Lecture 6 Semicoductor physics IV The Semicoductor i Equilibrium
Equilibrium, or thermal equilibrium No exteral forces such as voltages, electric fields. Magetic fields, or temperature gradiets are actig o the semicoductor. All properties of the semicoductor will be idepedet of time i this case.
Goal The cocetratio of electros ad holes i the coductio ad valece bads with the Fermi-Dirac probability fuctio ad the desity of quatum states. The properties of a itrisic semicoductor. The properties of a semicoductor with impurities (dopats). 3
CHARGE CARRIERS IN SEMICONDUCTORS Two types of charge carrier, the electro ad the hole. The curret i a semicoductor is determied largely by the umber of electros i the coductio bad ad the umber of holes i the valece had. The distributio (with respect to eergy) of electros i the coductio bad is ( E) gc( E) ff( E) the desity of quatum states i the coductio bad the probability that a state is occupied by a electro The total electro cocetratio per uit volume i the coductio bad is ( E) de 4
The distributio (with respect to eergy) of holes i the valece bad is p( E) g ( E)[1 ff( E)] v the desity of allowed quatum states i the valece bad the probability that a state is ot occupied by a electro The total hole cocetratio per uit volume is foud by itegratig this fuctio over the etire valace-bad eergy. p p( E) de 5
The locatio of Fermi eergy E F At T=K, valece bad is full ad the coductio bad is empty i a itrisic semicoductor (o impurities ad o lattice damage i crystal). E E E v F c At T>K, the valece electros gai eergy ad a few move to coductio bad ad leave empty states. Electros ad holes are created i pairs by the thermal eergy. The umber of electros i the coductio bad is equal to the umber of holes i the valece bad. Fig (b) The splittig of the 3s ad 3p states of silico ito the allowed ad forbidde eergy bads.
* 3/2 4 (2 m ) gc( E) E E 3 h * 3/2 4 (2 mp ) gv( E) E 3 v E h c If we assume that the electro ad hole effective masses are equal, the g c (E) ad g v ( E ) are symmetrical fuctios about the midgap eergy. The fuctio f F (E) for E > E F is symmetrical to 1 - f F (E) for E < E F about the eergy E = E F. f F 1 ( E) E EF 1 exp( ) The areas represetig electro ad hole cocetratios are equal E F i the middle of badgap eergy 7 Fig Desity of states fuctios, Fermi-Dirac probability fuctio, ad areas represetig electro ad hole cocetratios for the case whe E, is ear the midgap eergy
The ad p Equatios The thermal-equilibrium cocetratio of electros gc( E) ff( E) de The lower limit of itegratio is E c The upper limit of itegratio should be the top of the allowed coductio bad eergy. However, sice the Fermi probability fuctio rapidly approaches zero with icreasig eergy we ca take the upper limit of itegratio to be ifiity. 8
* 3/2 4 (2 m ) gc( E) E E 3 h If (E c - E F ) >> k T, the (E - E F ) >> f F 1 [ ( EEF )] ( E) exp ( E EF ) 1 exp * 3/2 4 (2 m ) ( EEF) c 3 o E E exp[ ] de h E C c Boltzma approximatio Let E Ec * 3/2 4 (2 m ) ( EcEF) 1/2 exp[ ] exp( ) 3 h 1/2 exp( ) d 1 2 d * 2 m 3/2 ( EcEF) 2 2( ) exp[ ] h 9
The thermal-equilibrium electro cocetratio i the coductio bad ( Ec EF) N c is called the effective desity states fuctio i the * coductio bad. 2 m 3/2 The thermal-equilibrium cocetratio of holes i the valece bad is p g ( E)[1 f ( E)] de v N c 2( ) 2 h * 3/2 4 (2 mp ) gv( E) E 3 v E h 1 1 ff( E) ( EF E) 1 exp N F c exp[ ] 1
11 If ( EF Ev) 1 ( EF E) 1 ff( E) exp[ ] ( EF E) 1 exp E v * 3/2 4 (2 mp) ( EF E) 3 v h p E E exp[ ] de The lower limit of itegratio is take as mius ifiity istead of the bottom of the valece bad sice the probability fuctio of holes approach zero whe eergy is mius ifiite. p ' ( Ev E) 4 (2 m ) ( E E ) exp[ ] ( ) exp( ) * 3/2 p F v 3 h p * 3/2 2 mp ( EF Ev) 2 exp 3 h ' 1/2 ' ' d
The thermal-equilibrium cocetratio of holes i the valece bad p N v ( EF Ev) exp[ ] The effective desity of states fuctio i the valece bad is * 2 mp Nv 2 2 h The magitude of N v is also o the order of 1 19 cm -3 at T = 3 K for most semicoductors. 3/2 12
The effective desity of states fuctios, N c ad N v, are costat for a give semicoductor material at a fixed temperature. Table Effective desity of states fuctio ad effective mass values N c * 2 m 2 2 h 3/2 N v * 2 mp 2 2 h 3/2 13
14 Example: Calculate the probability that a state i the coductio bad is occupied by a electro ad calculate the thermal equilibrium electro cocetratio i silico at T= 1 K. Assume the Fermi eergy is.25 ev below the coductio bad. The value of N c for silico at T = 1 K is N c = 2.8 x 1 19 cm -3. Solutio: f F The probability that a eergy state at E = E c is occupied by a electro is give by 1 ( Ec EF).25 ( Ec ) exp[ ] exp( ) 6.431 Ec EF 1 exp( ).259 ( Ec EF).25 Nc exp[ ] (2.81 )exp( ) 1.81 cm.259 19 15 3 The probability of a state beig occupied ca be quite small, but the fact that there are a large umber of states meas that the electro cocetratio is a reasoable value. 5
Example Calculate the thermal equilibrium hole cocetratio i silico at T= 4 K. Assume that the Fermi eergy is.27 ev above the valece bad eergy. The value of N v for silico at T = 3 K is N v = 1.4 1 19 cm -3. Solutio Nv 4 (1.4 1 )( ) 1.61 3 The hole cocetratio is p cm 19 19 3 ( E E ).27 Nv.3453 15 3 6.431 cm F v 19 exp[ ] (1.6 1 )exp( ) 15
The Itrisic Carrier Cocetratio For a itrisic semicoductor, The cocetratio of electros i the coductio bad i is equal to the cocetratio of holes i the valece bad p i. The Fermi eergy level is called the itrisic Fermi eergy, or E F = E Fi. EFi Ec Ev EFi i Ncexp[ ] p pi i Nv exp[ ] N N 2 i c v EFi Ec Ev EFi exp[ ]exp[ ] 2 Ev E E c g i NcNv exp[ ] NcNv exp[ ] E g is the badgap eergy 16
For E g = 1.12 ev, i = 6.95 x 1 9 cm -3 from the equatio for silico at T = 3 K. The commoly accepted value of i for silico at T = 3 K is approximately 1.5 1 1 cm -3 This theoretical fuctio does ot agree exactly with experimet. Fig The itrisic carrier cocetratio of Ge, Si, ad GaAs as a fuctio of temperature. 17
The Itrisic Fermi-Level Positio Sice the electro ad hole cocetratios are equal N If we take the atural log of both sides of this equatio * 1 3 mp EFi ( Ec Ev ) l( ) * 2 4 m 1 ( E E ) E 2 The midgap eergy c ( ) ( ) exp[ E c E Fi Fi v ] Nvexp[ E E ] 1 1 Nv EFi ( Ec Ev ) l( ) 2 2 N c N N c v midgap v c * 2 mp 2 2 h * 2 m 2 2 h 3/2 3/2 18
3 EFi Emidgap 4 * mp l( ) m * The itrisic Fermi level is m m m m * * p m * * p m * * p exactly i the ceter of the badgap below the ceter of the badgap. above the ceter of the badgap. 19
Example To calculate the positio of the itrisic Fermi level with respect to the ceter of the badgap i silico at T = 3 K. The desity of states effective carrier masses i silico are Solutio m 1.8 m m.56m * * p The itrisic Fermi level with respect to the ceter of the * badgap is 3 m 3.56 E E l( ) (.259)l( ) Fi midgap p 4 * m 4 1.8 E E.128eV 12.8meV Fi midgap 2 If we compare 12.8 mev to 56 mev, which is oe-half of the badgap eergy of silico, we ca, i may applicatios, simply approximate the itrisic Fermi level to be i the ceter of the badgap.
DOPANT ATOMS AND ENERGY LEVELS The real power of semicoductors is realized by addig small, cotrolled amouts of specific dopat, or impurity, atoms. The doped semicoductor, called a extrisic material addig a group V elemet, such as phosphorus 21 The phosphorus atom without the door electro is positively charged. At very low temperatures, the door electro is boud to the phosphorus atom.
The door electros jump to the coductio bad with thermal eergy Fig The eergy-bad diagram showig (a) the discrete door eergy state ad (b) the effect of a door state beig ioized. 22 The electro i the coductio bad ca ow move through the crystal geeratig a curret, while the positively charged io is fixed i the crystal. The door impurity atoms add electros to the coductio bad without creatig holes i the valece bad. The resultig material is referred to as a -type semicoductor.
Addig a group III elemet, such as boro, as a substitutio impurity purity to silico. Oe covalet bodig positio appears to be empty Fig Valece electros may gai a small amout of thermal eergy ad move about i the crystal. The "empty" positio associated with the boro atom becomes occupied, ad other valece electro positios become vacated. These other vacated electro positios ca he thought of as holes i the semicoductor material. 23
Fig The eergy-bad diagram showig (a) the discrete acceptor eergy state ad (b) the effect of a acceptor state beig ioized. Acceptor atom gets electros from the valece bad with thermal eergy. If a electro were to occupy this "empty" positio, its eergy would have to be greater tha that of the valece electros. The acceptor atom ca geerate holes i the valece had without geeratig electros i the coductio bad. This type of semicoductor material is referred to as a p-type material 24
Ioizatio Eergy Eergy required to elevate the door electro ito the coductio bad. Bohr theory The most probable distace of a electro i a hydroge atom from the ucleus from quatum mechaics is the same as Bohr radius. The coulomb force of attractio betwee the electro ad io equal to the cetripetal force of the orbitig electro. This coditio give a steady orbit. 25 e m v 2 * 2 2 4 r r
If we assume the agular mometum is also quatized, the we ca * write m r v is a positive iteger 26 2 2 h 2 * 2 Substitute v e m v 4 * mr ito r 2 * 2 4 r r me The assumptio of the agular mometum beig quatized leads to the radius beig quatized The Bohr radius is 4 2 2 me ormalize the radius of the door orbital to that of the Bohr radius r 2 m r ( ) rest mass of a electro * a m the relative dielectric costat of the semicoductor material a.53a effective mass of the electro i the semicoductor.
If we cosider the lowest eergy state i which = 1, ad if we cosider silico i which r = 11.7 ad the coductivity effective mass is m*/m =.26. the we have that r 1 a 45 r 1 = 23.9 A. This radius correspods to approximately four lattice costats silico. Recall that oe uit cell i silico effectively cotais eight atoms, so the radius of the orbitig door electro ecompasses may silico atoms. => The door electro is ot tightly boud to the door atom. 27
The total eergy of the orbitig electro is give by E T V 28 The kietic eergy is Sice T * m r v T The potetial eergy is The total eergy is 1 2 * 4 me 2( ) (4 ) 2 2 m v * 2 V r 2 2 4 * 2 me 2 * 4 e m e 4 r ( ) (4 ) E T V 2 2 * 4 me 2( ) (4 ) 2 2
I silico the ioizatio eergy is E = -25.8 mev, much less tha the badgap eergy of silico. This eergy is the approximate ioizatio eergy of the door atom. Table Impurity ioizatio eergies i silico ad germaium Germaium ad silico have differet relative dielectric costats ad effective masses, resultig i differet ioizatio eergy. 29
THE EXTRINSIC SEMICONDUCTOR A material has impurity atoms. Oe type of carrier will predomiate. The Fermi eergy will chage as dopat atoms are added. type: the desity of electros is greater tha the desity of holes >p majority carrier: electros; miority carrier: holes the Fermi eergy is above the itrisic Fermi eergy p type: the desity of holes is greater tha the desity of electros <p majority carrier: holes; miority carrier: electros the Fermi eergy is below the itrisic Fermi eergy 3
type Fig. Desity of states fuctios. Fermi-Dirac probability fuctio, ad areas represetig electro ad hole cocetratios for the case whe E F is above the itrisic Fermi eergy. 31
p type Fig Desity of states fuctios, Fermi-Dirac probability fuctio, ad areas represetig electro ad hole cocetratios for the case whe E F is below the itrisic Fermi eergy. 32
The electro cocetratio i extrisic semicoductor N The itrisic carrier cocetratio i c ( Ec EFi ) ( EF EFi ) exp[ ] N c ( Ec EFi ) exp[ ] EF EFi i exp[ ] p i ( EF EFi ) exp[ ] 33
The ad p Product p N N c v ( Ec EF ) ( EF Ev ) exp[ ]exp[ ] Eg p NcNvexp[ ] p 2 i The product of ad p is always a costat for a give semicoductor material at a give temperature. The equatio is ivalid if the Boltzma approximatio is ot valid sice it is derived by the Boltzma approximatio 34 E E ad E E c F F v
35 The Fermi-Dirac Itegral If the Boltzma approximatio (E-E F >>) does ot hold. 1/2 4 * 3/2 ( E E ) 3 (2 ) c de m h E E E F c 1 exp( ) If we agai make a chage of variable ad let E E c EF Ec F * 1/2 2mp 3/2 d 4 ( ) 2 h 1exp( ) The itegral is defied as F 1Τ2 (η F ) = η 1Τ2 dη 1 + exp(η η F ) F Fermi-Dirac itegral
The Fermi-Dirac itegral, is a tabulated fuctio. F 1 2 d 1/2 F ( F ) 1 e if F The Fermi eergy is actually i the coductio bad. 36 / E E F C F Fig The Fermi-Dirac itegral as a fuctio of the Fermi eergy
p 2 m ( ) 1 exp( ) * ' 1/2 ' p 3/2 d 4 ( ) 2 ' ' h F ' Ev E E E ' v F F if ' F the Fermi level is i the valece bad. 37
38 Example To calculate the electro cocetratio usig the Fermi-Dirac itegral. Let = 2 so that the Fermi eergy is above the coductio bad by approximately 52 mev at T = 3 K. Solutio F * 1/2 2m 3/2 d 2 4 ( ) 2 NtF1/2 ( F) h 1exp( ) F For silico at 3K, N c = 2.8 1 19 cm -3 The Femi-Dirac itegral has a value of F 1/2 (2) = 2.3 2 19 19 3 (2.8 1 )(2.3) 7.27 1 cm With the Boltzma approximatio N c ( Ec EF) exp( ) o = 2.8 1 2 cm -3
STATISTICS OF DONORS AND ACCEPTORS The probability fuctio of electros occupyig the door state is Nd d 1 Ed EF 1 exp( ) 2 d is the desity of electros occupyig the door level 39 N d is the door cocetratio E d is the eergy of the door level The factor 1/2 i this equatio is a direct result of the spi factor. Each door level has two possible quatum states (spi orietatios). The isertio of a electro ito oe quatum state, however, precludes puttig a electro ito the secod quatum state.
The desity of electros occupyig the door level is equal to the door cocetratio mius the cocetratio of ioized doors N N d The cocetratio of holes i the acceptor states Na pa Na Na 1 EF Ea 1 exp( ) g N a is the cocetratio of acceptors N a- is the cocetratio of ioized acceptors d d g is a degeeracy factor( 简并因子 ) g=4 i silico ad gallium arseide. 4
41 Complete Ioizatio ad Freeze-Out If Ed EF Nd Ed EF d 2Nd exp( ) 1 E exp( d EF ) 2 The Boltzma approximatio is also valid The ratio of electro i the door state to the total umber of electros i the coductio bad plus door state. d d N c ( Ec EF) exp( ) ( Ed EF) 2Nd exp( ) ( Ed EF ) ( Ec Ed ) 2Nd exp( ) Ncexp( )
d d 1 Nc ( Ec Ed ) 1 exp( ) N d Small value The factor (E c - E d ) is just the ioizatio eergy of the door electros. At room temperature, the door states are essetially completely ioized. => All door impurity atoms have doated a electro to the coductio bad. At room temperature, there is also essetially complete ioizatio of the acceptor atoms. =>each acceptor atom has accepted a electro from the valece bad. 42
Complete Ioizatio at T=3 K Fig Eergy-bad diagrams showig complete ioizatio of (a) door states ad (b) acceptor states. Partial ioizatio of door or acceptor atoms whe K<T<3K 43
Freeze-Out at T=K Coductio bad Coductio bad Valece bad Valece bad Fig Eergy-bad diagram at T = K for (a) -type ad (b) p-type semicoductors. N d+ = =>Each door state must cotai a electro. => E F >E d N a- = =>Each acceptor state does ot cotai electro. => E F >E v 44
CHARGE NEUTRALITY I thermal equilibrium, the semicoductor crystal is electrically eutral. =>The electros are distributed amog the various eergy states, creatig egative ad positive charges, but the et charge desity is zero. A compesated semicoductor is oe that cotais both door ad acceptor impurity atoms i the same regio. A -type compesated semicoductor occurs whe N d > N a, A p-type compesated semicoductor occurs whe N a > N d. If N d = N a, it is a itrisic material. 45
Equilibrium Electro ad Hole Cocetratios The charge eutrality coditio is N p N a ( N p ) p ( N ) a a d d : total electro cocetratio = thermal electros + door electros p : total hole cocetratio = thermal holes + acceptor holes p a : the hole cocetratio i acceptor states d : the electro cocetratio i door states d 46 Fig Eergy-bad diagram of a compesated semicoductor showig ioized ad u-ioized doors ad acceptors
If we assume complete ioizatio, d ad p a are both zero N p N a 2 i Na N Nd Na Nd Na ( ) 2 2 d d 2 2 i Two factor to impact --- the cocetratio of impurity atoms --- the itrisic carrier cocetratio 47
48 Fig Eergy-bad diagram showig the redistributio of electros whe doors are added
Complete ioizatio i N c ( Ec EFi ) exp[ ] 49 Fig Electro cocetratio versus temperature showig the three regios: partial ioizatio, extrisic, ad itrisic. As the temperature icreases, additioal electro-hole pairs are thermally geerated so that the i term may begi to domiate.
Similarly 2 i a p N p N Na Nd Na Nd p ( ) 2 2 d 2 2 i If N a -N d >> i, the p N N a d 2 2 i i p ( N N ) a d 5
POSITION OF FERMI ENERGY LEVEL The positio of the Fermi eergy level is a fuctio of the dopig cocetratios ad as a fuctio of temperature N c ( Ec EF) exp[ ] N c Ec EF l where Nd Na Nd Na ( ) 2 2 2 2 i For a -type semicoductor, N d >> i the Nd 51 N c Ec EF l Nd As the door cocetratio icreases, the Fermi level moves closer to the coductio bad. EF EFi i exp[ ] E E F Fi l i
For a p-type semicoductor N v EF Ev l p If we assume that Na>>Ni N v EF Ev l Na As the acceptor cocetratio icreases, the Fermi level moves closer to the valece bad. The differece betwee the itrisic Fermi level ad the Fermi eergy i terms of the acceptor cocetratio E E p Fi F l i 52
Fig Positio of Fermi level for a (a) -type ad (b) p-type semicoductor. Fig Positio of Fermi level as a fuctio of door cocetratio ( type) ad acceptor cocetratio (p type). 53
E E F Fi l i i E E p Fi F l i N c ( Ec EFi ) exp[ ] Fig Positio of Fermi level as a fuctio of temperature for various dopig cocetratios. As T icreases, i will icrease E F E Fi 54
Summary The cocetratio of electros ad holes g ( E) f ( E) de p g ( E)[1 f ( E)] de c F Usig the Maxwell-Boltzma approximatio v F ( Ec EF) exp[ ] Nc p N v ( EF Ev) exp[ ] The itrisic carrier cocetratio is N N 2 i c v Eg exp[ ] 55
The cocept of dopig the semicoductor with door atoms ad acceptor atoms to form -type or p-type material. The fudametal relatioship of the hole ad electro 2 cocetratio is p i The electro ad hole cocetratios is a fuctio of impurity dopig cocetratios Nd Na Nd Na ( ) 2 2 2 2 i Na Nd Na Nd p ( ) 2 2 2 2 i The positio of the Fermi eergy level is a fuctio of impurity dopig cocetratio E E F Fi l i E E p Fi F l i 56
Homework 5 Cosider germaium with a acceptor cocetratio of Na = 1 15 cm -3 ad a door cocetratio of N d =. Plot the positio of the Fermi eergy with respect to the itrisic Fermi level as a fuctio of temperature over the rage 2K T 5K