CHAPTER 4 – THE SEMICONDUCTOR IN EQUILIBRIUM

1 PHYSICAL ELECTRONICS(ECE3540)Brook Abegaz, Tennessee Technological University, Fall 2013 Friday, September 20, 2013 Tennessee Technological University1 CHAPTER 4 THE SEMICONDUCTOR IN EQUILIBRIUMC hapter 4 The SEMICONDUCTOR in EQUILIBRIUM CHAPTER 3: considering a general crystal andapplying to it the concepts of quantum mechanicsin order to determine a few of the characteristics ofelectrons in a single-crystal lattice. CHAPTER 4:apply these concepts specifically to asemiconductor material. CHAPTER 4:use thedensity of quantum statesin theconduction bandand thedensity of quantum statesin thevalence bandalong with theFer mi-Diracprobability functionto determine theconcentrationof electrons and holesin the conduction andvalence bandsFriday, September 20, 2013 Tennessee Technological University2 EQUILIBRIUM , or thermal EQUILIBRIUM , implies thatno externalforcessuch as voltages, electric fields, magnetic fields, ortemperature gradients are acting on the SEMICONDUCTOR .

2 All properties of the SEMICONDUCTOR will beindependent oftimein this case. Anintrinsic SEMICONDUCTOR = a pure crystal with no impurityatoms or defects. The electrical properties of an intrinsic SEMICONDUCTOR can bealtered in desirable ways by adding controlled amounts ofspecific impurity atoms, calleddopant atoms,to the crystal, thuscreating anextrinsic SEMICONDUCTOR . Adding dopant atoms changes the distribution of electronsamongtheavailableenergystates,s otheFer mi energ ybecomes a function ofthe type and concentration of , September 20, 2013 Tennessee Technological University3 CHAPTER 4 The SEMICONDUCTOR in EquilibriumFriday, September 20, 2013 Tennessee Technological University4 CHAPTER 4 The SEMICONDUCTOR in EquilibriumFig.

3 1: Thermal Excitation caused jumping of an electronto CB. There is a corresponding holecreated in VBwhere the electron was located. Current is the rate at which charge flows. In a SEMICONDUCTOR ,two types of chargecarriers,the electronandthe hole,cancontribute to a current. Since thecurrentinasemiconductorisdetermined largely by the number ofelectronsin the conduction band and the number ofholes in the valence band ,animportantcharacteristics is thedensity of these , September 20, 2013 Tennessee Technological University5 Charge Carriers in SemiconducductorsEquilibrium Distribution of Electrons and Holes The distribution (with respect to energy ) ofelectrons in the conduction band is given bythedensity of allowed quantum statestimestheprobability that a state is occupied by an ( ) where fF(E) is the Fermi-Dirac probability functionand gc(E)

4 Is the density of quantum states in theconduction band . The total electron concentration per unit volume intheconductionbandisthenfoundbyintegrat ingEquation ( ) over the entire , September 20, 2013 Tennessee Technological University6 (E)f (E)g =n(E)FcEquilibrium Distribution of Electrons and Holes The distribution (with respect to energy ) ofholes in the valence bend is the density ofallowed quantum states in the valence handmultiplied by the probability that a state is notoccupied by an ( ) The total hole concentration per unit volume is found by integrating this function over the entire valence- band , September 20, 2013 Tennessee Technological University7(E)] f -(E)

5 [1g = p(E)FvEquilibrium Distribution of Electrons and Holes An ideal intrinsic SEMICONDUCTOR is a puresemiconductor with no impurity atoms and nolattice defects in the crystal ( pure Silicon). For an intrinsic SEMICONDUCTOR at T = 0K, allenergy states in the valence band are filled withelectrons and all energy states in the conductionband are empty of electrons. TheFermienergymust,therefore,besomewhere betweenEcand Ev(The Fermienergydoes not need to correspond to an allowedenergy.)Friday, September 20, 2013 Tennessee Technological University8 The noand poEquations Theequationforthethermal-equilibriumconc entration of electrons may be found byintegrating Equation ( ) over the conduction bandenergy, as:eq.]

6 ( ) Applying the Boltzmann approximation to theFermi energy calculation, the thermal-equilibriumdensity of electrons in the conduction band is:eq. ( )Friday, September 20, 2013 Tennessee Technological University9 dEEfEgnFc)()(0dEkTEEEEhmnFcEnc])(exp[)() 2(432/3*0 The noand poEquations Solving the integral, substituteNcas theeffective density ofstates function in the conduction ( ) If m* = mo, then the value of the effective density ofstates function at T = 300 K is Nc= ,which is the value of Ncfor most semiconductors . If theeffective mass of the electronislarger or smaller thanmo,thenthe value of the effective density of states functionchanges accordingly, but is still of the same order ofmagnitude.

7 Thethermal- EQUILIBRIUM electron concentrationin theconduction band is:eq. ( )Friday, September 20, 2013 Tennessee Technological University102/32*)2(2hkTmNnc ])(exp[0kTEENnFcc The noand poEquations The thermal- EQUILIBRIUM concentration of holes inthe valence band is found by integrating Equation( ) over the valence band energy as:eq. ( )eq. ( ) Defining theeffective density of states function in thevalence band :eq. ( ) The thermal- EQUILIBRIUM concentration of holes inthe valence band is then:eq. ( )Friday, September 20, 2013 Tennessee Technological University11])(exp[0kTEENpvFv 2/32*)2(2hkTmNpv dEEfEgpFv)](1)[(0)exp(11)(1kTEEEfFF The noand poEquations Theeffective density of states functions,Ncand Nvareconstantfor a given SEMICONDUCTOR material at afixedtemperature.

8 The values of the density of states function and of theeffective masses forSilicon,Gallium Arsenide,andGermaniumare: The thermal EQUILIBRIUM concentrations of electrons inthe conduction band and of holes in the valence bandare directly related to the effective density of statesconstants and to the Fermi energy , September 20, 2013 Tennessee Technological University12 Table Effective Density of States Function and Effective Mass ValuesNc(cm 3)Nv(cm 3)mn*/m0mp* * * * * * * Intrinsic Carrier Concentration Foranintrinsicsemiconductor,theconcentra tion of electrons in the conductionband(n) is equal to theconcentration of holesin the valence band (p).

9 These parameters are usually referred to as theintrinsic electron concentration(ni)andintrinsichole concentration(pi). TheFermi energy level for the intrinsicsemiconductoris calledthe intrinsic Fermienergy,orEf= , September 20, 2013 Tennessee Technological University13 The Intrinsic Carrier Concentration For an intrinsic SEMICONDUCTOR :eq. ( )eq. ( )eq. ( )eq. ( )Friday, September 20, 2013 Tennessee Technological University14])(exp[0kTEENnnFicci ])(exp[0kTEEN nppvFivii ])(exp[.])(exp[2kTEEkTEENNnvFiFicvci ])(exp[])(exp[2kTENNkTEENN ngvcvcvci The Intrinsic Carrier Concentration For a given SEMICONDUCTOR material at a constanttemperature, the value ofniis a constant, andindependent of the Fermi , September 20, 2013 Tennessee Technological University15 Table Commonly Accepted Va l u e s o f niat T=300 Kni(cm-3) *1010 Gallium * *1013 Fig.

10 2: Intrinsic carrier concentration niwith respect to change in of Intrinsic semiconductors High Electron Mobility Transistor High resistivity substrate for RF circuits Amorphous Si Solar CellsFriday, September 20, 2013 Tennessee Technological University16 Fig. 3: Structure of a Solar nopoProduct Using the general expressions fornoandpo:eq. ( ) which is simplified as:eq. ( ) Thus,forasemiconductorinthermalequilibri um, theMass Action Lawstates:eq. ( )Friday, September 20, 2013 Tennessee Technological University17])(exp[])(exp[00kTEEkTEENNpn vFFcvc ])(exp[00kTENN pngvc 200inpn the Fermi energy is below theconduction band . The value of Ncfor Silicon at T=300 KisNc= Calculate theprobability that a state in the conduction band isoccupied by an electron and calculate the thermalequilibrium electron concentration in silicon at T=300 that the Fermi energy is above Nvfor Siliconat T = 300K is Nv= thermal EQUILIBRIUM hole concentration insilicon at T = 400 , September 20, 2013 Tennessee Technological probability that an energy state at E = Ecis occupied by an electron is given by : The electron concentration is given by.