# First Time User Guide to Carrier Statistics Lab on nanohub.org Ver. 2

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1 Network for Computational Nanotechnology (NCN) UC Berkeley, Univ.of Illinois, Norfolk State, Northwestern, Purdue, UTEP First Time User Guide to Carrier Statistics Lab on nanohub.org Ver. 2,Saumitra Raj Mehrotra, Gerhard Klimeck, and Ben Haley University West Lafayette, IN 4796, USA

2 Topics discussed Types of particles :» Fermions» Bosons Primer on Semiconductors & doping. Distribution function for electrons & holes. What is Carrier Statistics Lab?» Inputs / Ouputs» Capabilities What happens when you hit simulate? Types of Temperature sweeps. Study of carrier concentration variation» with temperature variation at fixed doping. Limitations of the tool. Few words about the tool. References.

3 Types of particles FERMIONS Each particle occupy only one state. They obey Pauli s Exclusion* principle. Their distribution is governed by Fermi-Dirac Statistics (explained later). They have half-integer spin. S =n/2, where n is integer. E.g.: Electrons, neutrons Two types of particles exist Energy States BOSONS Any number of particles can occupy a state. Do not obey Pauli s Exclusion* principle. Their distribution is governed by Bose-Einstein Relation. They have integer spin. S = n, where n is integer. E.g.: Photons, Phonons. *Pauli s Exclusion : No two fermions can occupy the same quantum state at the same time.

4 n-type ( Phosphorus or Grp 5 doped) Ec Primer on Semiconductors & Doping intrinsic (No doping) Ec p-type (Boron or Grp 3 doped) Energy Eg Ei Ef Ev Ei Ef Ev Ei Ec = Conduction Band, Ev = Valence Band, Ei = Intrinsic level =.5(Ec+Ev), Ef = Fermi level., Eg = Ec- Ev is the Band Gap of Semiconductor, like Eg(Si) = 1.12eV At T = K, Energy states Ef are completely FILLED. States Ef are completely EMPTY. At T>K, Electrons jump from Ev to Ec, leaving empty states in Ev called HOLES (p), creating a few occupied states in Ec (ELECTRONS (n)) Electron & Hole (Fermions) distribution obtained using Fermi-Dirac or Maxwell-Boltzmann (under special case, discussed in next slide) If Ef = Ei then n = p ; INTRINSIC SEMICONDUCTOR If Ef closer to Ev then p > n; P-TYPE SEMICONDUCTOR If Ef closer to Ec then n > p, N-TYPE SEMICONDUCTOR If (Ec-Ef) or (Ev-Ef) >3KT non-degenerate doping ; else ; degenerate doping

5 FERMI DIRAC (FD) DISTRIBUTION Electron & Hole distribution function Two types of distribution functions (for electrons) Efs = Fermi level = here K B = Boltzmann constant MAXWELL-BOLTZMANN (MB) DISTRIBUTION f FD = 1/[1+exp([E-E fs ]/K B T)] Provides occupation density of fermions at all temperature & doping. Like Step function at low temp (T < 4K) Spreads out as T increases. f MB = exp([e- E fs ]/K B T)] Provides occupation density of particles when T is high enough and density is low (nondegenerate doping). For Ec-Efs >> KT; FD becomes MB. Spreads out as T increases., for Images

6 Carrier Statistics Lab:» A MATLAB based tool. What is Carrier Statistics Lab?» Provides electron and hole distribution in common semiconductors.» Temperature dependent Fermi-level and electron, hole and ionized dopant calculation.» Allows Fermi-Dirac and Maxwell-Boltzmann non-degenerate statistics to be used.» Tool developed at Purdue University Part of the teaching tools on nanohub.org. Developers :» / Purdue University.» Gerhard Klimeck / Purdue University.

7 [a] [c] [b] Inputs : Material and distribution model selection F FD = g i /(1+exp [(Ef Ei)/KT] ) F MB = g i exp [(Ef Ei)/KT] Two types of distribution available. Ef, Ei and gi are Fermi level, Energy level and degeneracy of energy band Ei. Select the fermi level method Material options available. Material decides the e- & h+ mass & in turn the m dos & Density Of States Under doping fermi level is calculated by setting total charge density to zero in semiconductor. (Charge neutrality) Images from Carrier Statistics Lab on nanohub.org

8 Inputs : Temperature and Energy Range Two options available for temperature : [1] Fixed temperature : Calculates distribution at single temperature. Set [Sweep the Temp] to NO and enter T value in K. [2] Temperature Sweep : Allows to see the variation in distribution function with temperature. Set [Sweep the Temp] to YES Allowed inputs : Range of energy within which distribution function is evaluated. Supply minimum & maximum values in ev Images from Carrier Statistics Lab on nanohub.org

9 What happens when you just hit SIMULATE? Runs with Default Inputs : [Distribu)on func)on]: Fermi-Dirac [Material] : Silicon [Fermi- level Selec)on]: User defined [Fermi level value]:.56 ev (wrt Ev) [Sweep Temperature] : No [Temperature] : 3 K [Min Energy]: -.4 ev [Max Energy]: 1.52 ev [1] [2] [4] [3] Default outputs Images from Carrier Statistics Lab on nanohub.org

10 Constant Doping Na = 1e14cm-3, Nd = 1 cm-3. Maintaining constant doping; Efs is varying with temperature. Temperature Sweep Constant Fermi level Ev Efs =.1 ev, Ev = ev Maintaining constant Efs; doping is varying with temperature. As T inc. distribution shifts higher since Efs is inc. with T( discussed later) T inc As T inc. distribution shifts higher around same Energy point since Efs is constant. Efs =.1eV Exercise : Perform similar study with Maxwell Boltzmann function. Images from Carrier Statistics Lab on nanohub.org

11 Temperature Sweep : constant doping Input setting : [1] [Fermi Level Selection] = Dope the semiconductor. [2] Set the doping values for Na = 1e14 cm -3 & Nd = 1cm -3. [3] [Sweep the temperature] = yes [4] Set [Min Temp ] = 5K ; [Max Temp] = 6K, [Steps] = 15, Material = Si. Energy(eV) Freeze-out Intrinsic Extrinsic Temperature (K) High temperature fully -Ionized Increasing Ionization of Na with temperature Low temperature less -Ionized Temperature (K) T<12K: Dopants not fully ionized. p = Na - [ < Na ] (Freeze out) 12K<T<44K: Dopants completely ionized. p = Na (Na - =Na) (Extrinsic) T > 44K: n = p = ni (p > Na) (Intrinsic) Images from Carrier Statistics Lab on nanohub.org

12 Temperature Sweep : constant doping contd. Mobile carrier density Intrinsic carrier (ni) density p Intrinsic n Extrinsic ni = T = 3K Freeze-out Temperature Temperature n,p and ni concentration vary with temperature. p = Na - ; n = ni 2 /Na - Partial Ionization of dopants (Freeze out) p = Na ; n = ni 2 /Na Complete Ionization of dopants (Extrinsic) n=p=ni High T more intrinsic carrier generated to maintain charge neutrality(intrinsic) Exercise : Perform similar study with Maxwell Boltzmann distribution and other materials. Images from Carrier Statistics Lab on nanohub.org

13 Limitations of the tool The tool cannot handle degenerate carrier statistics.» Fails for heavy doping (Na or Nd > 1e2 cm -3 ). Cannot simulate Bose-Einstein distribution. Results can be error prone for very small temperature for (T < 4K) Ea and Ed (acceptor and dopant energy levels) are assumed to be 5meV from the bandedge. Always keep checking the tool web-page for latest features, releases and bug-fixes at :

14 Few words regarding the tool Use this tool to learn about electron-hole distribution in semiconductors. Feel free to post about (on tool webpage)» the bugs» new features you want. Contact the developers in case you want to collaborate for some work using this tool. Suggested Exercises : [1] Perform a study on variation of electron and hole concentration with temperature at : (a) fixed Fermi level. (b) using Maxwell Boltzmann distribution. [2] find out the doping limit at T=3K where MB and FD give same result for electron and hole concentration.

15 Material Parameters used in the tool* Si: m m E electron hole g ( m ( m ) ( ev ) = 1.17 ) = = T T T T T 7 2 T 2 Appendix Valence band= ev for all materials Ge m m E electron hole g ( m ( ev ) ( m ) = ) = = T T GaAs electron m ( m m E hole g ( m ) ) = = ( ev ) = Ref*: Semiconductor Device Fundamentals, Robert Pierret, Addison-Wesley T T

16 Appendix Other formulas/quantities: Electron and hole density calculation Nc = 2. π. m 2. electron 2 ( 2. π. ). kt 3/ 2 n = Nc. e ( E F E c )/ kt Nv = 2. π. m 2. hole ( 2. π. ). kt 2 3/ 2 p = Nv. e ( E V E F )/ kt Acceptor (above VB) and Donor (below CB) energy levels E E a d = 5meV = 5meV

17 Appendix Dopant ionization and charge neutrality condition: N N + D A p + N Nd = [ ( Ed E f )/ kt ] e Na = [ ( Ed E f )/ kt ] e + D = n + N A

18 URLs on distribution functions :» References [1]» (very good applets on semiconductors.) Books & Notes:» Physics of Semiconductor Devices, S. M. Sze. New York: Wiley, 1969, ISBN ; 2nd ed., 1981, ISBN ; 3rd ed., with Kwok K. Ng, 26, ISBN » Semiconductor Device Fundamentals, Robert Pierret, Addison- Wesley. ISBN-1: ISBN-13: » Raseong Kim; Mark Lundstrom (28), "Notes on Fermi-Dirac Integrals (3rd Edition)",

19 References [2] Homework assignment using the tool.» ; Saumitra Raj Mehrotra; Gerhard Klimeck (28), "Homework Exercise on Fermi-Dirac and Maxwell-Boltzmann Distributions, Link for the tool :»

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