TCAD Mobility Models

1. Theory

• Definition: TCAD mobility models give the carrier mobility at every mesh point as a function of doping, temperature, the field perpendicular to the current near interfaces, and the field along the current (Allpix; COMSOL).
• Low-field bulk mobility: Phonon scattering sets the lattice mobility, and ionized impurities reduce it at high doping, described by Masetti or Arora (Caughey-Thomas type) fits (Allpix; nextnano).
• Surface mobility: In inversion layers, acoustic phonons and surface roughness lower the mobility further as the perpendicular field rises, combined by Matthiessen’s rule in the Lombardi model (COMSOL).
• High-field mobility: At high driving fields the drift velocity saturates near $1\times 10^7$ $cm/s$, described by the Caughey-Thomas form with Canali parameters (Allpix).
• Composition: Simulators combine the effects, typically using the low-field and surface terms as the input mobility of the high-field model (Allpix; COMSOL; background knowledge for the ordering).

2. Working Principle

• Scattering rates add: Independent scattering mechanisms add their rates, so the inverse mobilities add (COMSOL; background knowledge for the rate argument).
• Impurity scattering: Coulomb scattering from ionized dopants dominates above about $1\times 10^{17}$ $\mathrm{c}{\mathrm{m}}^{-3}$, halving the electron mobility there (Allpix; computed).
• Interface effects: A stronger gate field pulls carriers against the oxide, where roughness and surface phonons scatter them (COMSOL; background knowledge for the mechanism).
• Velocity saturation: Carriers heated by the field emit optical phonons, which caps the velocity (background knowledge; Allpix).

3. Applications

• MOSFET simulation: Lombardi with Masetti and Canali for drain current and saturation (COMSOL; Allpix).
• Particle detectors: Jacoboni-Canali and Hamburg models for charge collection in silicon sensors (Allpix).
• Power and bipolar devices: Doping- and temperature-dependent mobility for drift regions and bases (Allpix; Temperature Scaling page).
• Calibration: Fitting surface-mobility parameters to measured effective mobility (COMSOL; background knowledge).

Low-Field Bulk Mobility

Name	Formula	Related Formulae	Variables/Constants
Masetti doping-dependent mobility	${\mu }_n\left(N\right)={\mu }_0+\frac{{\mu }_{max}-{\mu }_0}{1+{\left(N/C_r\right)}^{\alpha }}-\frac{{\mu }_1}{1+{\left(C_s/N\right)}^{\beta }}$	${\mu }_p\left(N\right)={\mu }_0{e}^{-P_c/N}+\frac{{\mu }_{max}}{1+{\left(N/C_r\right)}^{\alpha }}-\frac{{\mu }_1}{1+{\left(C_s/N\right)}^{\beta }}$	$\begin{array}{r}N:totalionizedimpurityconcentration(|\\ N_D-N_A|inAllpix,N_D+N_{A}innextnano)\end{array}$ ${\mu }_0,{\mu }_1:lowandcorrectionmobilities$ ${\mu }_{max}:latticemobility$ $C_r,C_s:referenceconcentrations$ $\alpha ,\beta :exponents$ $\begin{array}{r}{P}_c:holecut-offconcentration;Allpix:\\ arsenic-dopedelectrons{\mu }_0=52.2,{\mu }_{max}=\\ 1417(T/300{)}^{-2.5},C_r=9.68\times 10^{16},\alpha =0.68,{\mu }_1=\\ 43.4,C_s=3.43\times 10^{20},\beta =2.0;phosphorus{\mu }_0\\ =68.5,{\mu }_{max}=1414,C_r=9.20\times 10^{16},\alpha =\\ 0.711,{\mu }_1=56.1,C_s=3.41\times 10^{20},\beta =1.98;\\ boronholes{\mu }_0=44.9,{\mu }_{max}=470.5(T/300{)}^{-2.2},\\ C_r=2.23\times 10^{17},\alpha =0.719,{\mu }_1=29.0,C_s=\\ 6.1\times 10^{20},\beta =2.0,P_c=9.23\times 10^{16};nextnano:\\ thesamevaluesfromDESSIS2001except\\ electron{C}_s=3.34\times 10^{20}(seetheQ/A),valid\\ at300\mathrm{K}only(mobilitiesin\mathrm{c}{\mathrm{m}}^2/(\mathrm{V}\mathrm{s}),\\ concentrationsin\mathrm{c}{\mathrm{m}}^{-3})\end{array}$ $\begin{array}{r}(computed,arsenicandboronsets:\\ electrons1404,1359,727,108and61\\ \mathrm{c}{\mathrm{m}}^2/(\mathrm{V}\mathrm{s})andholes469,461,319,73and\\ 50\mathrm{c}{\mathrm{m}}^2/(\mathrm{V}\mathrm{s})at1\times 10^{14},1\times 10^{15},1\times 10^{17},1\times 10^{19}\\ and1\times 10^{20}\mathrm{c}{\mathrm{m}}^{-3})\end{array}$
Arora (Caughey-Thomas type) mobility	$\mu \left(N,T\right)={\mu }_{min}+\frac{{\mu }_0}{1+{\left(N/N_{ref}\right)}^{\alpha }}$	${\mu }_{min}=88{\left(\frac{T}{300}\right)}^{-0.57},{\mu }_0=7.40\times 10^8{T}^{-2.33},N_{ref}=1.26\times 10^{17}{\left(\frac{T}{300}\right)}^{2.4},\alpha =0.88{\left(\frac{T}{300}\right)}^{-0.146}$	${\mu }_{min}:high-dopinglimit$ ${\mu }_0:phonon-limitedrange$ $N_{ref}:referencedoping$ $\begin{array}{r}\alpha :exponent;Allpix:Aroraetal.(1982)\\ electronvaluesshownandholevalues{\mu }_{min}=\\ 54.3(T/300{)}^{-0.57},{\mu }_0=1.36\times 10^8{T}^{-2.23},N_{ref}=\\ 2.35\times 10^{17}(T/300{)}^{2.4};COMSOL:Aroraphononand\\ impuritymodelinitsmobilitylist;\\ Bortolossi:low-fieldmobilitiesof1417\\ and470.5\mathrm{c}{\mathrm{m}}^2/(\mathrm{V}\mathrm{s})inaTCADtestcase\end{array}$ $\begin{array}{r}(computed:electrons1337,1322,777,114\\ and92\mathrm{c}{\mathrm{m}}^2/(\mathrm{V}\mathrm{s})andholes461,458,331,\\ 69and56\mathrm{c}{\mathrm{m}}^2/(\mathrm{V}\mathrm{s})at1\times 10^{14},1\times 10^{15},1\times 10^{17},\\ 1\times 10^{19}and1\times 10^{20}\mathrm{c}{\mathrm{m}}^{-3})\end{array}$

Surface Mobility

Name	Formula	Related Formulae	Variables/Constants
Lombardi surface (transverse-field) mobility	$\frac{1}{\mu }=\frac{1}{{\mu }_b}+\frac{1}{{\mu }_{ac}}+\frac{1}{{\mu }_{sr}}(\text{Matthiessen})$	${\mu }_{ac}=\frac{B}{E_{\perp }}+\frac{C{N}^{\lambda }}{T{E}_{\perp }^{1/3}},{\mu }_{sr}=\frac{\delta }{E_{\perp }^2}$	${\mu }_b:bulk(doping-dependent)mobility$ ${\mu }_{ac}:acoustic-phononsurfacemobility$ $\begin{array}{r}{\mu }_{sr}:surface-roughnessmobility,E_{\perp }:field\\ perpendiculartothecurrent\end{array}$ $\begin{array}{r}B,C,\lambda ,\delta :parameters(\delta inV/s);COMSOL:\\ Matthiesse{n}^{\prime }srulewithinput,\\ acoustic-phononandroughnessterms,{\mu }_{sr}=\\ \delta /E_{\perp }^2,anacoustictermbuiltfrom{\mu }_1,\\ E_{\perp }/E_{ref},T/T_{ref},{\mu }_{2}andN/N_{ref}(exactform\\ garbled),N=N_a^{-}+N_d^{+},a2-DMOSFET\\ example(5\mathrm{nm}oxide,N_A=5\times 10^{17},N_D=\\ 1\times 10^{19}\mathrm{c}{\mathrm{m}}^{-3})wherethemodellowerscurrent\\ andsharpenssaturation(acousticform\\ above,backgroundknowledgeafterLombardi\\ etal.)\end{array}$ $\begin{array}{r}(computed:doubling{E}_{\perp }quarters{\mu }_{sr};\\ combining1000,500and800\mathrm{c}{\mathrm{m}}^2/(\mathrm{V}\mathrm{s})\\ gives235\mathrm{c}{\mathrm{m}}^2/(\mathrm{V}\mathrm{s}))\end{array}$

High-Field Mobility

Name	Formula	Related Formulae	Variables/Constants
Caughey-Thomas / Canali high-field mobility	$\mu \left(E\right)=\frac{{\mu }_{low}}{{\left[1+{\left({\mu }_{low}E/v_m\right)}^{\beta }\right]}^{1/\beta }}$	$\mu \left(E\right)=\frac{v_m}{E_c}\frac{1}{{\left[1+{\left(E/E_c\right)}^{\beta }\right]}^{1/\beta }}(\text{Jacoboni-Canali})$	$\begin{array}{r}{\mu }_{low}:low-fieldmobility(forexample\\ Masetti)\end{array}$ $E:drivingfieldalongthecurrent$ $v_m:saturationvelocity$ $\beta :exponent$ $\begin{array}{r}{E}_c:criticalfield;Allpix:Jacoboni-Canali\\ withelectron{v}_m=1.53\times 10^9{T}^{-0.87}cm/s,E_c=\\ 1.01T^{1.55}\mathrm{V}\mathrm{cm},\beta =2.57\times 10^{-2}{T}^{0.66},holes\\ 1.62\times 10^8{T}^{-0.52},1.24T^{1.68}and0.46T^{0.17};Canali\\ withelectron{v}_m=1.43\times 10^9{T}^{-0.87};and\\ Masetti-Canalicombiningthetwo;COMSOL:\\ aCaughey-Thomashigh-fieldmodelwith\\ saturationvelocitiesinm/sand\\ selectabledrivingforces\end{array}$ $\begin{array}{r}(computedat300\mathrm{K}:electron{v}_m=1.07\times 10^7\\ (Jacoboni)and1.00\times 10^{7}cm/s(Canali),E_c=\\ 6980\mathrm{V}\mathrm{cm},\beta =1.11,giving1389,674,\\ 303and102\mathrm{c}{\mathrm{m}}^2/(\mathrm{V}\mathrm{s})andvelocities\\ 1.39\times 10^6,6.7\times 10^6,9.1\times 10^{6}and1.02\times 10^{7}cm/s\\ at1\times 10^3,1\times 10^4,3\times 10^{4}and1\times 10^5\mathrm{V}\mathrm{cm};holes\\ v_m=8.3\times 10^{6}cm/s,E_c=1.80\times 10^4\mathrm{V}\mathrm{cm},\beta =\\ 1.21)\end{array}$
Saturation velocity and its temperature dependence	$v_{sat}\left(T\right)=\frac{v_{sat,300}}{\left(1-A\right)+A\left(T/300\right)}$	${\mu }_e^{-1}=\frac{1}{{\mu }_{0,e}}+\frac{E}{v_{sat}}(\text{Hamburg, electrons})$	$v_{sat,300}:saturationvelocityat300\mathrm{K}$ $\begin{array}{r}A:temperaturecoefficient;Allpix:Quay\\ etal.siliconvalues1.02\times 10^{7}cm/swithA\\ =0.74(electrons)and0.72\times 10^{7}cm/swithA\\ =0.37(holes);theHamburg<100>model\\ with{\mu }_{0,e}=1530(T/300{)}^{-2.42}and{v}_{sat}=\\ 1.03\times 10^7(T/300{)}^{-0.226}cm/sforelectrons,{\mu }_{0,h}=\\ 464(T/300{)}^{-2.20}andaquadratichigh-field\\ holetermwith{E}_0=2640\mathrm{V}\mathrm{cm};mobility\\ temperaturelawsontheTemperature\\ Scalingpage\end{array}$ $\begin{array}{r}(computed:theQuayelectronlawgives\\ 1.02\times 10^{7}at300\mathrm{K}and8.2\times 10^{6}cm/sat400\mathrm{K}\\ (a20percentdrop))\end{array}$

Important Questions

Ques: How much does doping reduce silicon mobility?

By more than an order of magnitude at heavy doping. With the Masetti parameters for arsenic-doped electrons and boron-doped holes listed by Allpix, the electron mobility is 1404 $\mathrm{c}{\mathrm{m}}^2/(\mathrm{V}\mathrm{s})$ at $1\times 10^{14}$ $\mathrm{c}{\mathrm{m}}^{-3}$, 727 at $1\times 10^{17}$, 108 at $1\times 10^{19}$ and 61 at $1\times 10^{20}$, and the hole mobility 469, 319, 73 and 50 (computed). The Arora form gives similar values, 1337, 777, 114 and 92 for electrons (computed), but it lacks Masetti’s extra high-doping term $-{\mu }_1/[1+(C_s/N{)}^{\beta }]$, which lowers the mobility above about $1\times 10^{20}$ $\mathrm{c}{\mathrm{m}}^{-3}$, so the two differ most there, 61 against 92. Electron mobility halves near $1\times 10^{17}$ $\mathrm{c}{\mathrm{m}}^{-3}$, close to the reference concentration $C_r=9.68\times 10^{16}$. The Masetti model is valid at 300 $\mathrm{K}$ in nextnano’s implementation, while Arora carries explicit temperature exponents. Reference: Allpix Squared collaboration, ‘Charge Carrier Mobility’, Allpix Squared documentation, CERN (models after Jacoboni et al. 1977, Canali et al. 1975, Masetti et al. 1983, Arora et al. 1982, the Hamburg model 2015, Ruch and Kino, and Quay et al.) (fetch-tool summary); nextnano GmbH, ‘mobility_masetti’, nextnano++ database documentation (parameters from the DESSIS manual 2001; equation placeholder only) (fetch-tool summary).

Ques: Is the Masetti electron Cs $3.43\times 10^{20}$ or $3.34\times 10^{20}$?

The sources disagree by a digit transposition. Allpix lists $C_s=3.43\times 10^{20}$ $\mathrm{c}{\mathrm{m}}^{-3}$ for arsenic-doped electrons, while nextnano, quoting the DESSIS manual, gives $3.34\times 10^{20}$ $\mathrm{c}{\mathrm{m}}^{-3}$; all other electron values (52.2, 1417, $9.68\times 10^{16}$, 0.68, 43.4, 2.0) agree. The two differ by 2.7 per cent (computed), and since $C_s$ only enters the high-doping correction, the computed mobility at $1\times 10^{20}$ $\mathrm{c}{\mathrm{m}}^{-3}$ changes by under 1 $\mathrm{c}{\mathrm{m}}^2/(\mathrm{V}\mathrm{s})$ (computed), so the practical impact is negligible. Masetti et al.’s original arsenic value is generally quoted as $3.43\times 10^{20}$ $\mathrm{c}{\mathrm{m}}^{-3}$ (background knowledge), and the phosphorus value of $3.41\times 10^{20}$ in Allpix is close to it, so $3.43\times 10^{20}$ is more likely correct and $3.34\times 10^{20}$ a transcription slip (my reading). Reference: Allpix Squared collaboration, ‘Charge Carrier Mobility’, Allpix Squared documentation, CERN (models after Jacoboni et al. 1977, Canali et al. 1975, Masetti et al. 1983, Arora et al. 1982, the Hamburg model 2015, Ruch and Kino, and Quay et al.) (fetch-tool summary); nextnano GmbH, ‘mobility_masetti’, nextnano++ database documentation (parameters from the DESSIS manual 2001; equation placeholder only) (fetch-tool summary).

Ques: Is the lattice-limited electron mobility of silicon 1417 or about 1530?

Both are used, from different fits. Masetti’s arsenic set, also used in Bortolossi’s TCAD test case, has ${\mu }_{max}=1417$ $\mathrm{c}{\mathrm{m}}^2/(\mathrm{V}\mathrm{s})$, and the phosphorus set 1414, while the Hamburg model lists 1530 $\mathrm{c}{\mathrm{m}}^2/(\mathrm{V}\mathrm{s})$ for <100> electrons and Jacoboni-Canali’s low-field limit $v_m/E_c$ is $1.07\times 10^7/6980=1534$ $\mathrm{c}{\mathrm{m}}^2/(\mathrm{V}\mathrm{s})$ at 300 $\mathrm{K}$ (computed). Masetti fitted resistivity data across doping levels, while Jacoboni and the Hamburg group fitted drift velocities in high-purity silicon (background knowledge for the data types). The 8 per cent difference therefore reflects the purpose: the Masetti values describe doped device regions, and the higher values describe lightly doped detector silicon (my reading). For devices doped above about $1\times 10^{15}$ $\mathrm{c}{\mathrm{m}}^{-3}$ the difference shrinks, because impurity scattering already dominates. Reference: Allpix Squared collaboration, ‘Charge Carrier Mobility’, Allpix Squared documentation, CERN (models after Jacoboni et al. 1977, Canali et al. 1975, Masetti et al. 1983, Arora et al. 1982, the Hamburg model 2015, Ruch and Kino, and Quay et al.) (fetch-tool summary); A. Bortolossi, ‘3D Finite Element Drift-Diffusion Simulation of Semiconductor Devices’, Politecnico di Milano thesis (2014) (fetch-tool summary).

Ques: How does the Lombardi model describe inversion-layer mobility?

As three scattering mechanisms added by Matthiessen’s rule. COMSOL writes $1/\mu =1/{\mu }_{in}+1/{\mu }_{ac}+1/{\mu }_{sr}$, where ${\mu }_{in}$ is the doping-dependent bulk mobility, ${\mu }_{ac}$ an acoustic-phonon term that falls with the perpendicular field and temperature, and ${\mu }_{sr}=\delta /E_{\perp }^2$ a surface-roughness term, with $\delta$ in V/s. Doubling the perpendicular field cuts the roughness term four times (computed), so roughness dominates at high gate voltage and phonons at low. Combining 1000, 500 and 800 $\mathrm{c}{\mathrm{m}}^2/(\mathrm{V}\mathrm{s})$ gives 235 $\mathrm{c}{\mathrm{m}}^2/(\mathrm{V}\mathrm{s})$ (computed), lower than any single term. In COMSOL’s 2-D MOSFET example, with a 5 $\mathrm{nm}$ oxide and $5\times 10^{17}$ $\mathrm{c}{\mathrm{m}}^{-3}$ channel doping, the model lowers the current and sharpens saturation compared with constant mobility, reducing mobility near the gate. Reference: COMSOL, ‘Lombardi Surface Mobility’ application model documentation, Semiconductor Module 6.4 (equations garbled in the extract; after Lombardi et al., IEEE Trans. CAD 7(11), 1988) (fetch-tool summary); COMSOL, ‘Mobility Models’, Semiconductor Module User’s Guide 6.4 (equations as images; parameter values not in the text) (fetch-tool summary).

Ques: How does high-field mobility produce velocity saturation?

Through $\mu (E)={\mu }_{low}/[1+({\mu }_{low}E/v_m{)}^{\beta }{]}^{1/\beta }$, so the velocity $\mu E$ tends to $v_m$. With the Jacoboni-Canali silicon electron parameters listed by Allpix at 300 $\mathrm{K}$, $v_m=1.07\times 10^7$ $cm/s$, $E_c=6980$ $\mathrm{V}\mathrm{cm}$ and $\beta =1.11$, the mobility falls from 1389 $\mathrm{c}{\mathrm{m}}^2/(\mathrm{V}\mathrm{s})$ at $1\times 10^3$ $\mathrm{V}\mathrm{cm}$ to 674 at $1\times 10^4$, 303 at $3\times 10^4$ and 102 at $1\times 10^5$ $\mathrm{V}\mathrm{cm}$, while the velocity rises from $1.39\times 10^6$ to $6.7\times 10^6$, $9.1\times 10^6$ and $1.02\times 10^7$ $cm/s$ (computed). Holes saturate more softly, with $v_m=8.3\times 10^6$ $cm/s$ and $E_c=1.80\times 10^4$ $\mathrm{V}\mathrm{cm}$ (computed). In a 100 $\mathrm{nm}$ channel at 1 $\mathrm{V}$ the average field is $1\times 10^5$ $\mathrm{V}\mathrm{cm}$ (computed), so short-channel devices operate near saturation, which is why the driving-field choice matters. Reference: Allpix Squared collaboration, ‘Charge Carrier Mobility’, Allpix Squared documentation, CERN (models after Jacoboni et al. 1977, Canali et al. 1975, Masetti et al. 1983, Arora et al. 1982, the Hamburg model 2015, Ruch and Kino, and Quay et al.) (fetch-tool summary).

Ques: What is the saturation velocity of electrons in silicon?

About $1.0\times 10^7$ $cm/s$ at room temperature, with sources spreading by 7 per cent. Allpix’s model set gives $1.07\times 10^7$ $cm/s$ from Jacoboni et al., $1.00\times 10^7$ from Canali et al., which changes only the prefactor from $1.53\times 10^9$ to $1.43\times 10^9$, $1.03\times 10^7$ from the Hamburg model and $1.02\times 10^7$ from Quay et al. (computed at 300 $\mathrm{K}$). The spread reflects different samples and fitting ranges rather than physics (my reading), and $1.0\times 10^7$ $cm/s$ is the usual design value (background knowledge). The velocity falls with temperature because phonon emission increases: Quay’s law with A = 0.74 gives $8.2\times 10^6$ $cm/s$ at 400 $\mathrm{K}$, 20 per cent lower (computed), and holes saturate at about $0.72\times 10^7$ $cm/s$ with a weaker temperature dependence, A = 0.37. Reference: Allpix Squared collaboration, ‘Charge Carrier Mobility’, Allpix Squared documentation, CERN (models after Jacoboni et al. 1977, Canali et al. 1975, Masetti et al. 1983, Arora et al. 1982, the Hamburg model 2015, Ruch and Kino, and Quay et al.) (fetch-tool summary).

References

1. Allpix Squared collaboration, “Charge Carrier Mobility,” Allpix Squared documentation, CERN. Available: https://allpix-squared.docs.cern.ch/docs/06_models/02_carrier_mobility
2. nextnano GmbH, “mobility_masetti,” nextnano++ database documentation. Available: https://nextnano.com/nextnanoplus/software_documentation/database/mobility_masetti.htm
3. A. Bortolossi, “3D Finite Element Drift-Diffusion Simulation of Semiconductor Devices,” Laurea Magistrale thesis, Politecnico di Milano, 2014. Available: https://www.politesi.polimi.it/retrieve/a81cb05a-eff6-616b-e053-1605fe0a889a/2014_7_Bortolossi.pdf
4. COMSOL, “Lombardi Surface Mobility,” Semiconductor Module application model documentation, version 6.4. Available: https://doc.comsol.com/6.4/doc/com.comsol.help.models.semicond.lombardi_surface_mobility/models.semicond.lombardi_surface_mobility.pdf
5. COMSOL, “Mobility Models,” Semiconductor Module User’s Guide, version 6.4. Available: https://doc.comsol.com/6.4/doc/com.comsol.help.semicond/semicond_ug_semiconductor.6.57.html

Related pages in this vault: Generation-Recombination Models, Temperature Scaling, Band-Gap Narrowing and Quantum Corrections, Drift-Diffusion Device Equations, Hydrodynamic and Energy-Balance Transport, Hall Effect System, Spreading Resistance Profiler (SRP), Four-Point Probe, Thermal and Electro-Thermal Simulation.