor A. Schenk.

K.-H. Kao, W. G. Vandenberghe, G. Groeseneken, and K. De Meyer are withthe Interuniversity Microelectronics Center (IMEC), 3001 Leuven, Belgium,and also with the Department of Electrical Engineering, Katholieke Universiteit(K.U.), 3001 Leuven, Belgium (e-mail: kaofrank@imec.be).

A. S. Verhulst is with the Interuniversity Microelectronics Center (IMEC),3001 Leuven, Belgium.

B. Sorée is with the Interuniversity Microelectronics Center (IMEC), 3001Leuven, Belgium, and also with the Department of Physics, UniversiteitAntwerpen, 2020 Wilrijk, Belgium.

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TED.2011.2175228

Fig. 1. Schematic band structure of (a) Ge and (b) Si at room temperature.

Si source with a smaller band gap material, e.g., germaniumfor n-channel TFETs (nTFETs) [5]–[15]. In addition, all-GeTFETs are being studied [16], [17]. So far, most research onGe-based TFETs has merely considered the indirect transitions[6]–[17], and the default parameter file of the TCAD simulationtool [18] only specifies the Ge parameters for the indirect BTBTtransition. At room temperature, however, the energy differencein the conduction band edges at the L and Γ valleys of Geis only 0.14 eV (see Fig. 1) [19]. Moreover, direct transitionsare experimentally observed to dominate the BTBT currentin Ge diodes [20], [21]. The appropriate BTBT parametersfor Si1−xGex with varying Ge mole fraction have not beeninvestigated.

In this paper, we start with the theoretical Si, Ge and un-strained Si1−xGex parameter calculations of Kane’s direct andindirect BTBT model in Section II. Based on these calculatedparameters, we discuss the BTBT generation rate as a func-tion of the electric field in the uniform electric field limit. InSection III, we present the impact of the direct band gap of Geon two different Ge-based TFETs by simulations and discussthe influence of quantum confinement.

II. THEORETICAL BTBT CALCULATION

A. Parameter Calculation

In this section, we theoretically calculate the Adir/Bdir andAind/Bind parameters of Kane’s direct and indirect BTBTmodels in different tunneling directions ([100], [110], and[111]) for Si, Ge and unstrained Si1−xGex, where the minimumband gap as well as the direct band gap at the Γ point areconsidered, as shown in Fig. 1.

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KAO et al.: DIRECT AND INDIRECT BAND-TO-BAND TUNNELING IN GE-BASED TFETs 293

TABLE IPHYSICAL PARAMETERS USED IN THEORETICAL CALCULATIONS OF Adir, Bdir AND Aind, Bind FOR Si1−xGex AT VARIOUS Ge MOLE FRACTIONS.

ALL EFFECTIVE MASSES ARE IN THE UNIT OF FREE ELECTRON MASS mo. NOTE THAT THE ELECTRON EFFECTIVE MASSES mT AND mL ARE

CONSTANT IN 0 ≤ x ≤ 0.8 AND THE HOLE EFFECTIVE MASSES mhh AND mlh DECREASE LINEARLY WITH INCREASING x [41], [42]

In the uniform electric field limit, the widely used Kane’smodel to determine the BTBT generation rate G per unitvolume is given by [18], [22]

G = A

(F

F0

)P

exp(−B

F

)(1)

where F0 = 1 V/cm; P = 2 and 2.5 for the direct and indirectBTBT, respectively; A and B are the Kane parameters; and F isthe electric field. Prefactor A and exponential factor B for directand indirect transitions can be expressed by [18], [22], [23]

Adir =

[gπm

1/2r (qF0)2

9h2(EΓ

g

)1/2

](9π2

)(2)

Bdir =π2m

1/2r

(EΓ

g

)3/2

qh(3)

Aind =g(mcmv)3/2(1 + 2NTA)D2

TA(qF0)5/2

221/4h5/2m5/4r ρεTAE

7/4g

(4)

Bind =27/2πm

1/2r E

3/2g

3qh(5)

where g is a degeneracy factor detailed later in this section;mr is the reduced tunneling mass; q is the elementary charge;h is Planck’s constant; EΓ

g is the direct band gap at the Γpoint; mv (mc) is the valence (conduction) band densityof states effective mass; NTA = 1/[exp(εTA/kT ) − 1] isthe occupation number of the transverse acoustic phonon attemperature T , where k is the Boltzmann constant; DTA is thedeformation potential of transverse acoustic phonons; ρ is themass density; εTA is the transverse acoustic phonon energy;and Eg is the minimum band gap. Note that we are using thesame formula [see Eqs. (2)–(5)] as given by Sentaurus Device[18] except for a correction to the prefactor Adir, which is heremultiplied by (9/π2) according to [23]. For the indirect model,a more complete formula fully incorporating the impact of theemitted or absorbed phonon is described in [24].

Table I lists all parameters of unstrained Si1−xGex with vari-ous Ge mole fraction x. Mass density ρ is a function of Ge molefraction x as given by ρ = 2.329 + 3.493x − 0.499x2 [25]. Theestimated deformation potential of transverse acoustic phonons,DTA, for Si is 2.45 × 1010 eV/m [26], and by performing ananalogous calculation, the value for Ge is 0.8 × 1010 eV/m[27]. The transverse acoustic phonon energy εTA for Si(19 meV) and Ge (8.6 meV) are extracted from [8]. The εTA

of Si1−xGex with various Ge concentrations is determined bylinear interpolation since the variation of εTA is almost linear

with increasing Ge mole fraction [28]. Note that we only takethe transverse acoustic phonons into account because they havethe highest phonon occupation number and the smallest phononenergy and therefore provide the main contribution in Si andGe tunnel diodes [29]. For indirect transitions, the conductionand valence band density of state effective mass mc and mv

are respectively given by (mLm2T )1/3 and (m3/2

lh + m3/2hh )2/3,

where mL and mT are the electron longitudinal and transverseeffective masses and mlh and mhh are the light and heavy holeeffective masses, respectively [30].

Table II lists degeneracy factor g, reduced tunneling massmr, and lightest electron mass me for direct and indirect BTBTin different tunneling directions. mr is appropriate to the lighthole mlh and the lightest electron effective mass me along a cer-tain tunneling direction (mr = memlh/me + mlh) [10], [22]because the lightest carriers have the dominant contributionto the tunneling current. For direct BTBT, mr is determinedby assuming that the electron effective mass is the same asthe light-hole effective mass according to the k · p theory [31].Degeneracy factor g is expressed as

g = 2 × gv × gc (6)

which consists of the electron spin degeneracy factor 2 and thevalence and conduction band valley degeneracy. As mentionedabove, since we only consider the lightest carriers, gv is equalto 1 (light hole band), and gc depends on the tunneling direction(see Table II) [32].

Table III lists the theoretically calculated Aind, Bind, Adir,and Bdir by (2)–(5) for unstrained Si1−xGex with variousGe mole fractions x. The deviation between our theoreticalvalue Bdir (∼6.04 MV/cm) for direct transitions in Ge andthe prediction of [33] (∼5.3 MV/cm) and the experimentalcalibration of [21] (∼5.7 MV/cm) is small, namely about 14%and 6%, respectively. This deviation is expected to originatemainly from the uncertainty on the value of reduced mass mr.Existing parameters Aexi and Bexi provided in the simulationtool are 4 × 1014 cm−3 · s−1 and 19 MV · cm−1 for Si and9.1 × 1016 cm−3 · s−1 and 4.9 MV · cm−1 for Ge, respectively[18]. The existing parameters of Si are extracted from experi-mental data of a p-n tunneling diode [34]. The calibration forGe has been done for the indirect BTBT transitions based onp-n diodes and theoretical calculations [8], [33] although theprocedure used by the TCAD software to extract the existingparameters is not fully transparent. In the next subsection,the BTBT generation rate based on existing and theoreticallycalculated parameters is compared.

294 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 59, NO. 2, FEBRUARY 2012

TABLE IIDEGENERACY FACTOR g, ELECTRON EFFECTIVE MASS me, AND REDUCED TUNNELING MASS mr USED IN THEORETICAL CALCULATIONS OF Adir,

Bdir AND Aind, Bind FOR Si1−xGex IN DIFFERENT TUNNELING DIRECTIONS. NOTE THAT THE BAND STRUCTURE OF BULK Si1−xGex

EXHIBITS A Si-LIKE (Ge-LIKE) CONDUCTION BAND MINIMA AT Δ(L) WITH SIX (EIGHT) EQUIVALENT VALLEYS WHEN Ge MOLE

FACTION x IS LESS (MORE) THAN 0.85. ALL EFFECTIVE MASSES ARE IN THE UNIT OF FREE ELECTRON MASS mo

TABLE IIITHEORETICALLY CALCULATED AND EXISTING PARAMETERS A AND B FOR THE NONLOCAL PATH BTBT MODEL FOR UNSTRAINED

Si1−xGex AT VARIOUS Ge MOLE FRACTIONS. A AND B ARE IN THE UNIT OF cm−3 · s−1 AND MV · cm−1, RESPECTIVELY

Fig. 2. BTBT generation rates of Si and Ge as a function of the uniformelectric field with theoretically calculated Adir, Bdir and Aind, Bind fordirect and indirect models and with existing Aexi, Bexi for the indirect model.The tunneling direction is along [100].

B. BTBT Generation Rate Calculation

The BTBT generation rate as a function of the uniformelectric field is plotted in Fig. 2 for the indirect and directband gap transitions in pure Si and pure Ge according to (1).It is expected that the BTBT generation rate for direct bandgap transitions dominates over the rate for indirect band gap

transitions, if the energy values of the direct and indirect bandgap are the same and if all other relevant parameters, such asthe electron and hole masses, are the same. This is becausea transition from the valence to the conduction band thatrequires interaction with a phonon, as in an indirect band gapmaterial, is less likely than such a transition that occurs withoutinteraction with a phonon, as in a direct band gap material.The BTBT generation rate of Si for indirect transitions deter-mined by our theoretically calculated parameters and by theexisting parameters match very well, whereas our theoreticallycalculated direct BTBT rate is negligible compared with theindirect BTBT. This suggests that the indirect BTBT processdominates in Si. The BTBT generation rate determined by ourtheoretically calculated Adir, Bdir of Ge is always larger thanthe rate determined by Aind, Bind. This implies that when theband bending is such that direct transitions are allowed in Ge,the direct transitions will dominate over the indirect transitionsbeing in agreement with the experimental data [20], [21]. Fig. 2also shows that for Ge, the BTBT generation rate determinedby our theoretically calculated Adir, Bdir matches very wellwith the rate determined by the existing Ge parameters usedin an indirect BTBT setup. This implies that an indirect BTBTmodel has been widely used by many researchers and by thesimulation software, predicting BTBT generation rates with anintensity corresponding to a direct transition, even when theband bending is such that only indirect transitions are allowed.

KAO et al.: DIRECT AND INDIRECT BAND-TO-BAND TUNNELING IN GE-BASED TFETs 295

Fig. 3. BTBT generation rates of Si1−xGex as a function of the uniformelectric field with theoretically calculated parameters listed in Table III. Thetunneling direction is along [100].

Fig. 4. Minimum energy band gap and the highest valence and the lowestconduction band of unstrained Si1−xGex as a function of Ge mole fraction.

Fig. 3 presents the theoretically calculated BTBT generationrate as a function of the uniform electric field for unstrainedSi1−xGex with various Ge mole fractions for direct and indirecttransitions. At x = 0.5, the direct BTBT is significantly smallerthan the indirect BTBT rate up to 2 MV/cm. At x = 0.8,however, the direct and indirect BTBT rate have the sameorder of magnitude, despite the fact that the direct band gap of1.37 eV is still significantly larger than the indirect band gapof 0.86 eV (see Table I). Note that a relatively pronouncedgain of the indirect BTBT generation rate is observed be-tween Si0.2Ge0.8 and 100% Ge. This is attributed to an abruptdecrease in the indirect band gap while the Si1−xGex bandstructure transforms from a Si- to a Ge-like band structure atx = 0.85 [35], as shown in Fig. 4.

III. SIMULATIONS AND DISCUSSION

In this section, the impact of a correct use of parameterson performance predictions is illustrated based on device sim-ulations of two realistic TFET configurations. The direct andindirect BTBT models are activated for Si and Ge in the simu-lations. The calculated Aind, Bind and Adir, Bdir are specifiedfor Ge. For Si, the calculated Adir, Bdir are specified for the

direct BTBT model, but the existing Aexi, Bexi are used for theindirect BTBT model since the deviation from the theoreticallycalculated parameters is very small, as shown in Fig. 2.

A. Simulation Settings

Self-consistent device simulations are performed withSentaurus Device [18] using the Fermi–Dirac statistics model,the drift-diffusion carrier transport model, a doping-dependentmobility model, the high field velocity saturation model,the Auger and Shockley–Read–Hall generation/recombinationmodels, the doping-dependent band-gap-narrowing (BGN)model, and the dynamic nonlocal path BTBT model at 300 K.Note that the BTBT model allows tunneling across up to threedifferent band gaps simultaneously. This feature is used to si-multaneously determine tunneling across the direct and indirectband gaps. Since the carrier density at the semiconductor–oxideinterface is typically quite high, we also include the interface-orientation-dependent modified local-density approximation(MLDA) model to consider the carrier redistribution due tosurface quantization effects. At the end of this section, we willalso adjust the data to qualitatively include the impact of field-induced quantum confinement (FIQC) [36].

The first simulated structure, which is a Ge-source Si-TFET,is schematically presented in Fig. 5(a), and all device param-eters are indicated. An n-type doped drain concentration of1 × 1020 cm−3, a p-type doped source concentration Ns of1 × 1019 cm−3, an n-doped channel of 1 × 1015 cm−3 witha 40-nm channel length, a 50-nm body thickness, a 0.6-nmeffective oxide thickness, a 30-nm gate–source overlap, and a4.05-eV gate workfunction are specified in the simulations.Abrupt and uniform doping profiles are used. The secondsimulated structure, which is an all-Ge TFET [see Fig. 8(a)],is identical to the configuration in Fig. 5(a), except for areplacement of the channel and the drain material with Ge.

B. Results of Semiclassical Simulations

Fig. 5(b) shows the input characteristics of the Ge-sourcenTFET with the architecture shown in Fig. 5(a). The largegate–source overlap results in a lateral tunneling current atonset transitioning into a vertical tunneling current as thegate–source voltage increases. This is illustrated in Fig. 6 withthe generation rate contours: at small gate–source voltage [seeFig. 6(a)], the tunneling is lateral; at intermediate gate–sourcevoltage [see Fig. 6(b)], vertical tunneling builds up; and at largegate–source voltage [see Fig. 6(c)], the vertical tunneling isdominant. Fig. 5(b) shows that at large gate–source voltages,the indirect tunneling current (circle) is smaller than the directtunneling current (solid), as expected from Fig. 2. However, thedirect tunneling has a later onset by nearly 0.2 V because thedirect band gap is 0.14 eV larger than the indirect band gap.Comparing with the indirect BTBT current with the existingparameters (dashed), the tunneling current based on a correctuse of parameters is therefore significantly reduced at the smallgate–source voltages when only indirect BTBT is possible.Furthermore, the final tunneling current is also slightly smallerdue to the corresponding smaller BTBT generation rate in

296 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 59, NO. 2, FEBRUARY 2012

Fig. 5. (a) Simulated heterostructure device. (b) Input characteristics of(a) obtained by the indirect model with (dashed) existing parameters and(circle) theoretically calculated Aind, Bind and by direct and indirect modelswith (solid) Aind, Bind and Adir, Bdir. (c) Decomposition of the solid curveshown in (b) into three components as indicated in the legend and simulatedwith calculated parameters (V: vertical, L: lateral).

the uniform field limit (see Fig. 2) and the smaller BTBTregion resulting from the larger direct band gap. The latteris observable in Fig. 6(c) showing a higher BTBT generationrate but smaller BTBT region associated with direct transitionshaving a longer tunneling length. To visualize this more clearly,Fig. 7 presents the band diagram and the BTBT generationrate along a 1-D cut perpendicular to the oxide interface atthe middle of the source–gate overlap in Fig. 6(c). There is aclear shoulder on the hole generation rate curve (dash) witha shorter tunneling length associated with a smaller band gap:the hole generation area is therefore larger for the indirect bandgap transitions. A less pronounced shoulder can be also seen inthe electron generation curve (solid) indicating a larger electrongeneration area for the direct band gap transitions.

Fig. 6. Electron and hole BTBT generation rate contours (cm−3 · s−1) ofthe n-channel device in Fig. 5(a) are shown in (a) for Vgs = 0.2 V, in (b) forVgs = 0.35 V, and in (c) for Vgs = 1 V resulting from simulations with directand indirect BTBT models corresponding to the solid curve in Fig. 5(b). Thedrawing, which is a segment of the full TFET, is to scale with the 30-nm gate–source overlap as size reference.

Fig. 7. (Right vertical axis) Energy band diagram and (left vertical axis)BTBT generation rate of (solid) electrons and (dash) holes of the configurationin Fig. 6(c) along a 1-D cut perpendicular to the oxide interface at the middle ofthe source–gate overlap. The (in)direct conduction band is denoted with a dash(solid) curve. The shaded (striped) region shows the (in)direct tunneling withlonger (shorter) tunnel lengths and a narrower (wider) generation region.

In Fig. 5(c), the input characteristic with direct and indirectmodels corresponding to the solid curve in Fig. 5(b) is split inits different components: indirect lateral, indirect vertical, anddirect vertical tunneling. The indirect lateral tunneling curve isobtained with a structure identical to that in Fig. 5(a) exceptfor the gate–source overlap, which is reduced to 0 nm. In thissimulation, only BTBT across the indirect band gap is allowed.The (in)direct vertical tunneling curve is obtained with a struc-ture identical to that in Fig. 5(a) except for the gate–channeloverlap, which is reduced to 0 nm. In this simulation, onlyBTBT across the (in)direct band gap is allowed. The sum of

KAO et al.: DIRECT AND INDIRECT BAND-TO-BAND TUNNELING IN GE-BASED TFETs 297

the subcomponents (not shown) is very close to the solid curve.This implies that the modified gate electrode and the tunnelingcurrent itself only have a limited impact on the electrostaticprofile and therefore result in a very limited modification of thedevice current. The latter is in agreement with a previous work,where the tunneling current prediction based on a nonself-consistent analytical model is in very good agreement with theself-consistently simulated current as long as the current is notvery high [37].

Note that at a gate–source voltage of about 0.3 V, there is asmall kink in the dashed and solid curves in Fig. 5(b), whichresults from the transition from lateral tunneling leakage fromsource to channel into vertical tunneling in the source perpen-dicular to the gate, as can be observed from the decompositionin Fig. 5(c). This lateral leakage current is tunneling across aheterojunction. Since the simulator is limited to either take adirect BTBT model or an indirect BTBT model along the entiretunneling path [18], and since the direct band gap of Si is toohigh to play a role, the lateral tunneling current is based onthe indirect BTBT parameters in both materials. More insightin heterojunction tunneling is needed to determine whether itwould not be more meaningful to use the direct-band-gap Geparameters for the heterojunction tunneling.

Fig. 8(b) shows the input characteristics of the all-Ge nTFETwith the structure shown in Fig. 8(a). Similar results in Fig. 5(b)are observed at high gate–source voltages. At low gate–sourcevoltages, the current is larger due to the larger drain–sourceleakage current and the increased lateral tunneling, which areboth due to the smaller band gap of Ge together with theavailability of direct BTBT transitions into the channel and thedrain. Fig. 8(c) shows a split of the solid curve in Fig. 8(b) in itsdifferent components: direct and indirect lateral tunneling anddirect and indirect vertical tunneling. The different curves areobtained in a similar way to the curves in Fig. 5(c). This split-up shows that the kink at 0.5 V in the solid curve is associatedwith the transition from direct lateral BTBT to direct verticalBTBT. Direct lateral tunneling is dominant over indirect lateraltunneling, even near onset of lateral tunneling. The earlier onsetby about 0.14 V of the indirect component is present at verylow current levels but not observable in the configuration witha 40-nm channel length. For a device with an 80-nm channellength, the leakage currents are smaller, and it can be seenthat the indirect lateral tunneling has an onset that is about0.14 V earlier than the direct lateral tunneling (not shown).Note that the predictions of the lateral tunneling component inFigs. 5(c) and 8(c) should not be compared quantitatively, sincethe direct Ge band gap cannot contribute to the lateral tunnelingin the heterostructure due to a limitation of the simulationsoftware, as discussed before, and since the stress associatedwith the Ge–Si heterostructure is not included in the analysis[6], [38], [39]. The latter choice is made because stress is notproperly implemented in the BTBT model, and an in-depthstudy of the impact of the stress is beyond the scope of thispaper. In particular, the simulation software uses the specifiedparameters A and B of the BTBT model, and these parametersare constants along the tunneling paths despite the fact that theband gap and the effective mass are varying in a heterostructureconfiguration due to the nonuniform stress.

Fig. 8. (a) Simulated all-Ge TFET. (b) Input characteristics of the all-GeTFET obtained by the indirect model with (dashed) existing parameters and(circle) theoretically calculated Aind, Bind and by direct and indirect modelswith (solid) Aind, Bind and Adir, Bdir. (c) Decomposition of the solid curveshown in (b) into four components as indicated in the legend and simulated withcalculated parameters (V:vertical, L:lateral).

The delayed onset of direct BTBT versus indirect BTBT withabout 0.14 V is also expected in the presence of nonabrupt dop-ing profiles in the source (e.g., 1-D doping gradient along thegate dielectric interface). Note that this might not be observableas clear kinks in the characteristics due to the degradation of thesubthreshold slope and the averaging effect associated with thegraded doping profile.

C. Impact of FIQC

The MLDA model is included in simulations to considerquantum confinement near an oxide interface, which reducesthe local carrier density and decreases the tunneling current.In fact, FIQC also creates subbands and gradually raises the

298 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 59, NO. 2, FEBRUARY 2012

Fig. 9. (a) Subband energy at the onset of vertical tunneling and (b) onset voltage of vertical tunneling with different electron effective masses as a function ofthe energy band gap.

Fig. 10. Input characteristics including the shifted vertical tunneling components according to the back-of-the-envelope calculation of FIQC (see Fig. 9) for(a) Ge-source Si TFETs and (b) all-Ge TFETs.

subband energy with increasing gate bias, which has beenshown to significantly delay the onset of vertical tunneling [36].We will now include FIQC in our analysis.

According to [36], the ground-state subband energy of the αvalley Eα

sub in the triangular well approximation is given by

Eαsub = −a0

(h2F 2

ox,α

8π2m∗α

)1/3

(7)

where a0 ≈ −2.3381 is the first zero of the Airy function, andm∗

α is the electron effective mass in the α valley. The forceacting on the electron at the semiconductor–oxide interfaceFox,α at the onset of vertical tunneling is

Fox,α =

√2q2Ns

εs

(Eα

g + Eαsub

)(8)

where εs is the semiconductor permittivity, and Eαg is the

tunneling band gap between the valence band and the α valleyof the conduction band. Fig. 9(a) shows the subband energy fortwo different electron effective masses at the onset of verticaltunneling; this is when the subband energy level aligns withthe valence band level along the [100] as a function of Eα

g .The subband energy decreases with decreasing Eα

g due to theweaker electric field at the onset of vertical tunneling. Thesubband energy increases with decreasing electron effectivemass as a result of stronger quantum confinement. Note that

mΓ = 0.044m0 and mLe = 0.117m0 are the electron effective

mass at the Γ valley (direct band gap) and the electron effectivemass in the L valley of Ge along the [100] tunneling direction,respectively (see Table II).

For the device operation, the onset voltage also needs to bedetermined [see Fig. 9(b)]. Onset voltage V α

onset of the verticaltunneling can be written as

V αonset = Eα

g + Eαsub + Fox,αTox

εs

εox− VFB (9)

where VFB is the flat-band voltage between the gate work-function and the Fermi level of the source region. The thirdterm in (9) is the potential drop across the oxide, in whichTox and εox are the physical oxide thickness and oxide per-mittivity, respectively. As shown in Fig. 9(b), the onset voltageof vertical tunneling is decreased with decreasing Eα

g andincreasing electron effective mass. The differences in the on-set voltage when including FIQC are ΔV L

onset = 0.563 V atEL

g = 0.66 eV − ΔEBGN = 0.645 eV and ΔV Γonset = 0.883 V

at EΓg = 0.8 eV − ΔEBGN = 0.785 eV. (In the Γ and L

valleys, Eαg are extracted from simulations, which includes

doping-induced BGN effect ΔEBGN = 0.015 eV for Ns =1E19 cm−3.)

Manually shifting the vertical direct and indirect tunnelingcomponents shown in Fig. 5(c) by ΔV α

onset, Fig. 10(a) showsthe input characteristics of the Ge-source Si TFET consideringFIQC. The dashed line with existing parameters is the same

KAO et al.: DIRECT AND INDIRECT BAND-TO-BAND TUNNELING IN GE-BASED TFETs 299

as in Fig. 5(b) and is added to highlight the difference inpredictions when incorrect assumptions are made. The solidline is the sum of the lateral tunneling and the shifted verticaltunneling from the valence band to the first subband determinedby the light electron effective mass 3mLmT /(2mL + mT ) in[100] direction (see Table II) in the L valley (indirect BTBT)and the first subband in the Γ valley (direct BTBT). It is clearthat predictions are significantly affected by correctly includingthe direct band gap of Ge. The onset of direct tunneling isdelayed by two effects: first, the delay of about 0.2 V due tothe larger band gap as discussed above and second, the relativedelay compared with the indirect tunneling of about 0.3 Vdue to the smaller effective tunneling mass at Γ resulting in alarger subband energy derived in Fig. 9(b). This last relativeshift allows to observe pure indirect tunneling in Ge over alarger range of electric fields than is possible in a Ge diode,and therefore, calibration of the indirect BTBT in Ge should bemore straightforward with the TFET configuration.

The same manual shifting has been applied to the all-GeTFET, as shown in Fig. 10(b). The vertical indirect BTBTassociated with the first subband in the L valley is lower thanthe lateral direct BTBT. The transition from lateral to verticalBTBT can therefore be only observed for the first subband inthe Γ valley.

Note that the orientation of the lateral tunneling path withrespect to the gate oxide varies with gate bias. Lateral tunnelingstarts with a long tunneling path that is rather parallel to theoxide–channel interface at low gate bias, such that less FIQC isexpected. With increasing gate bias, the tunneling direction tiltsmore toward the gate oxide, and FIQC is expected to increase.Given that a varying impact of FIQC is not realistic with theavailable software [40], we have assumed for the calculationsin Fig. 10 that there is no FIQC for the lateral tunneling.

IV. CONCLUSION

We theoretically calculate the parameters A and B of Kane’sdirect and indirect BTBT models for different tunneling di-rections ([100], [110], and [111]) for Si, Ge and unstrainedSi1−xGex. A relatively pronounced gain in the indirect BTBTgeneration rate between Si0.2Ge0.8 and Ge is attributed to anabrupt decrease in band gap while Si1−xGex transforms fromSi- to Ge-like material at x = 0.85. The direct BTBT contri-bution in unstrained Si1−xGex becomes equally important tothe indirect BTBT when the Ge concentration is about 80%.For 100% Ge, the calculation of the BTBT generation rate inthe uniform electric field limit reveals that direct tunneling al-ways dominates. We show that less optimistic predictions for aGe-source TFET upon using the correct parameters and includ-ing the Ge direct band gap. The onset of direct versus indirectBTBT is delayed by about 0.5 V in the vertical tunneling com-ponent due to the larger direct band gap and the larger directsubband energy resulting from FIQC. The Ge-source TFETshould therefore be a more appropriate vehicle to calibratethe strength of the indirect Ge BTBT process than Ge diodes,where calibration of the indirect branch is limited to a 0.14-Vvoltage window.

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Kuo-Hsing Kao (S’11) was born in Taipei, Taiwan,on November 10, 1982. He received the M.S. de-gree from National Chiao Tung University, Hsinchu,Taiwan, in 2008. He is currently working at Interuni-versity Microelectronics Center, Leuven, Belgiumas a Ph.D. student of the Katholieke UniversiteitLeuven, Leuven.

His research interests include the working prin-ciple of the tunnel FET and their modeling andsimulation.

Anne S. Verhulst received the M.S. degree from theKatholieke Universiteit Leuven, Leuven, Belgium, in1998 and the Ph.D. degree from Stanford University,Stanford, CA, in 2004, both in electrical engineer-ing. She performed her master thesis at Interuni-versity Microelectronics Center (IMEC), Leuven onthe topic of Scanning Capacitance Microscopy. Inher Ph.D. thesis, she investigated optical pumpingtechniques to increase the polarization in nuclear-spin-based quantum computers, and she performedpart of her Ph.D. research, from April 1999 to May

2001, at the IBM Almaden Research Center, CA.Since March 2005, she has been a Researcher at IMEC in the Nanotech-

nology group. Her research topic has included the areas of nanowires andcarbon nanotubes, involving investigation of applications in nanoelectronics.She currently focuses on the tunnel field-effect transistor.

William G. Vandenberghe (S’05) was born inTielt, Belgium, on January 31, 1985. He receivedthe M.S. degree in electrical engineering fromthe Katholieke Universiteit (K.U.) Leuven, Leuven,Belgium, in 2007. He is currently working at In-teruniversity Microelectronics Center, Leuven, as aPh.D. student of the K.U. Leuven.

His research interests include the working princi-ple of the tunnel FET and the quantum mechanicaltransport mechanisms present in the tunnel FET.

Bart Sorée (M’08) was born in Antwerp, Belgium,on March 11, 1973. He received the Civil Engi-neering Physics and Theoretical Physics degreesfrom the Universiteit Gent, Ghent, Belgium, in 1998and 1999, respectively, and the Ph.D. degree fromthe Katholieke Universiteit (K.U.) Leuven, Leuven,Belgium, in 2003. In his Ph.D. thesis titled “Dissi-pative quantum transport using balance equations,”he solved several fundamental theoretical problemsrelated to conductance quantization and dissipation.

In 2004, he joined the Interuniversity Microelec-tronics Center, Leuven, where he is currently a Senior Researcher in thePhysics, Modeling, and Simulation group. He has contributed to the researchof carbon nanotubes, graphene, planar and nanowire devices, junctionlesstransistors, and tunneling field effect devices by investigating electronic andvibrational properties of nanoconfined systems, dissipative (low-field) andballistic transport of charge carriers in devices, tunneling theory, and quantumconfinement effects. In 2011, he also joined the Condensed Matter Theorygroup in the Physics Department, University of Antwerp, Antwerp, Belgium, asa part-time Professor. His research interests include quantum transport, physicsand modeling of devices, nonequilibrium statistical mechanics, and identifyingnovel and emerging device concepts and materials.

Dr. Sorée is a member of the Belgian Physical Society and the AmericanPhysical Society.

KAO et al.: DIRECT AND INDIRECT BAND-TO-BAND TUNNELING IN GE-BASED TFETs 301

Guido Groeseneken (S’80–M’89–SM’95–F’05) re-ceived the M.Sc. degree in electrical and mechanicalengineering and the Ph.D. degree in applied sciencesfrom the Katholieke Universiteit (K.U.) Leuven,Leuven, Belgium, in 1980 and 1986, respectively.

In 1987, he joined the R&D Laboratory ofInteruniversity Microelectronics Center (IMEC),Leuven, where he is responsible for research inreliability physics for deep submicrometer CMOStechnologies. From October 2005 to April 2007,he was also responsible for the IMEC Post CMOS

Nanotechnology program within IMEC’s core partner research program. In2001, he also became Professor at the K.U. Leuven, where he is responsiblefor the Erasmus Mundus Master in Nanoscience and Nanotechnology. He hasauthored or coauthored more than 300 publications in international scientificjournals and in international conference proceedings and six book chapters. Heis the holder of ten patents in his fields of expertise.

Dr. Groeseneken became an IMEC Fellow in 2007. He has made contribu-tions to the fields of nonvolatile semiconductor memory devices and technologyand reliability physics of VLSI technology. Recently, he has also worked oncharacterization and reliability of high-k gate dielectrics, FinFET, and Ge-basedMOSFETs; integration of carbon nanotubes for interconnect applications; andtunneling MOSFET nanowire devices. He has served as a technical programcommittee member of several international scientific conferences, amongwhich the IEEE International Electron Device Meeting (IEDM), the EuropeanSolid State Device Research Conference, the International Reliability PhysicsSymposium, the IEEE Semiconductor Interface Specialists Conference, andthe EOS/ESD Symposium. From 2000 to 2002, he also acted as EuropeanArrangements Chair of IEDM. In 2005, he was the General Chair of theInsulating Films on Semiconductor conference, organized in Leuven.

Kristin De Meyer (S’73–M’79–SM’00–F’11) re-ceived the M.Sc. degree in electrical engineering andthe Ph.D. degree from the Katholieke Universiteit(K.U.) Leuven, Leuven, Belgium, in 1974 and 1979,respectively.

From November 1979 to November 1980, she wasgranted an IBM World Trade Postdoctoral Fellow-ship and was with the IBM T. J. Watson ResearchCenter, Yorktown Heights, NY, working on the de-velopment of electrically alterable read-only mem-ory devices using Si-rich oxides. At the end of 1980,

she returned to the K.U. Leuven as a Senior Research Assistant and, in October1982, a Research Associate with the National Fund for Scientific Research(NFWO). With the foundation of the Interuniversity Microelectronics Center(IMEC), Leuven, her activities moved from the K.U. Leuven to the IMEC,where she was initially a Research Associate of the NFWO. Since October1989, she has been a Regular Employee with the IMEC, where, from 1985 to1992, she was the Head of the group on process and device modeling and simu-lation, working on process optimization techniques, parameter extraction, ana-lytical and numerical device modeling, and computational physics. In addition,statistical process control activities were monitored through her group. FromJanuary 1993 to December 2002, she was in charge of deep-submicrometermetal–oxide–semiconductor technology and exploratory field-effect devices.In January 2003, she became the Strategic Coordinator of Doctoral Researchwith the IMEC. Since October 1986, she has been also a part-time Professorwith ESAT-INSYS, K.U. Leuven. She was the Coordinator for IMEC in theESPRIT 962 EVEREST, STORM, NOVA, ULTRA, VAHMOS2000 (prime),FASEM, and ULIS projects; in the IST SIGMUND (prime), SATURN (SaturnOrbit Insertion), and NESTOR projects; and in the Marie Curie EST EDITHproject. She was also involved in the ESPRIT ACCESS program, ADEQUAT,ACE, and IST HUNT. She also managed the HCM-SUSTAIN network on deep-submicrometer silicon technology with 21 partners from academia and researchinstitutes. She has authored or coauthored over 300 publications and organizedsummer courses, the Simulation of Semiconductor Processes and Devices 1998Conference, and the ULIS 2004 Workshop.

Dr. De Meyer is a member of the Flemish and Belgian Federal Councilfor Science Policy. She is currently the Editor for the IEEE ELECTRON

DEVICE LETTERS and a member of the International Technology Roadmap forSemiconductors working groups on process integration, devices and structures,and emerging research devices.