Band-Gap Narrowing and Quantum Corrections

1. Theory

• Definition: Band-structure and quantum corrections extend the classical drift-diffusion model for heavy doping and nanometre confinement: band-gap narrowing at high doping, Fermi-Dirac statistics when carriers become degenerate, and quantum corrections when the carrier wavefunction is confined near an interface or in a thin film (Klaassen; del Alamo; TU Wien dissertation).
• Band-gap narrowing: Many-body effects and impurity bands shrink the gap at high doping, raising the effective intrinsic density and the minority-carrier density (Klaassen; COMSOL; background knowledge for the mechanism).
• Degeneracy: Above roughly the effective density of states, the Boltzmann approximation overestimates the carrier density, so Fermi-Dirac integrals are used (del Alamo; computed).
• Quantum confinement: Inversion-layer carriers sit a nanometre away from the oxide and occupy discrete subbands, which lowers gate capacitance and raises threshold; density-gradient and local corrections mimic this without solving Schrodinger’s equation (TU Wien dissertation; Wagner et al.; background knowledge for the consequences).

2. Working Principle

• Band-edge shift: Narrowing is split between the conduction and valence bands and enters the carrier densities through the band edges (COMSOL).
• Statistics function: The carrier density follows $F_{1/2}(\eta )$, which grows more slowly than $e^{\eta }$ once the quasi-Fermi level approaches the band (del Alamo; computed).
• Quantum potential: The density-gradient model adds a correction potential that depends on the curvature of the density, pushing carriers away from abrupt barriers (TU Wien dissertation).
• Calibration: Correction parameters are fitted to Schrodinger-Poisson solutions (TU Wien dissertation; Nguyen et al.; Wagner et al.).

3. Applications

• Bipolar and solar-cell emitters: Band-gap narrowing sets the emitter saturation current and gain (Klaassen).
• Heavily doped contacts and source/drain: Fermi-Dirac statistics in degenerate regions (del Alamo; Farrell).
• Nanoscale MOSFETs: Quantum corrections for inversion charge, threshold and gate capacitance in bulk, double-gate and FinFET devices (Wagner et al.; Nguyen et al.).
• HEMTs and quantum wells: Subband physics, treated in compact form on the ASM-HEMT page (ASM-HEMT page).

Band-Gap Narrowing

Name	Formula	Related Formulae	Variables/Constants
Slotboom band-gap narrowing	$\Delta E_g=E_{ref}\left[\ln \frac{N}{N_{ref}}+\sqrt{{\ln }^2\frac{N}{N_{ref}}+C}\right]$	$E_c=E_{c0}-\alpha \Delta E_g,E_v=E_{v0}+\left(1-\alpha \right)\Delta E_g$	$\Delta E_g:band-gapnarrowing$ $N:doping(total{N}_D+N_{A}inCOMSOL)$ $E_{ref},N_{ref},C:parameters$ $\begin{array}{r}\alpha :fractiontakenbytheconductionband\\ (0.5forsiliconinCOMSOL);Klaassen\\ (1991):p-typevalues{E}_{ref}=9\mathrm{mV},N_{ref}=1\times 10^{17}\\ \mathrm{c}{\mathrm{m}}^{-3},C=0.5(SlotboomanddeGraaff)and\\ n-typevalues9.35\mathrm{mV},7\times 10^{17}\mathrm{c}{\mathrm{m}}^{-3},C=0\\ (delAlamoandco-workers),independentof\\ temperaturefrom250to400\mathrm{K},withdata\\ correctedforacommonintrinsic-density\\ parameterandanewminoritymobility\\ modelshowingnon/pdifference(seethe\\ Q/A);COMSOL:Slotboomwiththeupdated\\ parametersofKlaassenandothersinits\\ siliconlibrary,thesamenarrowingin\\ neutralanddepletedregions,andthe\\ Jain-RoulstonmodelwithcoefficientsA,\\ B,CforIII-Vmaterials,siliconand\\ germanium(band-edgesplit,background\\ knowledge)\end{array}$ $\begin{array}{r}(computed,9meV/1\times 10^{17}/0.5:0.96,6.4,42,\\ 83and125\mathrm{meV}at1\times 10^{16},1\times 10^{17},1\times 10^{18},1\times 10^{19}\\ and1\times 10^{20}\mathrm{c}{\mathrm{m}}^{-3};withtherevised6.92\mathrm{meV}\\ and1.3\times 10^{17}\mathrm{c}{\mathrm{m}}^{-3}(backgroundknowledge):\\ 0.66,3.4,29,61and92\mathrm{meV})\end{array}$
Effective intrinsic density	$n_{i,eff}^2=n_i^2\exp \left(\frac{\Delta E_g}{k_{B}T}\right)$	$n_0{p}_0=n_{i,eff}^2$	$\begin{array}{r}{n}_{i,eff}:effectiveintrinsicdensityincluding\\ narrowing\end{array}$ $\begin{array}{r}{k}_B:Boltzmannconstant(relation,\\ backgroundknowledge);the\\ minority-carrierdensityandhenceemitter\\ saturationcurrentrisebythesamefactor\\ (backgroundknowledge)\end{array}$ $\begin{array}{r}(computedwiththe9meVsetat300\mathrm{K}:\\ n_{i,eff}^2/n_i^2=1.28at1\times 10^{17},5.2at1\times 10^{18},25at\\ 1\times 10^{19}and124at1\times 10^{20}\mathrm{c}{\mathrm{m}}^{-3})\end{array}$

Carrier Statistics

Name	Formula	Related Formulae	Variables/Constants
Fermi-Dirac statistics	$n=N_c{F}_{1/2}\left(\eta \right),F_{1/2}\left(\eta \right)=\frac{2}{\sqrt{\pi }}{\int }_0^{\infty }\frac{\sqrt{x}dx}{1+e^{x-\eta }},\eta =\frac{E_{Fn}-E_c}{k_{B}T}$	$n\approx N_c{e}^{\eta }(\text{Boltzmann, }\eta \ll -1)$	$F_{1/2}:Fermi-Diracintegraloforderonehalf$ $\begin{array}{r}\eta :reducedquasi-Fermienergy;delAlamo:\\ n=N_c{F}_{1/2}((E_{fe}-E_c)/k_{B}T)andthe\\ Maxwell-Boltzmannlimit;Farrell:\\ Scharfetter-Gummelgeneralizationsfor\\ Fermi-DiracandBlakemorestatistics\\ (DiscretizationandNumericalSolution\\ page)\end{array}$ $\begin{array}{r}(computed:F_{1/2}=0.0489,0.765,2.82and\\ 8.84at\eta =-3,0,2and5,against{e}^{\eta }=\\ 0.0498,1,7.39and148;Boltzmann\\ overestimatesnby1.7percent,31per\\ cent,2.6timesand17times)\end{array}$

Quantum Corrections

Name	Formula	Related Formulae	Variables/Constants
Density-gradient quantum correction	$\gamma \left(\mathbf{x}\right)\approx \frac{{\hslash }^2}{12\lambda k_B{T}_L{m}^{\ast }}\left[{\nabla }^2\varphi -\frac{1}{2k_B{T}_L}{\left(\nabla \varphi \right)}^2\right]$	$n=N_c\exp \left(\frac{E_{Fn}-E_c-\gamma }{k_{B}T}\right),L_q=\sqrt{\frac{{\hslash }^2}{12m^{\ast }{k}_{B}T}}$	$\gamma :quantumcorrectionpotential(energy)$ $\hslash :reducedPlanckconstant$ $\lambda :fittingparameter$ $T_L:latticetemperature$ $m^{\ast }:effectivemass$ $\varphi :potentialenergyusedinthecorrection$ $\begin{array}{r}{L}_q:quantumlengthscale(nformand{L}_q,\\ backgroundknowledge);TUWien\\ dissertation:equation4.1as\\ reconstructedfromalttext,applied\\ separatelytoeachcarrier,theniterated\\ with\varphi +\gamma ,theerrorin\gamma beingofsecond\\ order,and\lambda fittedtoSchrodinger-Poisson\\ carrierdensitiesinaMOSstructure\end{array}$ $\begin{array}{r}(computed:L_q=0.97\mathrm{nm}for{m}^{\ast }=0.26m_{0}and\\ 1.14\mathrm{nm}for0.19m_{0}at300\mathrm{K})\end{array}$
Local and bound-state quantum corrections	$n_{QM}\left(z\right)=n_{cl}\left(z\right)\left[1-\exp \left(-\frac{z^2}{L_q^2}\right)\right](\text{MLDA-type})$	$\Delta t_{ox,eff}\approx \frac{{ϵ}_{ox}}{{ϵ}_{Si}}{z}_c(\text{charge-centroid shift})$	$\begin{array}{r}{n}_{QM},n_{cl}:quantum-correctedandclassical\\ densities\end{array}$ $z:distancefromtheinterface$ $\begin{array}{r}{z}_c:charge-centroidoffsetfromthe\\ interface(forms,backgroundknowledge);\\ TUWiendissertation:MLDAandIMLDAfrom\\ infinitesquare-wellsolutions,IMLDA\\ addingthenormalfieldbutvalidonlyfor\\ interfacedensitiesinMOSstructures,and\\ correctinterfaceconcentrationasthe\\ mostimportantrequirement;Nguyenetal.:\\ anIMLDAholecorrectionagreeingwith\\ Schrodinger-Poissonfrom200to500\mathrm{K}and\\ upto5\times 10^{18}\mathrm{c}{\mathrm{m}}^{-3},withacompatiblelocal\\ mobilitymodel;Wagneretal.:a\\ bound-statecorrectionfordouble-gate\\ MOSFETswithtabulatedeigenenergies,\\ matchingSchrodinger-Poissondensities,\\ inversionchargeandgatecapacitancefor\\ 5and10\mathrm{nm}films,withapurelyclassical\\ simulationoverestimatingdraincurrentin\\ a20\mathrm{nm}film\end{array}$ $\begin{array}{r}(computed:a1\mathrm{nm}centroidshiftadds\\ about0.33\mathrm{nm}ofequivalentoxide\\ thickness)\end{array}$

Important Questions

Ques: How large is band-gap narrowing in heavily doped silicon?

Tens of meV, enough to multiply the minority-carrier density many times. With the Slotboom parameters that Klaassen quotes for p-type silicon, 9 $\mathrm{mV}$, $1\times 10^{17}$ $\mathrm{c}{\mathrm{m}}^{-3}$ and C = 0.5, the narrowing is 6.4 $\mathrm{meV}$ at $1\times 10^{17}$, 42 at $1\times 10^{18}$, 83 at $1\times 10^{19}$ and 125 $\mathrm{meV}$ at $1\times 10^{20}$ $\mathrm{c}{\mathrm{m}}^{-3}$ (computed). Because $n_{i,eff}^2=n_i^2\exp (\Delta E_g/k_{B}T)$ (background knowledge), the minority density rises 5.2 times at $1\times 10^{18}$, 25 times at $1\times 10^{19}$ and 124 times at $1\times 10^{20}$ $\mathrm{c}{\mathrm{m}}^{-3}$ (computed). This raises the emitter saturation current of bipolar transistors and solar cells and lowers current gain. Klaassen reports the narrowing to be independent of temperature from 250 to 400 $\mathrm{K}$, and COMSOL applies it equally in neutral and depleted regions. Reference: D. B. M. Klaassen, ‘Physical Modelling for Bipolar Device Simulation’, Simulation of Semiconductor Devices and Processes Vol. 4 (SISDEP), eds. W. Fichtner and D. Aemmer, Hartung-Gorre, Zurich (1991) (Eq. 22 garbled in the extract) (fetch-tool summary); COMSOL, ‘Band Gap Narrowing’, Semiconductor Module User’s Guide 6.2 (equations as images) (fetch-tool summary).

Ques: Do n-type and p-type silicon have different band-gap narrowing?

Older data said yes; Klaassen’s correction says no. The Slotboom and de Graaff p-type parameters, 9 $\mathrm{mV}$, $1\times 10^{17}$ $\mathrm{c}{\mathrm{m}}^{-3}$, C = 0.5, and the del Alamo n-type parameters, 9.35 $\mathrm{mV}$, $7\times 10^{17}$ $\mathrm{c}{\mathrm{m}}^{-3}$, C = 0, give 83 and 50 $\mathrm{meV}$ at $1\times 10^{19}$ $\mathrm{c}{\mathrm{m}}^{-3}$ (computed), a 33 meV difference. Klaassen traced it to analysis: Slotboom and de Graaff used the majority electron mobility, while Swirhun and del Alamo used minority mobilities, and the groups assumed different intrinsic-density parameters. After correcting all data with a common value and his minority-mobility model, the corrected data showed no difference between n- and p-type. The common revised parameter set, 6.92 $\mathrm{meV}$ and $1.3\times 10^{17}$ $\mathrm{c}{\mathrm{m}}^{-3}$ (background knowledge), gives 61 $\mathrm{meV}$ at $1\times 10^{19}$ (computed), between the two old fits, so the revised set is more likely right for both types. Reference: D. B. M. Klaassen, ‘Physical Modelling for Bipolar Device Simulation’, Simulation of Semiconductor Devices and Processes Vol. 4 (SISDEP), eds. W. Fichtner and D. Aemmer, Hartung-Gorre, Zurich (1991) (Eq. 22 garbled in the extract) (fetch-tool summary).

Ques: Why must band-gap narrowing and mobility models be used as a matched set?

Because band-gap narrowing is not measured directly. Klaassen explains that published narrowing values were derived from electrical measurements of minority-carrier products, which require an assumed minority mobility; Slotboom and de Graaff used majority mobilities, others minority ones, and they also used different intrinsic-density parameters. A smaller recent value for that parameter, from Green, would raise the corrected narrowing by about 5 meV for both types. So the extracted narrowing absorbs whatever mobility and $n_i$ were assumed, and combining one group’s narrowing with another’s mobility double-counts or misses part of the effect. COMSOL’s silicon library accordingly uses Klaassen’s updated Slotboom parameters, consistent with his unified mobility model (background knowledge for the pairing). Reference: D. B. M. Klaassen, ‘Physical Modelling for Bipolar Device Simulation’, Simulation of Semiconductor Devices and Processes Vol. 4 (SISDEP), eds. W. Fichtner and D. Aemmer, Hartung-Gorre, Zurich (1991) (Eq. 22 garbled in the extract) (fetch-tool summary); COMSOL, ‘Band Gap Narrowing’, Semiconductor Module User’s Guide 6.2 (equations as images) (fetch-tool summary).

Ques: When are Fermi-Dirac statistics needed instead of Boltzmann?

When the quasi-Fermi level comes within a few $k_{B}T$ of the band edge. Del Alamo gives $n=N_c{F}_{1/2}((E_{fe}-E_c)/k_{B}T)$ with the Maxwell-Boltzmann form as its limit. At a reduced energy of -3, $F_{1/2}=0.0489$ against $e^{-3}=0.0498$, only 1.7 per cent apart; at the band edge, 0.765 against 1, so Boltzmann overestimates the density by 31 per cent; at +2 and +5 the overestimate is 2.6 and 17 times (computed). In silicon this matters above roughly $1\times 10^{19}$ $\mathrm{c}{\mathrm{m}}^{-3}$, near the effective density of states (background knowledge), as in source/drain regions, emitters and HEMT channels. Simulators then also need modified Scharfetter-Gummel fluxes, which Farrell compares, so that equilibrium stays exact. Reference: J. A. del Alamo, ‘6.720J/3.43J Integrated Microelectronic Devices’, Lecture 8, MIT OpenCourseWare (2007) (fetch-tool summary); P. Farrell et al., ‘Scharfetter-Gummel schemes for non-Boltzmann statistics’, ECMI 2016 talk, WIAS Berlin (fetch-tool summary); Discretization and Numerical Solution page of this vault.

Ques: What does the density-gradient model do?

It adds a quantum correction potential to the classical carrier densities. The TU Wien dissertation gives the correction as ${\hslash }^2/(12\lambda k_B{T}_L{m}^{\ast })$ times ${\nabla }^2\varphi -(\nabla \varphi {)}^2/(2k_B{T}_L)$, computed separately for electrons and holes and then iterated with the corrected potential, the error in the correction being of second order; $\lambda$ is fitted so that carrier densities in a MOS structure match a Schrodinger solution. The correction is significant over the quantum length $\sqrt{{\hslash }^2/(12m^{\ast }{k}_{B}T)}$, about 0.97 to 1.14 $\mathrm{nm}$ for effective masses of 0.26 and 0.19 free-electron masses at 300 $\mathrm{K}$ (computed), so it pushes inversion carriers about a nanometre away from the oxide. Because it is a differential equation, it works in arbitrary 2-D and 3-D geometries, unlike interface-only local models (background knowledge). Reference: ‘The Density Gradient Model’, Section 4.1, and ‘Quantum Correction Methods’, Chapter 4, doctoral dissertation, TU Wien (online 2016; author not named, URL path bina) (equations reconstructed from image alt text) (fetch-tool summary).

Ques: How accurate are local and bound-state quantum corrections?

Good where they are calibrated, limited elsewhere. The TU Wien dissertation notes that MLDA and IMLDA are based on infinite square-well solutions, IMLDA adding the interface-normal field, but that IMLDA only gives the corrected density at MOS interfaces. Nguyen et al. report an IMLDA hole correction agreeing with self-consistent Schrodinger-Poisson results from 200 to 500 $\mathrm{K}$ and up to $5\times 10^{18}$ $\mathrm{c}{\mathrm{m}}^{-3}$, paired with a compatible mobility model, both purely local and therefore fast and robust. Wagner et al. build a correction from tabulated bound states for double-gate MOSFETs that matches Schrodinger-Poisson densities, inversion charge and gate capacitance for 5 and 10 $\mathrm{nm}$ films, and show that a classical simulation of a 20 $\mathrm{nm}$ film overestimates the drain current. A 1 $\mathrm{nm}$ centroid shift adds about 0.33 $\mathrm{nm}$ of effective oxide (computed). Reference: ‘The Density Gradient Model’, Section 4.1, and ‘Quantum Correction Methods’, Chapter 4, doctoral dissertation, TU Wien (online 2016; author not named, URL path bina) (equations reconstructed from image alt text) (fetch-tool summary); C. D. Nguyen, C. Jungemann, B. Neinhues, B. Meinerzhagen, J. Sedlmeir and W. Molzer, ‘Compatible Hole Channel Mobility and Hole Quantum Correction Models for the TCAD Optimization of Nanometer Scale PMOSFETs’, Technical Proceedings of the 2003 Nanotechnology Conference, Vol. 2, 56-59 (2003) (abstract only) (fetch-tool summary); M. Wagner, M. Karner, J. Cervenka, M. Vasicek, H. Kosina, S. Holzer and T. Grasser, ‘Quantum correction for DG MOSFETs’, Journal of Computational Electronics 5, 397-400 (2006) (fetch-tool summary).

References

1. D. B. M. Klaassen, “Physical Modelling for Bipolar Device Simulation,” in Simulation of Semiconductor Devices and Processes, vol. 4, W. Fichtner and D. Aemmer, Eds. Zurich: Hartung-Gorre, 1991. Available: https://in4.iue.tuwien.ac.at/pdfs/sisdep1991/pdfs/Klaassen_2.pdf
2. COMSOL, “Band Gap Narrowing,” Semiconductor Module User’s Guide, version 6.2. Available: https://doc.comsol.com/6.2/doc/com.comsol.help.semicond/semicond_ug_semiconductor.6.60.html
3. J. A. del Alamo, “Carrier Drift and Diffusion (cont.), Carrier Flow,” Lecture 8, 6.720J/3.43J Integrated Microelectronic Devices, MIT OpenCourseWare, Spring 2007. Available: https://ocw.tau.edu.ng/courses/electrical-engineering-and-computer-science/6-720j-integrated-microelectronic-devices-spring-2007/lecture-notes/lecture8.pdf
4. P. Farrell, T. Koprucki, J. Fuhrmann, N. Rotundo and H. Doan, “Scharfetter-Gummel schemes for non-Boltzmann statistics,” ECMI 2016 talk, WIAS Berlin, 2016. Available: https://www.wias-berlin.de/people/peschka/files/ecmi2016_farrell.pdf
5. “The Density Gradient Model” (Section 4.1) and “Quantum Correction Methods” (Chapter 4), doctoral dissertation, Institute for Microelectronics, TU Wien (author not named on the pages). Available: https://iue.tuwien.ac.at/phd/bina/dissse14.html and https://www.iue.tuwien.ac.at/phd/bina/dissch4.html
6. C. D. Nguyen, C. Jungemann, B. Neinhues, B. Meinerzhagen, J. Sedlmeir and W. Molzer, “Compatible Hole Channel Mobility and Hole Quantum Correction Models for the TCAD Optimization of Nanometer Scale PMOSFETs,” in Technical Proceedings of the 2003 Nanotechnology Conference, vol. 2, pp. 56-59, 2003. Available: https://briefs.techconnect.org/?p=1932
7. M. Wagner, M. Karner, J. Cervenka, M. Vasicek, H. Kosina, S. Holzer and T. Grasser, “Quantum correction for DG MOSFETs,” Journal of Computational Electronics, vol. 5, pp. 397-400, 2006. Available: https://www.iue.tuwien.ac.at/pdf/ib_2007/JB2007_Wagner_2.pdf

Related pages in this vault: Drift-Diffusion Device Equations, Discretization and Numerical Solution, TCAD Mobility Models, ASM-HEMT Model, Thermal and Electro-Thermal Simulation, Hydrodynamic and Energy-Balance Transport, Fin Field Effect Transistor (FinFET), Quantum Mechanics Fundamentals.