Notes for ECE-606: Spring 03 Energy Bands & Carrier Densities Professor Mark Lundstrom Electrical and Computer Engineering Purdue University, West Lafayette, IN USA lundstro@purdue.edu /7/3 Key topics ) Energy band diagrams ) Equilibrium carrier densities 3) Equations of motion for a particle
Kroemer s lemma of proven ignorance Whenever I teach my semiconductor device physics course, one of the central messages I try to get across early is the importance of energy band diagrams. I often put this in the form of Kroemer s lemma of proven ignorance: If, in discussing a semiconductor problem, you cannot draw an Energy Band Diagram, this shows that you don t know what you are talking about. corollary: If you can draw one, but don t, then your audience won t know what you are talking about. Nobel Lecture, 000) 3 band bending? What happens when we apply a voltage to the gate? KE +V G E i V = 0 4
band bending E P.E. = ) = + ) qv ) P.E. = qv ) +V G E i V = 0 ) d d = q dv ) = qe d 5 band bending E E i ) = q E ref ) V ) E = d q d ) = d q d ) = de i q d ) 6
p), n), rho) E E i n ) = N C e ) k B T p ) = N V e ) k B T ) = q E ref ) V E = d q d ) ) = d q ) d ) ) de d = ρ ε ρ ) = ε de d = ε d q d = de i q d 7 Kroemer s lemma of proven ignorance Whenever I teach my semiconductor device physics course, one of the central messages I try to get across early is the importance of energy band diagrams. I often put this in the form of Kroemer s lemma of proven ignorance: If, in discussing a semiconductor problem, you cannot draw an Energy Band Diagram, this shows that you don t know what you are talking about. corollary: If you can draw one, but don t, then your audience won t know what you are talking about. Nobel Lecture, 000) 8
Key topics ) Energy band diagrams ) Equilibrium carrier densities 3) Equations of motion for a particle 9 carrier densities ) = f E ) k B T + e E D D D E) = m n * S = N C F 0 D π N D = m * k D B T C π n S = f 0 E) D D E)dE cm η F ) What happens at T = 0 K? 0
effect of temperature f E) k B T 0 T = 0 K f E) = ) k B T + e E T > T 0 0 T < T 0 E At T = 0 K ) = f E + e E ) k B T D D E) = m * D π n S = f 0 E) D D E)dE cm n S = m * D π N D C = ) * k B T π m D D n n S = D D E)dE cm S = N C η F E
law of mass action = E I = = n i hole = n i n i e E G k B T L n i 300K ) 0 0 cm -3 3 n-type semiconductor 3D) Epect: n0 = N C e ) k B T = N V e ) k B T >> n i =? 4
p-type semiconductor 3D) Epect: = N V e ) k B T holes = N C e ) k B T >> n i =? 5 any semiconductor in equilibrium = N C e ) k B T = N V e ) k B T hole = N C N V e E G k B T = n i n i = N C N V e E G k B T = n i 6
) Intrinsic: = = n i = 0 0 cm -3 ) N-type: eample = 0 7 cm -3 = 0 3 cm -3 Very curious! ) P-type: = 0 7 cm -3 = 0 3 cm -3 7 law of mass action R i = B = Bn i G i R i = E I G i = R i = Bn i hole 8
law of mass action R i = B = G i = Bn i = E I G R i i hole R can t change since G doesn t change, so as n increases, p must decrease so that the product is constant. 9 Carrier density vs. doping and temperature How do we compute this curve? Fig.. from R.F. Pierret, Semiconductor Device Fundamentals 0
space charge neutrality p n + N D + N A = 0 Need to know the ionized doping density any semiconductor in equilibrium ) = f E ) k B T + e E N D + N D = f E D ) = ) k B T? + e E D E D E V f E) = + e E ) k B T E A N A N A = f E ) = ) k B T? + e E A
statistics of donors/acceptors ) = f E ) k B T + e E N D + N D = ) k B T + g D e E D E D E V f E) = + e E ) k B T E A N A N A = ) k B T + g A e E A 3 Carrier density vs. doping and temperature p n + N D + N A = 0 N D + N D = N A N A = ) k B T + g D e E D ) k B T + g A e E A n = N C F / p = N V F / k B T k B T N V F / k B T N C F / k B T + + g D e E D ) k B T + g A e E A ) = 0 k B T 4
Carrier density vs. doping and temperature How do we compute this curve? How do we understand this curve? Fig.. from R.F. Pierret, Semiconductor Device Fundamentals 5 Carrier density vs. doping and temperature n p n i n N D n N D + Fig.. from R.F. Pierret, Semiconductor Device Fundamentals 6
carrier density vs. doping and temperature p n + N D + N A = 0 charge neutrality) p n + N D N A = 0 full ionization) np = n i n = N D N A + N D N A + n i / p = N A N D + N A N D + n i / p = n i n n = n i p 7 Key topics ) Energy band diagrams ) Equilibrium carrier densities 3) Equations of motion for a particle 8
motion in k-space / real space F e = d ) d k t) = k t 0) + q = d k ) = qe ) dt E t ) d t 0 υ g t) = E k t k ) r t) = r 0) + υ g t t )d t 0 equations of motion for semi-classical transport varies slowly on the scale of the electron s wavelength. no effective mass! 9