From Semiconductor Physics to Today's Information Technology R. Wördenweber SS 2015, Tuesday 12:00-13.30 Seminarraum des II. Physikalischen Instituts Energy Bands II & from Diodes to CMOS Si crystal CNT 28.4.2015 organic molecule e-mail: r.woerdenweber@fz-juelich.de Bloch-Theorem -- Boundary conditions • Periodic potential • Periodic boundary conditions Quasi free electrons ( x) ( x Na) eikNa ( x) ei 2n ( x) with n=1, 2, 3, …. • Kronig-Penney model 1D Strongly bounded electrons Kronig-Penney model Kronig-Penney Model -> Band theory Solution: f1() 1 2 sin o a sinh ob 1 cos o a cosh ob 1 cos k (a b) 2 (1 ) f2() 1 2 sin o a sin ob 1 cos o a cos ob 1 cos k (a b) 2 ( 1) [-1,1] 1.005 1.000 2 f() E<U f1() 0.995 0.990 2.10 2.15 2.20 = E/U k=0 f() 1 1 3 0 2 4 5 k=/(a+b) -1 E > Uo f2() -2 0 1 2 3 4 = E/U 5 6 7 oa=2 ob=3.14 0 E Uo E Uo Band theory & Brillouin-Zone E<U 2 4 f() 1 3 1 3 0 2 4 5 -1 2 1 Band theory: Energy band in crystal E > Uo -2 0 1 2 3 4 5 6 7 = E/U Reduced-zone representation 1. Brillouin-Zone: primitive Wigner-Seitz-Cell of the reciprocal lattice of the single crystal Extended-zone representation Band theory- Dynamic description: Particle velocity, Phase velocity, effective Mass Phase velocity: rate, at which the phase of the wave propagates in space (= velocity of the wave front) vp Group velocity: the velocity with which the overall shape of the waves' amplitudes — known as the modulation or envelope of the wave — propagates through space. vg k f d 1 dE dk dk center frequency: = E/ħ The effective Mass is defined as the analogy to the 2. Newton’s Law (F=ma). The quantummechanic description of electrons of the crystal in external electric field E gives the equation of motion: F m Compare to the classic (Newton‘s) mechanics * dvg dt with with m* effective Mass 1 1 d 2E 2 dk 2 Band theory- Dynamic description: Particle velocity, Phase velocity, effective Mass • For the free particle the Dispersion relation is quadratic E k2 => M*=const. (=me) m* < 0 • In the crystal the situation is more complex: The dispersion relation is in general not quadratic => the effective Mass depends on the velocity => ħk = <p> is the crystal momentum (Kristallimpuls) m* > 0 • Important for semiconductors: - Minimum of the conductivity band (curvature is positive => m* >0) - Maximum of the valence band (curvature is negative => m* < 0) m* > 0 Reflection of electrons at the ‚ion-frames‘: Far away from the Band edge the effective mass can also become negative or infinite, analog to the Bragg-Reflection (2dsinθ = nλ) in the 1D lattice: for small values of k, the electrons move according to their free mass me for greater values of k they will be reflected, until no effective acceleration due to the electric filed is possible: m* -> for more greater k-values the acceleration due to the external electric field leads to the acceleration in the opposite direction: m* < 0 m* < 0 Effective Mass: m* 1 2 d E 2 dk 2 1 Band theory - Remarks Charge carriers and current: - k-values: E N per Band (2 spin states -> 2N states (conditions)) Distance: k 2 N (a b) - completely full or empty energy levels do not contribute to the charge transfer Conductance band Valence band - The contribution of the almost full band (valence band) is equivalent to the transport of positively charged charge carriers (holes with mass m*) (Alternatively one could take the negative mass, but this will make the calculation (e.g. band borders) more complex) E-k Diagrams GaAs Si Ge a) direct / indirect transition b) Valence band: Conduction band - maxima at k=0 - 3 subbands: - 1st & 2nd are degenerated m* via energy-band curvature heavy-hole band & light-hole band - 3rd subband slightly reduced energy : split-off band - at k=0 shape and curvature of all subbands are essentially orientation independent Valence bands kz x 1st Brillouin zone fcc structure (diamond, zinc blende, e.g.: Si, Ge, GaAs, …) L ky x L kx L E-k Diagrams Ge GaAs Si c) Conduction band: - similar features for Ge, Si, GaAs - subbands with local and absolute minima: Ge: - mimimum along <111> - indirect - 8 equivalent minima Si: - at k=0.8(2/a) towards <100> - indirect - 6-fold symmetry ( 6 equivalent minima) GaAs: - minimum at zone center - direct - local minimum at <111> only 0.29eV higher can not be ignored for larger fields (e.g. Gunn-effect diode) Conduction band Valence bands GaAs Lecture 4: Energy Band Theory II - Fermi-Dirac statistics - band bending - doping Semiconductor Devices - Diode - different types and applications - basic principle of pn-diodes - fabrication - optical applications - FET to CMOS Fermi-Dirac statistics (1926): Average occupation of a system by Fermions (Pauli Exclusion Principle) at given energy: f E 1 1 e E f E EF kT 1 and kT with 1 Enrico Fermi (1901-1954) f(E) 1 e( E EF ) / kT E – EF (eV) Paul Dirac (1902–1984) Fermi-Dirac Statistics and Fermi Level: Equilibrium distribution of carriers: g c E f E Electrons: g v E 1 f E Holes: E EC E E 1-f(E) Electrons EC gc(E) gc(E)f(E) EF EV gv(E) EV f(E) Energy-bands conduction band gv(E)(1-f(E)) Holes Density of states Occupation factor Carrier distribution (Zustandsdichten) (Verteilungsfunktionen) (Besetzungsverteilung) valence band Equilibrium Distribution of Carriers: Energy band diagram Density of states Occupation factor Electrons: g c E f E Holes: g v E 1 f E EF above midgap Variation of Fermi level occupation ratio holes/electrons EF near midgap EF below midgap Carrier distribution from intrinsic charge carrier concentration to small for applications : e.g. Ge at 300K: ni 2.9 x 1013 cm-3 Si at 300K: ni 1.5 x 1010 cm-3 GaAs at 300K: ni 2.3 x 106 cm-3 (necessary: ni > 1015 cm-3 ) Doping with electric active defects (atoms) Donor Acceptor Additional: free electrons in conductance band free holes in valence band e.g. for Si: atom with higher valence e.g. P or As atom with lower valence e.g. B or Al typical doping density: ND > 1015 cm-3 Intrinsic carrier concentration [cm-3] Intrinsic charge carrier concentration: T [K] Doping: n-dopted Si e.g. P or As E Doping with electric active defects atoms with higher/lower valence x Donor p-dopted Si B or Al E x Acceptor Doping: Ionization energies of impurities in Ge, Si, GaAs: Ge Donor 0.66 eV Acceptor Si 1.12 GaAs 1.42 Doping elements S.M. Sze, Physics of Semiconductor Devices, 1967 Doping: n-doped Si ND = 1015/cm3 carrier concentration in doped semiconductors: Freeze out Extrinsic T-region n / ND ND :=density of donors Intrinsic T-region n / ND Temperature [K] Doping: - Fermi level doped Si ‚relative‘ Fermi level: ED EF – Ei [eV] Ec Conduction band Ei Intrinsic Fermi level P n-type p-type B ND Ev EA Temperature [K] experiment theory Valence band Fermi function and Fermi Level: ‚Band bending‘: E(x) : electrostatic potential valence band E x q (x) Electric field: (1D): E d E dx Charge density : (Poisson equation) conduction band dE dx E Electrostatic potential Electric field Charge density Fermi function and Fermi Level: ‚Band bending‘: Example: pn-junction p-type n-type Ei := intrinsic Fermi level (without defects and doping) EF := Fermi level (with doping) pn-junction Remark: Ei roughly in the middle between Ev and Ec (in equilibrium same number of electrons and holes) Position shifted by doping! E Electric field Charge density Lecture 4: Energy Band Theory II - Fermi-Dirac statistics - doping - band bending Semiconductor Devices - Diode - different types and applications - basic principle of pn-diodes - fabrication - optical applications - FET to CMOS Diodes: The word Diode comes from (gr.) δίοδος (díodos) - „Pass“, „Way“ Mechanical analog: back-pressure valve (Rückschlagventil) closed open History: 1874 First semiconductor component: rectifier (Gleichrichter) diode, Metal needle made of PbS by Ferdinand Braun 1906 G.W. Pickard’s Patent for crystal detector (rectifier diode, also called the Cat’s-whisker) Semiconductor (PbS) Diodes: The IV-curve of an ideal diode: Applications: - rectifiers ( Forward current) - stabilizators & limiters ( reverse voltage , Zener Diode) - High-frequency signal detection - Optical applications - … Types of diodes pn-diodes (e.g. rectifiers) PIN-Diodes (positive-intrinsic-negative diode, highfrequency applications) Varactors (p-i-n diodes, with controlled barriers) p Zener-Diodes (stabilization, limitation) n Schottky-Diodes (metal-semiconductor junction, fast rectifier) Gunn-diode (negative differential resistance NDR, microwave generation) p i n Tunnel-diode (high-frequency applications, Phonondetektor, -emitter) … Light diodes (LED), Laser diodes Photodiode OLED …. Solarcells LED Laser diode Fermi-Energy in Metals Separate metals Metals in contact Electron-transfer Fermi-Energy is equalized Example: Thermocouple ‚Band bending‘ pn – Diodes: Fermi-Energy in Semiconductors Fermi-Energy is equalized Distance EL-EF is predefined in volume pn – Diodes: Formation of the depletion zone Electrons Holes (Akzeptor-Rümpfe) (Verarmungszone) (Donor-Rümpfe) pn – junction: Space-charge region pn – junction: Currents Drift current jR(L) = Diffuse current jF(L) Conductance band: a) Diffuse- or Recombination current jF(L): - forward current - driven with the concentration gradient - against electric field - Recombination after time in p-region b) Drift-, Field- or Generation current jR(L) - reverse current - Thermal generation - Minor charge transfer p Diffuse current jF(V) n = Drift current jR(V) Valence band: (analog, but: movement of holes to higher energies is energetically favorable!) a) Diffuse current jF(V) b) Drift current jR(V) Drift- and Diffuse currents compensate each other! pn – junction: Field-, Potential and Band Structure Precondition: - thermodynamic balance (no external fields or temperature gradients) Schottky-approximation: SCZ - in the depletion zone (SCZ) all Donors and Acceptors are ionized - Transition of the charge density from the SCZ to the material’s volume is stepwise (stufenförmig) - out of the SCZ the total charge is = 0 0 for x < -wp = -eNA for -wp < x < 0 eND for 0 < x < -wp 0 for x > wn - out of the SCZ the semiconductor has no field : E(x) = 0 for x < -wp x > wn - out of the SCZ EL and EV are defined in the volume through the doping -wp 0 wn pn – junction: Poisson equation Electric field: Parallel plate capacitor Q E F Q – charge F – surface Addition of the plate with the thickness 𝛥x with the charge 𝛥Q: Addition of the field: Density of the charge carriers: Transfer to the continuum: E ( x) 1 o r ( x)dx pn – junction: Electric field E ( x) 1 o r ( x)dx Qualitative: (x) = 0 E(x) = constant = 0 (x) = const. E(x) linear in x The solution of the Poisson equation for Integration constant E(x) Boundary conditions: -wp 0 E(-0)=E(0) Analog for n: wn wp N A wn N D pn – junction: Schematic representation pn-diode Holes or Electrons concentration Charge carriers density Electric Field Electric Potential pn – junction: Field or Drift-current Dynamic equilibrium (schematically) Drift-current Diffuse-current Diffuse-current Drift-current pn – junction: (dynamic equilibrium) Diffuse-current (through the concentration gradient) dn Particle current Concentration gradient: j ( x) D dx j diff j diff n j diff p dn p dnn e Dn Dp dx dx Drift-current (through the el. field in SCZ) j drift jndrift j pdrift enn n n p p E Mobility Equillibrium: n j diff j drift 0 no external current Charge carrier are thermally injected in the SCZ SCZ pn – junction: with bias voltage Current ballance of Electrons: (analog for holes) p n: Drift-current: Electrons in p-SC are generated and transferred to the n-SC Limitation: thermal generation I Gen I drift n p: Diffuse-current: Does not depend on the potential Electrons from the n-SC diffuse over the potential barrier into the p-SC Limitation: Potential barrier VD-U I diff exp eVD U / kT - only a part travels over the barrier - Idiff depends on the applied voltage U! pn – junction: with bias voltage Diffuse current at the applied potential U: I ndiff (U ) Ae eVD U / kT Ae eVD / kT eeU / kT I ndiff (0)eeU / kT Diffuse current at U=0: I ndiff (0) Ae eVD / kT I ndrift (0) Drift-current (approx.) independent on U: (possible because of the thermal generation) I ndrift (U ) I ndrift (0) I ndiff (U ) I ndrifteeU / kT Total current of electrons: I n I ndiff I ndrift I ndrift eeU / kT 1 Total current of electrons and holes: (Holes analog to electrons) I ges I ndrift I pdrift eeU / kT 1 I gen eeU / kT 1 pn – junction: with bias voltage Diffuse current Drift-current I ges I gen eeU / kT 1 Pass direction 12 normalized current I/Igen 10 Block direction 8 6 4 2 0 -2 -10 -8 -6 -4 -2 0 2 4 6 8 normalized voltage U/(kT/e) 10 pn – junction: Production Doping of the metal Diffusing of the doped materials and structuring Structuring and then diffusing of the doped materials Structuring and Implantation Doping atoms have to plug into the crystal structure and be electrically active! pn-diodes for optical applications: - optical sensors (light detector/meter, position sensor, …. CCD camera) - solar cells - light emitting diode (LED) - standard LED LED - Laser diode (LLED) similar to LED electronically pumped laser - organic light-emitting diode (OLED) (Light Emitting Polymer (LEP) or Organic ElectroLuminescence (OEL) - ‘organic LED’ (polymer + transparent cond. (ITO) - flat, bendable, cheap - to be used for: monitors and screens (inst. of LCD or Plasma) electronic paper novel illumination Laser diode OLED SCZ pn-diodes for optical applications: Forward bias leads to: - recombination of electrons and holes - indirect transition: phonon assisted e.g., Si-diode - direct transition: emission of photon e.g., GaAs, GaP (green) wavelength: - direct transition - photon h c 1240nm EG EG [eV ] Semiconductor: 1eV < EG < 3 eV - indirect transition - phonon assisted = 400-1200nm (visible light: 400-700nm) pn-diodes for optical applications: Forward bias leads to: - recombination of electrons and holes Ge Si GaAs - indirect transition: phonon assisted e.g., Si-diode - direct transition: emission of photon e.g., GaAs, GaP (green) wavelength: h c 1240nm EG EG [eV ] Semiconductor: 1eV < EG < 3 eV = 400-1200nm (visible light: 400-700nm) pn-diodes for optical applications: h c 1240nm EG EG [eV ] Examples: GaAlAs : red (650nm) to IR (up to 1000 nm) GaAsP, AlInGaP: red, orange, yellow GaP: green SiC: 1st comm. blue LED (low efficiency) ZnSe: blue (no commercial use) InGaN/GaN: UV, blue and green CuPb: near IR (NIR) White LED: - combination of different colors - blue LED + luminescent converter (P layer) Engineering of material properties Spectra of red, green, blue, white LED blue LED (InGaN) pn-diodes for optical applications: Examples of application Advantages: - reduced power consumption - improved visibility - longer life time - reduced size - Integration - thin (flexible) foils (OLED) - on chip (optical data transfer) - …. LED, LLED, OLED are object to intense research