Carrier Statistics and State Distributions

Review 4 on Physical lectronics mportant Slide to watch without doing anything Carrier Statistics and State Distributions (Carriers, Fermi-Dirac Statistics in Solids, Fermi Level, Density of States, etc) Semiconductor ngineering by Prof Sungsik Lee

Carpark-Highway Analogy mpty lanes: Density of States for lectrons Conduction Band lectron transport Highway Band gap Road Gap Free (mobile) electrons one highway car one vacancy Free holes mpty places: Density of States for Holes Carpark Valence electrons Average Line Fermi level Valence Band Vacancy movement opposite to parked car s moving direction Hole transport Semiconductor ngineering by Prof Sungsik Lee

Top questions of this lecture What makes valence electrons excited into the? What is the Fermi level? How does the Density of States look like? How determines Free lectrons and Holes numbers? Are they the same each other all the time? What makes it different? And how does it relate to the Fermi level? Semiconductor ngineering by Prof Sungsik Lee

Free Carriers: Free lectrons and Holes What makes it free? For the xample semiconductor of Silicon: Temperature th = kt (at T = 0 K) = 0 ev th = kt (at T = 300 K) ~ 26m ev << g ~ 115 ev C V x g n 0 = 0 p 0 = 0 C V n 0 = 15x10 10 /cm 3 ~ 11 ev Fermi level Thermal ncreasing excitation T = 0 F T 0 Temperature Valence electrons p 0 = 15x10 10 /cm 3 1/10 12 1/10 12 Both cases at the thermal equilibrium No external effect (eg bias and Light) ntrinsic Semiconductor n 0 = p 0 F is exactly in the middle of g F Free electrons Free holes Free electrons notation: n 0 Free holes notation: p 0 Maximum n 0 ~ 5x10 22 /cm 3 Maximum p 0 ~ 5x10 22 /cm 3 Semiconductor ngineering by Prof Sungsik Lee

ntrinsic Semiconductor (n 0 = p 0 ) in Thermal quilibrium Carpark-Highway Analogy with Multi-roads (Band) For the xample semiconductor (bulk = 3D) of Silicon at T = 300 K: th = kt (at T = 300 K) ~ 26m ev << g ~ 115 ev Height n 0 = 15x10 10 /cm 3 Density of States for lectrons C C g c () Vacancies Vacancies Average Line Thermal excitation F F V g V () V p 0 = 15x10 10 /cm 3 Density of States for Holes Density of States (Vacancies)? How to determine the Carriers Number with F? Semiconductor ngineering by Prof Sungsik Lee

Physical Distribution of Vacancies (Rooms) for Thermally xcited Carriers: Density of States in nergy (3D Semiconductor) When T 0 C Density of States for lectrons C g c () V Thermal excitation F F V g V () 3D Semiconductor Density of States for Holes Semiconductor ngineering by Prof Sungsik Lee

Physical Distribution of Vacancies (Rooms) for Thermally xcited Carriers: Density of States in nergy (2D Semiconductor) When T 0 C V Thermal excitation F Density of States for lectrons C F V Density of States for Holes g c () g V () 2D Semiconductor Semiconductor ngineering by Prof Sungsik Lee

Physical Distribution of Vacancies (Rooms) for Thermally xcited Carriers: Density of States in nergy (1D Semiconductor) When T 0 Density of States for lectrons C C g c () V Thermal excitation F F V Density of States for Holes g V () 1D Semiconductor Semiconductor ngineering by Prof Sungsik Lee

Mathematical Derivation of 3D Density of States Supposing you have a symmetrical 3-D structure which is a sphere centered at the origin (0,0,0) in the k-space k x k z k 4π k 2 dk k y L : length in real space [m] v k : unit amount in k-space [rad 3 /m 3 ]! V a : total angular amount in real space [m 3 /rad 3 ] N total : total number of states! k : spatial frequency in the conduction band Volume for 3D Area for 2D Length for 1D 2 spins!! Differential elements! Semiconductor ngineering by Prof Sungsik Lee

Mathematical Derivation of 3D Density of States (continuous) k : spatial frequency in the conduction band = ( ) ħ!= ħ!= ħ! n u : number of states per unit volume [/cm 3 ] $ = = & ( )!= ħ ħ!=! (): Density of States [cm -3 ev -1 ] (): Density of States [cm -3 ev -1 ] ħ Same result as the Textbook Semiconductor ngineering by Prof Sungsik Lee

Summary of Key quations for DoS in for 3 different dimensions Parameters 3D (d=3) 2D (d=2) 1D (d=1) v k 4(2 / 12 [rad 3 /m 3 ] 2(2 12 [rad 2 /m 2 ] 2 12 [rad/m] V a = 36 2( 6 3 * 2( * [m 3 /rad 3 ] 2( / [m 2 /rad 2 ] 3 / 3 2( [m/rad] N total =2 36 2( 67 8 2 3* 2( *4(2/ 12 2 3/ 2( /2(2 12 2 3 2( 2 12 k [rad/m] 2+, (- - ) ħ / 2+, (- - ) ħ / 2+, (- - ) ħ / dk = f(d) 12 = +, 1 2ħ / 1- - - 12 = +, 1 2ħ / 1- - - 12 = +, 1 2ħ / 1- - - n u = 9 :;:<= 3 6 1 8( 2+, (- - ) +, 2( * ħ / 2ħ / 1- - - [cm -3 ] 4( 2+, (- - ) 2( / ħ / +, 2ħ / 1 1- - - [cm -2 ] 4 2( +, 2ħ / 1 1- - - [cm -1 ] = >? 1- ħ [cm -3 ev -1 ] ħ [cm -2 ev -1 ] ħ 5 [cm -1 ev -1 ] Derivation of DoS for 2D or 1D Midterm xam? Semiconductor ngineering by Prof Sungsik Lee

Density of States in 3D Semiconductor and Ambient Temperature Unchanged with Temp When T 0: Free Carriers excited When T = 0: No Carrier = C n 0 = p 0 0 n 0 = p 0 = 0 g c () = C ħ g c () F F F F V g V () V g V () = ħ Density of States (Vacancies) is unchanged with changing Temperature (but the effective mass may be changed) Semiconductor ngineering by Prof Sungsik Lee

Carrier Numbers in 3D Semiconductor and Ambient Temperature Changed with Temp When T = 100 K: Free Carriers excited When T = 300 K: More Carriers C n 0 = p 0 0 n 0 = p 0 0 g c () = C g c () F F F F V g V () V g V () n 0 = p 0 0 at T = 100 K < n 0 = p 0 0 at T = 300 K Who determines the carrier numbers as a function of Temperature? Semiconductor ngineering by Prof Sungsik Lee

Fermi Function nrico Fermi (talian) 1901-1954 Probability of Occupancy ABC D Semiconductor ngineering by Prof Sungsik Lee

Derivation of Fermi-Dirac Statistics (Appendix) Possible # of electrons distributed (occupied) at an nergy level () Degeneracy (total places = rooms) at an nergy level () ABC D Semiconductor ngineering by Prof Sungsik Lee

Fermi-Dirac ntegral and Carrier density (concentration) in volume H G d Area underneath DoS Outline Fermi function ABC D Carrier density [cm -3 ] = nergy of DoS finishs (DoS)(Fermi Function) d nergy of DoS begins Area underneath G d H DoS Outline Fermi function Semiconductor ngineering by Prof Sungsik Lee

Physical ndicator of Probability of xistence of Thermally xcited Carriers: Fermi Function with Fermi Level H G d Area underneath n 0 p 0 = n 0 Area underneath F G d H C F DoS V DoS Outline g c () g V () Outline F C V 05 Fermi function D Semiconductor ngineering by Prof Sungsik Lee ABC D 1 Fermi function

What happens to Fermi Function and Fermi Level in ntrinsic Semiconductor with increasing Temperature H G d Area underneath n 0 p 0 = n 0 Area underneath F G d H C F DoS V DoS Outline g c () g V () Outline F C V 05 Fermi function D Semiconductor ngineering by Prof Sungsik Lee ABC D 1 increasing Temperature Fermi function Textbook: Figure 41

Fermi-Dirac Statistics and Carrier Density Formula with Boltzmann s approximation ABC D BC D K BC D D D Analytical form BC D H G d G d H H G D d Boltzmann s Approximation G D d H D D BC D Analytical form Semiconductor ngineering by Prof Sungsik Lee

Boltzmann s approximation and ffective Density of States Analytical form BC D S D ffective DoS for R H G D d Boltzmann s Approximation G D d H K BC D D O BP, D BC D Analytical form S D R ffective DoS for Semiconductor ngineering by Prof Sungsik Lee

Physical Meaning of Fermi Level ( F ) and Charge Neutrality with Carrier Density lectron density BC D g n 0 p 0 = n 0 F BC D Hole density C F V To be p 0 = n 0 the ntrinsic SC Charge Neutrality g c () D D g V () F C V 05 U V 1 D D D = ( c - v ) / 2 = g / 2 Fermi level is exactly centered in the BG Semiconductor ngineering by Prof Sungsik Lee

N-type Semiconductor (example) and Fermi Level with Charge Neutrality p 0 = n 0 : ntrinsic Semiconductor p 0 < n 0 : n-type Semiconductor BC D g c () BC D g c () g F BC D C V g V () p 0 = n 0 = n i : ntrinsic Concentration U V g F BC D p 0 < n 0 n 0 p 0 = n i 2 : Mass action Law C V g V () U V W s this Correct? Semiconductor ngineering by Prof Sungsik Lee

Mass Action Law (Appendix) n 0 p 0 = n i 2 : Mass action Law K BC K BC S D D = ( c - v ) / 2 = g / 2 BC D BC D BC D BC BC D BC BC BC BC BC K K K K K n 0 p 0 = n i 2 Semiconductor ngineering by Prof Sungsik Lee

xtrinsic Semiconductor (doped semiconductor) and Fermi Level with Charge Neutrality p 0 < n 0 : n-type Semiconductor p 0 > n 0 : p-type Semiconductor BC D g c () BC D g c () N D + g Donor level BC D F C V g V () N A - g BC D onized accepter density U V A N D+ N A = onized donor density t is still zero! Acceptor level F C V g V () F is the reference energy level at which the charge neutrality is satisfied Semiconductor ngineering by Prof Sungsik Lee

ntrinsic vs xtrinsic Semiconductors Volume Charge Density - Neutrality U V A N D+ N A = where = 5k A OBS, N A = OBS = k A S, N D+ = S, 5k = k p 0 = n 0 : ntrinsic Semiconductor U =V A N D+ N A = where = 5k A OBS N A = OBS = = k A S N D+ = S = p 0 n 0 : xtrinsic Semiconductor U =V A N D+ N A = where = 5k A OBS N A = OBS ( lb) = k A S N D+ = S ( lb) U =V = U =V 5kA OBS k S A S OBS = Semiconductor ngineering by Prof Sungsik Lee

Donor level and Fermi function in n-type semiconductor (example) Nature of electron donor N D + J BC D g Donor level N D N D + C Occupied: Neutral as it is empty: Positive as it lost(donated) electron 5 N + D =m ( )! empty (ionized = unoccupied) () g c () () D F 05 profile: n D 1 () Donor Level: D J BC D V g V () occupied N D (neutral) 9 n D U =V A N D+ = N D+ = A =9 n exp ( - - q / 2r / Semiconductor ngineering by Prof Sungsik Lee

Top questions of this lecture What makes valence electrons excited into the? What is the Fermi level? How does the Density of States look like? What determines Free carriers and Holes numbers? Are they the same each other all the time? What makes it different? And how does it relate to the Fermi level? What happens to the charge neutrality? Semiconductor ngineering by Prof Sungsik Lee

Next Lecture Review on Physical lectronics ( 물리전자복습 ) Semiconductor ngineering by Prof Sungsik Lee