Budapest University of Technology and Economics Department of Electron Devices Microelectronics, BSc course Basic semiconductor physics

Budapest University of Technology and Economics Department of Electron Devices Microelectronics BSc course, Basic semiconductor physics © András Poppe & Vladimír Székely, BME-EET Summary of the physics required ► Charge carriers in semiconductors ► Currents in semiconductors ► Generation, recombination; continuity equation

Budapest University of Technology and Economics Department of Electron Devices Microelectronics BSc course, Basic semiconductor physics © András Poppe & Vladimír Székely, BME-EET Energy bands ► Resulting from principles of quantum physics Discrete energy levels: The discrete (allowed) energy levels of an atom become energy bands in a crystal lattice More atoms – more energy levels of electrons: There are so many of them that they form energy bands:

Budapest University of Technology and Economics Department of Electron Devices Microelectronics BSc course, Basic semiconductor physics © András Poppe & Vladimír Székely, BME-EET conductance band valance band Valance band, conductance band ► Valance band – these electrons form the chemical bonds  almost full ► Conductance bend – electrons here can freely move  almost empty v – valance band / uppermost full c – conductance band / lowermost empty

Budapest University of Technology and Economics Department of Electron Devices Microelectronics BSc course, Basic semiconductor physics © András Poppe & Vladimír Székely, BME-EET Electrons and holes ► Generation: using the average thermal energy ► Electrons: in the bottom of the conduction band ► Holes: in the top of the valance band ► Both the electrons and the holes form the electrical current conduction band valance band recombination generation Electron: negative charge, positive mass Hole:positive charge, positive mass

Budapest University of Technology and Economics Department of Electron Devices Microelectronics BSc course, Basic semiconductor physics © András Poppe & Vladimír Székely, BME-EET Conductors and insulators semiconductor insulator metal forbidden gap band gap For Si: W g = 1.12 eV for SiO 2 : W g = 4.3 eV 1 eV = 0.16 aJ = 0.16  J

Budapest University of Technology and Economics Department of Electron Devices Microelectronics BSc course, Basic semiconductor physics © András Poppe & Vladimír Székely, BME-EET Band structure of semiconductors Valance band conduction band GaAs: direct band  opto-electronic devices (LEDs) Si: indirect band indirect direct

Budapest University of Technology and Economics Department of Electron Devices Microelectronics BSc course, Basic semiconductor physics © András Poppe & Vladimír Székely, BME-EET Generation / recombination ~~~~> = W g /h Direct recombination may result in light emission (LEDs) W g Light absorption results in generation (solar cells) Experiment Spontaneous process: thermal excitation – jump into the conduction band / recombination  equilibrium

Budapest University of Technology and Economics Department of Electron Devices Microelectronics BSc course, Basic semiconductor physics © András Poppe & Vladimír Székely, BME-EET Crystalline structure of Si ► SiN=14 4 bonds, IV-th column of the periodic table ► Diamond lattice, lattice constant a=0.543 nm ► Each atom has 4 nearest neighbor real 3D simplified 2D undoped or intrinsic semiconductor

Budapest University of Technology and Economics Department of Electron Devices Microelectronics BSc course, Basic semiconductor physics © András Poppe & Vladimír Székely, BME-EET valance dopant: donor (As, P, Sb) ► Electron: majority carrier ► Hole: minority carrier conduction band valance band n-type semiconductor

Budapest University of Technology and Economics Department of Electron Devices Microelectronics BSc course, Basic semiconductor physics © András Poppe & Vladimír Székely, BME-EET valance dopant: acceptor (B, Ga, In) ► Electron: minority carrier ► Hole: majority carrier conduction band valance band p-type semiconductor

Budapest University of Technology and Economics Department of Electron Devices Microelectronics BSc course, Basic semiconductor physics © András Poppe & Vladimír Székely, BME-EET Calculation of carrier concentration possible energy states occupation probability concentrations electrons holes FD statistics:

Budapest University of Technology and Economics Department of Electron Devices Microelectronics BSc course, Basic semiconductor physics © András Poppe & Vladimír Székely, BME-EET Calculation of carrier concentration ► The results is: ► If there is no doping: n = p = n i  it is called intrinsic material = W i W F : Fermi-level

Budapest University of Technology and Economics Department of Electron Devices Microelectronics BSc course, Basic semiconductor physics © András Poppe & Vladimír Székely, BME-EET The Fermi-level ► Formal definition of the Fermi-level: the energy level where the probability of occupancy is 0.5: ► In case of intrinsic semiconductor this is in the middle of the band gap: ► This is the intrinsic Fermi-level W i electrons holes

Budapest University of Technology and Economics Department of Electron Devices Microelectronics BSc course, Basic semiconductor physics © András Poppe & Vladimír Székely, BME-EET Carrier concentrations ► Depends on temperature only, does not depend on doping concentration For silicon at 300 K absolute temperature n i = /cm 3 (10 electrons in a cube of size 0.01 mm x 0.01 mm x 0.01 mm) Mass action law

Budapest University of Technology and Economics Department of Electron Devices Microelectronics BSc course, Basic semiconductor physics © András Poppe & Vladimír Székely, BME-EET Carrier concentrations ► Si, T = 300 K, donor concentration N D = /cm 3 ► What are the hole and electron concentrations?  Donor doping  n  N D = /cm 3  Hole concentration:p = n i 2 /n = /10 17 = 10 3 /cm 3 ► What is the relative density of the dopants?  1 cm 3 Si contains 5  atoms  thus, / 5  = 2   The purity of doped Si is Problem

Budapest University of Technology and Economics Department of Electron Devices Microelectronics BSc course, Basic semiconductor physics © András Poppe & Vladimír Székely, BME-EET Carrier concentrations ► Just re-order the equations kT = 1.38  VAs/K  300 K =4,14  J = eV = 26 meV Thermal energy In doped semiconductors the Fermi-level is shifted wrt the intrinsic Fermi- level

Budapest University of Technology and Economics Department of Electron Devices Microelectronics BSc course, Basic semiconductor physics © András Poppe & Vladimír Székely, BME-EET Temperature dependence ni2 =ni2 = Problem ► How strong is it for Si?  15% / o C

Budapest University of Technology and Economics Department of Electron Devices Microelectronics BSc course, Basic semiconductor physics © András Poppe & Vladimír Székely, BME-EET Temperature dependence of carrier concentrations Problem Si, T = 300 K, donor dopant concentration N D = /cm 3 n  N D = /cm 3 p = n i 2 / n = /10 17 = 10 3 /cm 3  How do n and p change, if T is increased by 25 degrees? n  N D = /cm 3 – unchnaged n i 2 =  = 33   p = n i 2 / n = 33  /10 17 = 3.3  10 4 /cm 3 Only the minority carrier concentration increased!  T=16.5 o C  10 

Budapest University of Technology and Economics Department of Electron Devices Microelectronics BSc course, Basic semiconductor physics © András Poppe & Vladimír Székely, BME-EET Currents in semiconductors ► Drift current ► Diffusion current Not discussed:  currents due to temperature gradients  currents induced by magnetic fields  energy transport besides carrier transport  combined transport phenomena

Budapest University of Technology and Economics Department of Electron Devices Microelectronics BSc course, Basic semiconductor physics © András Poppe & Vladimír Székely, BME-EET Drift current No electrical field Thermal movement of electrons  = mobility m 2 /Vs There is E electrical field

Budapest University of Technology and Economics Department of Electron Devices Microelectronics BSc course, Basic semiconductor physics © András Poppe & Vladimír Székely, BME-EET Drift current  charge density v velocity (average) Differencial Ohm's law Specific electrical conductivity of the semiconductor Specific resistance

Budapest University of Technology and Economics Department of Electron Devices Microelectronics BSc course, Basic semiconductor physics © András Poppe & Vladimír Székely, BME-EET Carrier mobilities Si  n = 1500 cm 2 /Vs  p = 350 cm 2 /Vs Electric field [V/cm] Drift velocity [cm/s] electrons holes

Budapest University of Technology and Economics Department of Electron Devices Microelectronics BSc course, Basic semiconductor physics © András Poppe & Vladimír Székely, BME-EET Carrier mobilities ► Mobility decreases with increasing doping concentration ► At around room temperature mobilities decrease as temperature increases 300 K Si, holes  ~ T -3/2 Mobility

Budapest University of Technology and Economics Department of Electron Devices Microelectronics BSc course, Basic semiconductor physics © András Poppe & Vladimír Székely, BME-EET Diffusion current ► Reasons:  concentration difference (gradient)  thermal movement ► Proportional to the gradient ► D: diffusion constant [m 2 /s]

Budapest University of Technology and Economics Department of Electron Devices Microelectronics BSc course, Basic semiconductor physics © András Poppe & Vladimír Székely, BME-EET Total currents Einstein's relationship Thermal voltage

Budapest University of Technology and Economics Department of Electron Devices Microelectronics BSc course, Basic semiconductor physics © András Poppe & Vladimír Székely, BME-EET Generation, recombination ► Life-time: average time an electron spends in the conduction band ► Generation rate: g [1/m 3 s] ► Recombination rate: r [1/m 3 s] conduction band valance band recombination generation  n,  p 1 ns … 1  s

Budapest University of Technology and Economics Department of Electron Devices Microelectronics BSc course, Basic semiconductor physics © András Poppe & Vladimír Székely, BME-EET  N electron  V volume A surface Continuity equation

Budapest University of Technology and Economics Department of Electron Devices Microelectronics BSc course, Basic semiconductor physics © András Poppe & Vladimír Székely, BME-EET Diffusion equation

Budapest University of Technology and Economics Department of Electron Devices Microelectronics BSc course, Basic semiconductor physics © András Poppe & Vladimír Székely, BME-EET Example for the solition of the diff. eq.: p diffusion length