Lecture 10. Reading: Notes and Brennan Chapter 5

Transcription

1 Lecture tatstcal Mechancs and Densty of tates Concepts Readng: otes and Brennan Chapter 5 Georga Tech C Dr. Alan Doolttle

2 C Dr. Alan Doolttle Georga Tech How do electrons and holes populate the bands? Densty of tates Concept In lower level courses, we state that Quantum Mechancs tells us that the number of avalable states n a cubc cm per unt of energy, the densty of states, s gven by: ev cm tates of umber unt m m g m m g v v p p v c c n n c * * 3 2 * *, ( 2 (, ( 2 ( h h π π

3 How do electrons and holes populate the bands? Densty of tates Concept Thus, the number of states per cubc centmeter between energy and +d s Georga Tech g g c v ( d f ( d f otherwse c v and, and, But prevously derved ths for bulk materals. C Dr. Alan Doolttle

4 Georga Tech How do electrons and holes populate the bands? Dervaton of Densty of tates Concept Remnder from lecture 7... G( G( G( # of states per energy per volume 3 2 V 2 3π m 2m 2 3 π h 3 V k f V 2 2 π π 2m 2 h m 2 h Fnally, we can defne the densty of states functon: 2m 2 h V Applyng to the semconductor we must recognze m m* and snce we have only consdered knetc energy (not the potental energy we have - c G( * m 2m π h 2 3 * 3 2 c d d V C Dr. Alan Doolttle

5 How do electrons and holes populate the bands? Probablty of Occupaton (Ferm Functon Concept ow that we know the number of avalable states at each energy, how do the electrons occupy these states? We need to know how the electrons are dstrbuted n energy. Agan, Quantum Mechancs tells us that the electrons follow the Ferm-dstrbuton functon. f ( where k Boltzman cons tan t, T Temperature n ( F kt + e and Ferm energy (~ average energy n the crystal F Kelvn f( s the probablty that a state at energy s occuped -f( s the probablty that a state at energy s unoccuped Georga Tech C Dr. Alan Doolttle

6 How do electrons and holes populate the bands? Probablty of Occupaton (Ferm Functon Concept At TK, occupancy s dgtal : o occupaton of states above F and complete occupaton of states below F At T>K, occupaton probablty s reduced wth ncreasng energy. f( F /2 regardless of temperature. Georga Tech C Dr. Alan Doolttle

7 How do electrons and holes populate the bands? Probablty of Occupaton (Ferm Functon Concept At TK, occupancy s dgtal : o occupaton of states above F and complete occupaton of states below F At T>K, occupaton probablty s reduced wth ncreasng energy. f( F /2 regardless of temperature. At hgher temperatures, hgher energy states can be occuped, leavng more lower energy states unoccuped (-f(. Georga Tech C Dr. Alan Doolttle

8 How do electrons and holes populate the bands? Probablty of Occupaton (Ferm Functon Concept f( f ( ( F kt + e For < ( f -3kT: f( ~ -e -(-f/kt ~ +/-3 kt 3 kt 3 kt 3 kt 3 kt f.55 ev For > ( f +3kT: f( ~ e -(-f/kt ~ T K, kt.86 ev T3K, kt.259 T45K, kt [ev] Georga Tech C Dr. Alan Doolttle

9 How do electrons and holes populate the bands? Probablty of Occupaton We now have the densty of states descrbng the densty of avalable states versus energy and the probablty of a state beng occuped or empty. Thus, the densty of electrons (or holes occupyng the states n energy between and +d s: lectrons/cm 3 n the conducton band between nergy and +d g c (f(d f c and, Holes/cm 3 n the valence band between nergy and +d g v ([- f(]d f v and, otherwse Georga Tech C Dr. Alan Doolttle

10 How do electrons and holes populate the bands? Band Occupaton Georga Tech C Dr. Alan Doolttle

11 How do electrons and holes populate the bands? Intrnsc nergy (or Intrnsc Level f s sad to equal (ntrnsc energy when qual numbers of electrons and holes Georga Tech C Dr. Alan Doolttle

12 How do electrons and holes populate the bands? Addtonal Dopant tates Intrnsc: qual number of electrons and holes n-type: more electrons than holes p-type: more holes than electrons Georga Tech C Dr. Alan Doolttle

13 st What s an qulbrum Dstrbuton If all possble confguratons of a system are equally probable, then the most lkely dstrbuton s the dstrbuton that has the most degeneraces. In ths case, the degeneraces are not energy but the number of ways of reconfgurng the partcles to arrve at the same result. Consder a Craps system descrbed by the vector A+B> where A and B are -6 (dce. In small numbers, t s easy to see that the sum of dce, whle most lkely to be 7 (most probable combnaton of numbers on 2 de, many dfferent results can occur. However, what happens for a very large number of de descrbed by the state, A +B, A 2 +B2,...A, +B, >. The average par sum s almost always ~7. Thus, 7 s the equlbrum confguraton of ths system.e the most lkely arrangement of de pars. ntropy tells us that systems tend toward dsorder. Ths s merely a byproduct of the fact that a dsordered system has the largest number of ways the system can reconfgure tself. An Asde: tatstcal Mechancs s based on the assumpton of purely random partcle behavor. The Fundamental Postulate of tatstcal Mechancs states that all states n a system are equally lkely. Ths s a purely TRU assumpton n perfectly random systems. However, be careful how ths assumpton s appled n real systems where forces act between partcles. For example, havng 22 neutral partcles congregated n a nm 3 corner of the cm 3 block s just as lkely as specfc dstrbuton of these partcles throughout the cm 3 block. However, havng 22 electrons congregated n a nm 3 corner of the cm 3 block s not equally as lkely as on confguraton wth electrons equally dstrbuted throughout the block because the electrons are not random n that they exert forces on each other and thus do not congregate wthout added energy. Luckly, we are descrbng electron systems usng a constant total energy constrant so ths s not an ssue for us. Ths devaton from statstcal randomness s mportant n the formaton of hghly ordered materals such as crystals n that lowest energy confguratons can overrde a systems natural tendency to maxmze entropy. Georga Tech C Dr. Alan Doolttle

14 But where dd we get the ferm dstrbuton functon? Probablty of Occupaton (Ferm Functon Concept lectrons, belong to a class of partcles (ncludng protons and neutrons known as Fermons. Fermons are characterzed as quantum mechancal partcles wth ½ ntegral spn whch thus must obey the Paul xcluson Prncple. The number of Fermons s always conserved n a closed system and each Fermon s ndstngushable from any other Fermon of lke type.e. all electrons look alke. o two Fermons can occupy the same quantum mechancal state. Consder a system of total electrons spread between allowable states. At each energy,, we have avalable states wth of these states flled. As they are Fermons, electrons are ndstngushable from one another. Constrants for electrons (fermons: ( ach allowed quantum state can accommodate at most, only one electron (2 Σ constant; the total number of electrons s fxed (3 total Σ ; the total system energy s fxed A smple test as to whether a partcle s ndstngushable (statstcally nvarant s when two partcles are nterchanged, dd the electronc confguraton change? Georga Tech Fgure after eudeck and Perret Fgure 4.5 C Dr. Alan Doolttle

15 Georga Tech But where dd we get the ferm dstrbuton functon? Probablty of Occupaton (Ferm Functon Concept Consder only one specfc energy, : How many ways, W, can we arrange at each energy,, ndstngushable electrons nto the avalable states. W! (!! (!! ( & & & 4 xample: The number of unque ways of arrangng 3 electrons n 5 states at energy s: 5, 3 W ! 5!!3! or or +2 + C Dr. Alan Doolttle

16 But where dd we get the ferm dstrbuton functon? Probablty of Occupaton (Ferm Functon Concept When we consder more than one level (.e. all s the number of ways we can arrange the electrons ncreases as the product of the W s. W W (!!! +2 If all possble dstrbutons are equally lkely, then the probablty of obtanng a specfc dstrbuton s proportonal to the number of ways that dstrbuton can be constructed (n statstcs, ths s the dstrbuton wth the most ( complexons. For example, nterchangng the blue and red electron would result n two dfferent ways (complexons of obtanng the same dstrbuton. The most probable dstrbuton s the one that has the most varatons that repeat that dstrbuton. To fnd that maxmum, we want to maxmze W wth respect to s. Thus, we wll fnd dw/d. However, to elmnate the products and factorals, we wll frst take the natural log of the above then we wll take d([w]/d. Before we do that, we can use trg s Approxmaton to elmnate the factorals. Georga Tech ( W (! ([ ]! (! + C Dr. Alan Doolttle

17 But where dd we get the ferm dstrbuton functon? Probablty of Occupaton (Ferm Functon Concept Usng trg s Approxmaton, for large x, (x! ~ (x(x x so, Georga Tech ( W (! ( [ ]! (! ( W ( ( ( [ ] + ( ( ( W ( ( ( [ ] ( Collectng lke terms, ow we can maxmze W wth respect to s. otng that s ndependent of and that snce d((x/dx/x, then d(wdw/w when dw, d([w]. d d ettng the dervatve equal to... (4 d d ( W ( W ( W [ ( [ ] + ( ] ( W d d + C Dr. Alan Doolttle

18 C Dr. Alan Doolttle Georga Tech But where dd we get the ferm dstrbuton functon? Probablty of Occupaton (Ferm Functon Concept From our orgnal constrants, (2 and (3, we get... ( ( a constant, a constant, total d d Usng the method of undetermned multplers (Lagrange multpler method we multply the above constrants by constants α and β and add (addng zeros to equaton 4 to get... ( ( d d β α ( for all that requres can take on any value, and whch snce d W d β α β α β α

19 C Dr. Alan Doolttle Georga Tech But where dd we get the ferm dstrbuton functon? Probablty of Occupaton (Ferm Functon Concept Ths fnal relatonshp can be solved for the rato of flled states, per states avalable... ( kt F e f e f ( kt and kt - we have semconductors, of n the case of ( F β α β α β α

20 What other Dstrbutons are Possble? Bose-nsten Dstrbuton- Probablty of Occupaton for Bosons and Planckans Bosons are characterzed as quantum mechancal partcles wth ntegral spn whch thus DO OT obey the Paul xcluson Prncple. Two types of Bosons are possble and Brennan has a restrctve defnton of both (note most texts descrbe Planckans as a subset of Bosons whle Brennan adopts a separate defnton for Bosons and Planckans: Brennan defnes a Boson as an ndstngushable (.e. all Bosons look alke partcle whose number s conserved n a closed system. He further defnes a Planckan as an ndstngushable partcle whose number s OT conserved. The number of Planckans n a system s unlmted (except for total unversal energy constrants. Unlke Fermons, any number of Bosons/Planckans can occupy the same quantum mechancal state. Bosons/Planckans tend to collect n the same state at low temperatures formng Bose-nsten Condensates. Consder a system of total electrons spread between allowable states. At each energy,, we have avalable states wth of these states flled. As defned by Brennan, Bosons (number conserved are mportant n nuclear, atomc and hgh energy physcs whle Planckans (number unlmted are mportant n photoncs and phononcs. Constrants for Bosons: ( ach allowed quantum state can accommodate any number of Bosons. (2 Σ constant; the total number of electrons s fxed (3 total Σ ; the total system energy s fxed Constrants for Planckans: ( ach allowed quantum state can accommodate any number of Bosons. (2 total Σ ; the total system energy s fxed Bosons are named after atyendra ath Bose who constructed the orgnal theory for photons n 92 whch was generalzed to other partcles by nsten n 924 Georga Tech C Dr. Alan Doolttle

21 How many ways, W, can we arrange at each energy,, ndstngushable Bosons nto the avalable states. W What other Dstrbutons are Possble? Bose-nsten Dstrbuton- Probablty of Occupaton for Bosons and Planckans Total # Permutatons of Bns and Partcles (# Indstngushable Permutatons of Bns(# Indstngushable Permutatons of Partcles 3 & 5 3 & 5 ( +! W 2 & 4 (!! xample: The number of ways of arrangng 3 Bosons n 5 states at energy s: 5, 3 ( +! (3 + 5! W 35!! 5!3! ( ( xample: The number of ways of arrangng 2 Bosons n 3 states at energy s: 3, 2 W ( +! (!! ( 3 (2 + 3!!2! 6 or or Georga Tech C Dr. Alan Doolttle

22 What other Dstrbutons are Possble? Bose-nsten Dstrbuton- Probablty of Occupaton for Bosons and Planckans When we consder more than one level (.e. all s the number of ways we can arrange the electrons ncreases as the product of the W s. W W ( ( +!!! +2 If all possble dstrbutons are equally lkely, then the probablty of obtanng a specfc dstrbuton s proportonal to the number of ways that dstrbuton can be constructed (n statstcs, ths s the dstrbuton wth the most ( complexons. For example, nterchangng the blue and red electron would result n two dfferent ways (complexons of obtanng the same dstrbuton. The most probable dstrbuton s the one that has the most varatons that repeat that dstrbuton. To fnd that maxmum, we want to maxmze W wth respect to s. Thus, we wll fnd dw/d. However, to elmnate the products and factorals, we wll frst take the natural log of the above then we wll take d([w]/d. Before we do that, we can use trg s Approxmaton to elmnate the factorals. Georga Tech ( W ([ + ]! ([ ]! (! + C Dr. Alan Doolttle

23 What other Dstrbutons are Possble? Bose-nsten Dstrbuton- Probablty of Occupaton for Bosons and Planckans Usng trg s Approxmaton, for large x, (x! ~ (x(x x so, ( W ( [ + ]! ( [ ]! (! ( W [ + ] ( [ + ] [ + ] ( ( [ ] + ( ( ( W [ + ] ( [ + ] ( ( [ ] ( Collectng lke terms, + ow we can maxmze W wth respect to s. otng that s ndependent of and that snce d(wdw/w when dw, d([w]. Georga Tech d d ( W ( W ( W [ ( [ + ] + ( ] ettng the dervatve equal to... (4 d d ( W + d d C Dr. Alan Doolttle

24 What other Dstrbutons are Possble? Bose-nsten Dstrbuton- Probablty of Occupaton for Bosons and Planckans For Bosons or Planckans we have dfferent constrants, (2 and (3, we get... Both Constrants for Bosons, a constant total, a constant d ( d ( For Bosons: Usng the method of undetermned multplers (Lagrange multpler method we multply the above constrants by constants α and β and add (addng zeros to equaton 4 to get... d ( W α d β d whch requres that ( + ( + α β α β d for all Only Constrant for Planckans Georga Tech C Dr. Alan Doolttle

25 C Dr. Alan Doolttle Georga Tech Ths fnal relatonshp can be solved for the rato of flled states, per states avalable... ( ( kt and kt - we have semconductors, of n the case of ( F + + kt F e f e f β α β α β α What other Dstrbutons are Possble? Bose-nsten Dstrbuton- Probablty of Occupaton for Bosons and Planckans For Bosons:

26 What other Dstrbutons are Possble? Bose-nsten Dstrbuton- Probablty of Occupaton for Bosons and Planckans For Bosons or Planckans we have dfferent constrants, (2 and (3, we get... Both Constrants for Bosons, a constant total, a constant d ( d ( For Planckans: Usng the method of undetermned multplers (Lagrange multpler method we multply the above constrants by OLY the constant β and add (addng zero to equaton 4 to get... ( β d Only Constrant for Planckans d ( W whch requres that + + β β d for all Georga Tech C Dr. Alan Doolttle

27 C Dr. Alan Doolttle Georga Tech Ths fnal relatonshp can be solved for the rato of flled states, per states avalable... ( ( kt we have semconductors, of n the case of ( + kt e f e f β β β What other Dstrbutons are Possble? Bose-nsten Dstrbuton- Probablty of Occupaton for Bosons and Planckans For Planckans