Chapter 4: Semiconductor Physics

Transcription

1 Chater 4: Semioutor Physis Crystal strutures of solis ergy ba strutures of solis Charge arriers i semioutors Carrier trasort PHYS53 Chater our (II) 1

2 quilibrium Distributio of letros a Holes The esities of eletros a holes are relate to the esity-of-states futio a the ermi istributio futio. g f g 1 f The total eletro/hole oetratio er uit olume is fou by itegratig the orresoig futio oer the etire outio/alee ba eergy. We ee to etermie the ermi eergy i orer to fi the thermalequilibrium eletro a hole oetratios. We first osier a itrisi semioutor. A ieal itrisi semioutor is a ure semioutor without imurities or lattie efets. At T = K, all eergy states i the alee ba are fille with eletros a all eergy states i the outio ba are emty. The ermi eergy must be somewhere betwee a. At T > K, the umber of eletros i the outio ba is equal to that of holes i the alee ba. PHYS53 Chater our (II)

3 quilibrium Distributio of letros a Holes g 4 m h 3 * 3/ g 4 m h 3 * 3/ If m * = m *, the ermi eergy is at the miga eergy. If m * m *, the ermi leel for a itrisi semioutor will slightly shift away from the miga eergy. PHYS53 Chater our (II) 3

4 The a quatios g f Beause the ermi istributio raily aroahes zero with ireasig eergy, therefore If ( ) >>, the ( ) >>, so that 1 f ex 1 ex If we let PHYS53 Chater our (II) 4 * 3/ m 4 ex 3 h g f

5 The * 3/ m 4 1/ ex 3 ex h The itegral is a gamma futio. So 1/ We efie a arameter as ex 3/ * m ex h * m h The ex PHYS53 Chater our (II) 5 3/ 1 is alle the effetie esity-of-states futio i the outio ba.

6 is alle the effetie esity-of-states futio i the outio ba. If we were to assume that m * = m, the at T = 3 K, / m 3 or holes: g 1 f ex 1 1 f 1 1 ex 1 ex If ( ) >>, the ( ) >>. 1 f ex PHYS53 Chater our (II) ex

7 PHYS53 Chater our (II) 7 h m 3 3/ * ex 4 If we let ' 1/ 3 3/ * ex ex 4 ' ' ' h m h m 3/ * ex If we let 3/ * h m ex the is alle the effetie esity-of-states futio i the alee ba. The

8 PHYS53 Chater our (II) 8 The Itrisi Carrier Coetratio i i i ex i i i ex ex We use i a i to eote eletro a hole oetratios i a itrisi semioutor. Sie i = i, we usually use i to eote either the itrisi eletro or hole oetratio. The ermi leel for a itrisi semioutor is alle the itrisi ermi eergy, i. i i ex g i ex ex

9 At T = 3 K,, a i are ostat for a gie semioutor material at a gie temerature. PHYS53 Chater our (II) 9

10 PHYS53 Chater our (II) 1 3/ * h m 3/ * h m g i ex

11 The Itrisi ermi Leel Positio We a alulate the ermi leel ositio sie the eletro a hole oetratios are equal for a itrisi semioutor: i i i 1 miga miga ex i i i 1 l * 3 m l * 4 m * 3 m l * 4 m l ex PHYS53 Chater our (II) 11 i If m * = m *, the itrisi ermi leel will be i the eter of the baga. If m * > m *, the itrisi ermi leel will be slightly aboe the eter. If m * < m *, the itrisi ermi leel will be slightly below the eter of the baga.

12 Door Atoms a ergy Leels Real ower of semioutors is realize by aig otrolle amouts of seifi oat, or imurity atoms. The oe semioutor is alle a extrisi material. Doig is the rimary reaso that we a fabriate arious semioutor eies. A a grou V elemet, suh as hoshorus, to silio as a substitutioal imurity. The grou V elemet has fie alee eletros. our of these will otribute to the oalet boig with silio atoms, leaig the fifth more loosely bou to the hoshorus atom. PHYS53 Chater our (II) 1

13 Door Atoms a ergy Leels At ery low temeratures, the extra eletro is bou to the hoshorus atom. Howeer, it shoul be lear that the eergy require to eleate the extra eletro ito the outio ba is osierably smaller tha that for the eletros iole i the oalet boig. The eletro eleate ito the outio ba a moe through the rystal to geerate a urret, while the ositiely harge hoshorous atoms are fixe i the rystal. This tye of atom is alle a oor imurity atom. Door atoms a eletros to the outio ba without reatig holes i the alee ba. The resultig material is referre to as a -tye semioutor. PHYS53 Chater our (II) 13

14 Aetor Atoms a ergy Leels Cosier aig a grou III elemet, suh as boro, whih has three alee eletros. Oe oalet boig ositio is emty. If a eletro were to ouy this emty ositio, its eergy woul hae to be greater tha that of the alee eletros, sie the et harge of the B atom woul beome egatie. Howeer, the eletro ouyig this emty ositio oes ot hae suffiiet eergy to be i the outio ba, so its eergy is far smaller tha the outio ba eergy. The emty ositio assoiate with the B atom a be ouie a other alee eletro ositios beome aate. These other aate eletro ositios a be thought of as holes. PHYS53 Chater our (II) 14

15 Aetor Atoms a ergy Leels The hole a moe through the rystal to geerate a urret, while the egatiely harge boro atoms are fixe i the rystal. The grou III atom aets a eletro from the alee ba a so is referre to as a aetor imurity atom. The aetor atom a geerate holes i the alee ba without geeratig eletros i the outio ba. This tye of semioutor materials is referre to as a -tye semioutor. A extrisi semioutor will hae either a reoerae of eletros (-tye) or a reoerae of holes (-tye). PHYS53 Chater our (II) 15

16 Ioizatio ergy We a alulate the aroximate istae of the oor eletro from the oor imurity io, a also the aroximate eergy require to eleate the oor eletro ito the outio ba. This eergy is referre to as the ioizatio eergy. We will use the Bohr moel of the atom for these alulatios. The ermittiity of the semioutor material istea of the ermittiity of free sae will be use, a the effetie mass of the eletro will be use. rom the Coulomb fore of attratio beig equal to the etrietal fore of the orbitig eletro: 4 e r * m r Assume that the agular mometum is quatize, PHYS53 Chater our (II) 16

17 = 1,,3, 4 The Bohr raius is 4.53 The r a m r * m If we osier = 1 state, a if we osier silio, for whih r = 11.7, m * /m =.6, the we hae r a 1 45 r1.39m or silio, a =.543 m. PHYS53 Chater our (II) 17

18 The kieti eergy is The otetial eergy is 4 4 The total eergy is 4 The lowest eergy state of the hyroge atom is = 13.6 ev. or silio, the lowest eergy is = 5.8 mev ( g = 1.1 ev at T = 3 K). PHYS53 Chater our (II) 18

19 Quatum otrol of sigle atoms: the grou state of the oor is withi.1 m a o-iteratig, but the exite states exa to seeral aometers a strogly iteratig. Soli-state imlemetatio of oheret otrol of imurity waefutios ature 1, 465, 157. PHYS53 Chater our (II) 19

20 Grou III-V Semioutors The oor a aetor imurities i III-V omou semioutors is more omliate tha those i Si. Whe we talk about oors or aetors i III-V semioutors, we ee to kow for whih atoms (III or V) imurity atoms are substitute. or examle, for Si atoms i GaAs semioutor, if Si atoms relae Ga atoms, Si imurities will at as oors. But if Si atoms relae As atoms, they will at as aetors. or GaAs PHYS53 Chater our (II)

21 quilibrium Distributio of letros a Holes i xtrisi Semioutors Aig oor or aetor imurity atoms to a semioutor will hage the istributio of eletros a holes i the material. Sie the ermi eergy is relate to the istributio futio, the ermi eergy will hage as oat atoms are ae. I geeral, whe > miga, the esity of eletros is larger tha that of holes, a the semioutor is -tye. PHYS53 Chater our (II) 1

22 quilibrium Distributio of letros a Holes i the xtrisi Semioutor I geeral, whe < miga, the esity of eletros is smaller tha that of holes, a the semioutor is -tye. ex ex The aboe are geeral equatios for a i terms of the ermi eergy. The alues of a will hage with the ermi eergy,. PHYS53 Chater our (II)

23 xamle: osier silio at T = 3 K so that = m 3 a = m 3. If we assume the ermi eergy is.5 ev below the outio ba, alulate the thermal equilibrium oetratios of eletros a holes. The baga eergy of silio is 1.1 ev. Solutio:.5eV.87eV ex ex m 4.61 m Commet: eletro a hole oetratios hage by orers of magitue from the itrisi arrier oetratios (at 3 K, i = m 3 ) as the ermi eergy hages by a few teths of a ev. I a -tye semioutor, >, eletros are referre to as majority arriers a holes as miority arriers. I a -tye semioutor, >, holes are referre to as majority arriers a eletros as miority arriers. PHYS53 Chater our (II) 3 3 3

24 PHYS53 Chater our (II) 4 i i ex ex We a erie aother form of the equatios for the thermalequilibrium oetratios of eletros a holes: i i ex ex i i ex i i i i ex ex

25 The Prout ex ex g ex i The rout of a is always a ostat for a gie semioutor material at a gie temerature. It is oe of the fuametal riiles of semioutors i thermal equilibrium. It is imortat to kee i mi that the aboe equatio is erie usig the Boltzma aroximatio. We a thik of the itrisi oetratio i simly as a arameter of the semioutor material. PHYS53 Chater our (II) 5

26 The ermi-dira Itegral g f If we efie 4 h 3 * 3/ m * m 4 h 3/ 1 ex a 1/ 1 ex 1/ 1/ 1 ex ermi-dira itegral PHYS53 Chater our (II) 6

27 or holes: g 1 f 4 h 3 * 3/ m If we efie ' a 1 ex ' We ee to use the ermi- Dira itegral whe is aboe or below. * m 4 h 3/ ' 1/ ' 1 ex ' ' PHYS53 Chater our (II) 7

28 Degeerate a oegeerate Semioutors Whe isussig oors a aetors, the oetratio of oat atoms is assume to be small omare to the esity of host atoms. Imurities itroue isrete, o-iteratig oor a aetor eergy states i a - tye a -tye semioutor, resetiely. These tyes of semioutors are referre to as oegeerate semioutors. As the imurity oetratio is irease, the istae betwee imurity atoms ereases a the eletros from imurity atoms will begi to iterat. Whe this ours, the isrete oor or aetor eergy leel will ombie ito a ba of eergies. As the imurity oetratio is further irease, the ba of oor or aetor states wies a oerlas with the outio ba bottom or the alee ba to. Whe the oetratio of eletros i the outio ba exees the esity of states, the ermi eergy lies withi the outio ba. This tye of semioutors is alle egeerate -tye semioutors. Whe the oetratio of holes exees the esity of states, the ermi eergy lies i the alee ba. This tye of semioutors is alle egeerate -tye semioutors. PHYS53 Chater our (II) 8

29 Degeerate a oegeerate Semioutors oegeerate Degeerate PHYS53 Chater our (II) 9

30 Statistis of Doors a Aetors The robability of the oor eergy leel beig ouie is 1 1 ex ex ah oor leel a be emty, otai oe eletro of either si, or two eletros of oosite sis. Howeer, the Coulomb reulsio of two loalize eletros raises the eergy of the oubly ouie leel so high that ouble ouatio is essetially rohibite. This is the reaso for the fator of ½ aearig i the robability futio. is the esity of eletros ouyig the oor leel. is the oetratio of oor atoms. + is the oetratio of ioize oors. a 1 1 g a ex a g is a egeeray fator. The grou state g is ormally take as 4 for the aetor leel i Si a GaAs beause of the etaile ba struture. a a a PHYS53 Chater our (II) 3

31 PHYS53 Chater our (II) 31 Comlete Ioizatio a reeze-out ex ex ex If ( ) >>, we the hae ex ex 1 1 ex We a etermie the eretage of eletros i the oor state omare with the total umber of eletros: Beause ex 1 1

32 or hoshorus-oe silio at T = 3 K, = m 3, = 1 16 m 3, a the ioizatio eergy is.45 ev: 1.4.4% ex At room temerature, the oor states are almost omletely ioize, whih is also true for the aetor states at room temerature. Comlete ioizatio PHYS53 Chater our (II) 3

33 At T = K, all eletros are i their lowest eergy state. or a -tye semioutor, eah oor state must otai a eletro, therefore, =. ex T / The ermi leel is aboe the oor leel for a -tye semioutor at T = K. Similarly, the ermi leel will be below the aetor leel for a - tye semioutor at T = K. reeze-out (T = K) PHYS53 Chater our (II) 33

34 Charge eutrality i Comesate Semioutors A omesate semioutor is oe that otais both oor a aetor imurity atoms i the same regio. A -tye omesate semioutor ours whe > a a a -tye omesate semioutor ours whe a >. If a =, we hae a omletely omesate semioutor. Charge eutrality oitio: a a a PHYS53 Chater our (II) 34

35 PHYS53 Chater our (II) 35 quilibrium i Comesate Semioutors Uer omlete ioizatio, a a are both zero. We the hae a i a i a i Beause therefore The aboe equatio is use to alulate the eletro oetratio i a -tye semioutor, or whe > a. i a a

36 Cosier silio at 3 K: i = m 3, = 1 16 m 3, a a =. Majority arrier m Miority arrier i m 16 1 As we a oor imurity atoms, there is a reistributio of eletros amog aailable eergy states. A few of the oor eletros will fall ito the emty states i the alee ba to aihilate some of the itrisi holes. Beause of this reistributio, i the outio ba is ot simly equal to the oor oetratio lus the itrisi eletro oetratio. PHYS53 Chater our (II) 36

37 Cosier Ge at 3 K: i = m 3, = m 3, a a =. Majority arrier m Miority arrier i m Silio with = m 3. The itrisi arrier oetratio i is a ery strog futio of temerature. As the temerature is irease, the semioutor will eetually lose its extrisi harateristis, a the itrisi oetratio will begi to omiate. PHYS53 Chater our (II) 37

38 PHYS53 Chater our (II) 38 quilibrium i Comesate Semioutors Uer omlete ioizatio, a a are both zero. We hae a i a a i The aboe equatio is use to alulate the hole oetratio i a - tye semioutor, or whe a >. i a a

39 Cosier silio at 3 K: i = m 3, a = 1 17 m 3, a =. Majority arrier m Miority arrier i m 17 1 Cosier silio at T =3 K: i = m 3, = 1 16 m 3, a a = m m i m We ee remember that the miority arrier oetratio is alulate from = i oe the majority arrier oetratio is etermie. PHYS53 Chater our (II) 39

40 ermi Leel i Doe Semioutors If the Boltzma aroximatio is ali, for -tye semioutors, we hae ex l / A slightly ifferet exressio: i As or the oor oetratio ireases, the ermi leel moes loser to the outio ba. PHYS53 Chater our (II) 4 i ex i l Cosier silio at T = 3 K: i = m 3, = 1 16 m 3, a a =, = m m C.59l ev i /

41 ermi Leel i Doe Semioutors If the Boltzma aroximatio is ali, we hae for -tye semioutors ex l / A slightly ifferet exressio: i As or the aetor oetratio ireases, the ermi leel moes loser to the alee ba. i i ex i l / -tye, > i -tye, < i PHYS53 Chater our (II) 41

42 ermi Leel ersus Doig Coetratio & Temerature Kee i mi that we hae assume that Silio at the Boltzma aroximatio is ali. T = 3 K As the temerature is irease, the ermi leel moes loser to the itrisi ermi leel. At the low temerature where freeze-out ours, the ermi leel goes aboe for -tye semioutors a below a for -tye semioutors. PHYS53 Chater our (II) 4

43 Releae of the ermi ergy A imortat oit is that i thermal equilibrium, the ermi leel is a ostat throughout the material system. This oit is ery imortat for our isussio of jutios a other semioutor eies. The ermi eergy of material A is at A. The ermi eergy of material B is at B. A is higher tha B. If these two materials are brought ito otat (ery outie), eletros will flow from material A ito B, util thermal equilibrium is reahe. This thermal equilibrium will our whe the ermi eergy is the same i the two materials. PHYS53 Chater our (II) 43

44 Reaig Materials D. A. eame, Semioutor Physis a Deies: Basi Priiles, Irwi, Bosto, MA 116, 199, Chater 4, The Semioutor i quilibrium. PHYS53 Chater our (II) 44

45 Chater 4: Semioutor Physis Crystal strutures of solis ergy ba strutures of solis Charge arriers i semioutors Carrier trasort PHYS53 Chater our (II) 45

46 Hole Drift Curret Desity The rift urret esity ue to holes is: J rf e The equatio of motio for a hole i the resee of a eletri fiel is: * m a e If the eletri fiel is ostat, the we exet the eloity to irease liearly with time. Howeer, harge artiles i semioutors are iole i ollisios with ioize imurity atoms a with thermally ibratig lattie atoms. Whe the ollisio haes, the harge artile loses most or all of its eergy. It will begi to aelerate a gai eergy util it is agai iole i a satterig roess. Throughout this roess, the artile will gai a aerage rift eloity, whih, for low eletri fiels, is iretly roortioal to the eletri fiel. The J rf e e PHYS53 Chater our (II) 46

47 rf J e letro Drift Curret Desity The same isussio hols for eletros: J rf e e The oetioal rift urret ue to eletros is also i the same iretio as the alie eletri fiel ee though the eletro moemet is i the oosite iretio. letro a hole mobilities are futios of temerature a oig oetratios. PHYS53 Chater our (II) 47

48 Total Drift Curret Desity The total rift urret esity is the sum of the iiiual eletro a hole rift urret esities: J rf e Calulate the rift urret esity of a Ge samle at T = 3 K with =, a = 1 16 m 3, i = m 3, = 39 m /(Vs), a = 19 m /(Vs) uer a alie eletri fiel of = 5 V/m. The semioutor is -tye: The rift urret will be ue rimarily to the majority arrier i a extrisi semioutor. J J rf rf a i e a PHYS53 Chater our (II) 48 e i 1 16 m m A/m 3

49 Hole a letro Mobilities The motio of a hole is relate to a eletri fiel by: If we assume the iitial rift eloity to be zero, the we hae: m * t et * m e The hole uergoes raom thermal motio uer o alie eletri fiel. I the resee of a eletri fiel, there will be a et rift of the hole i the iretio of the eletri fiel. PHYS53 Chater our (II) 49

50 Hole a letro Mobilities If there is a mea time betwee ollisios a this mea time is ieeet o the eletri fiel, the the mea eak eloity just before a ollisio is: The aerage rift eloity will be: eak e * m 1 e * m If we osier the aerage ollisio time oer the raom thermal motio, the fator of ½ will be elimiate. The hole mobility is the gie by: The same aalysis alies for eletros: e m * PHYS53 Chater our (II) 5 e m *

51 Phoo or Lattie Satterig Two ollisio or satterig mehaisms affet the arrier mobility: hoo or lattie satterig, a ioize imurity satterig. Atoms ibrate raomly about their lattie ositios aboe K. The thermal ibratios ause a isrutio of the otetial futio, resultig i a iteratio betwee eletros or holes a the ibratig lattie atoms. The mobility ue to lattie satterig ereases with ireasig temeratures. or silio PHYS53 Chater our (II) 51

52 Ioize Imurity Satterig The other ollisio mehaism is ioize imurity satterig. Doat imurities are ioize at room temerature a a Coulomb iteratio exists betwee eletros or holes a ioize imurities. This Coulomb iteratio roues satterig or ollisios a also alters the eloity harateristis of harge arriers. As the imurity oetratio is irease, the mobility ue to ioize imurity satterig ereases. PHYS53 Chater our (II) 5

53 Total Mobility of Multile Satterig Mehaisms If L is the mea time betwee ollisios ue to lattie satterig, the t/ L is the robability of a lattie satterig eet ourrig i a ifferetial time t. Likewise, t/ I is the robability of a ioize imurity satterig eet ourrig i the ifferetial time t with I beig the mea time betwee ollisios ue to ioize imurity satterig. If these two satterig roesses are ieeet, the the total robability of a satterig eet ourrig i the ifferetial time t is the sum of the iiiual eets: t t t I L is the mea time betwee ay satterig eet. rom the efiitio of mobility, we a write: 1 1 With two or more ieeet satterig mehaisms, the ierse mobilities a, whih meas that the et mobility ereases. I 1 L PHYS53 Chater our (II) 53

54 Coutiity a Resistiity J rf e e 1 or itrisi semioutors: 1 e i e PHYS53 Chater our (II) 54

55 PHYS53 Chater our (II) 55 Ohm s Law for a Semioutor If we hae a bar of a semioutor with a oltage alie, urret is roue. A I J IR I A L I A L V L V L V A I

56 letro Coetratio a Coutiity ersus Temerature I the mi-temerature rage, or the extrisi rage, oors are Silio omletely ioize a the eletro = 1 15 m 3 oetratio remais essetially ostat. Beause the mobility is a futio of temerature, so the outiity aries with temerature i this rage. At higher temeratures, the itrisi arrier oetratio omiates the eletro oetratio as well as the outiity. I the lower temerature rage, freeze-out begis to our. The eletro oetratio a outiity erease with ereasig temeratures. PHYS53 Chater our (II) 56

57 Veloity Saturatio We hae so far assume that the mobility is ieeet of the eletri fiel. Howeer, this is ot true. The mobility a beome saturate at higher eletri fiels or ee exhibit a egatie alue. 1 mth 3 T 3K Mea thermal eloity ~ m/s There is a liear ariatio of the eloity with the eletri fiel at low eletri fiels. The sloe of the ure is the mobility. The rift eloity saturates at high eletri fiels. or GaAs, as the eletri fiel is irease, the rift eloity reahes a eak a the ereases. This egatie sloe reresets a egatie ifferetial mobility. The egatie ifferetial mobility roues a egatie ifferetial resistae. PHYS53 Chater our (II) 57

58 egatie Differetial Mobility i GaAs m * =.55m m * =.67m The small effetie eletro mass at the lower alley leas to a large mobility. As the eletri fiel is irease, the eergy of eletros ireases a eletros a be sattere ito the uer alley, where they hae a larger effetie mass. The larger effetie mass leas to a smaller mobility. This iter-alley trasfer mehaism results i a ereasig aerage rift eloity of eletros with ireasig eletri fiels, or the egatie ifferetial mobility. PHYS53 Chater our (II) 58

59 letro Diffusio Curret Desity A seo mehaism that a iue a urret i a semioutor is iffusio. Assume a eletro oetratio ariatio i oe imesio. The temerature is uiform so that the aerage thermal eloity of eletros is ieeet of x. if the istae l is the mea free ath of a eletro, that is, the aerage istae a eletro traels betwee ollisios (l = th ), the o aerage, oe half of the eletros moig to the right at x = l a oe half of the eletros moig to the left at x = l will ross the x = lae. We will osier the et eletro flow i the +x iretio: 1 1 th l l th 1 l l th PHYS53 Chater our (II) 59

60 J x if letro Diffusio Curret Desity 1 th l x ed x J e e th l x l x x D is alle the eletro iffusio oeffiiet. It has a uit of m /s a is a ositie quatity. If /x is ositie, whih meas the eletro oetratio ireases with ireasig x, the eletros will moe i the egatie x iretio. Sie eletros hae a egatie harge, the urret iretio will be i the ositie x iretio. th l PHYS53 Chater our (II) 6

61 J x if ed Hole Diffusio Curret Desity x D is alle the hole iffusio oeffiiet. It has a uit of m /s a is a ositie quatity. If /x is ositie, meaig that the hole oetratio ireases with x, the holes will moe i the egatie x iretio. Sie holes are ositiely harge artiles, the urret iretio will be i the egatie x iretio. PHYS53 Chater our (II) 61

62 J Total Curret Desity We hae four urret mehaisms i a semioutor: eletro rift a iffusio, hole rift a iffusio. or a oe-imesioal ase, we hae: e x e x ed ed x Geeralize the total urret to the three-imesioal ase: x J e e ed ed The mobility gies a iiatio of how well a harge arrier moes i a semioutor ue to the fore of a eletri fiel. The iffusio oeffiiet gies a iiatio of how well a artile moes i a semioutor as a result of a esity graiet. The mobilities a iffusio oeffiiets are ot ieeet arameters. The exressio for the total urret otais four terms. ortuately i most situatios, we will oly ee to osier oe term at ay oe time at a artiular oit i a semioutor. PHYS53 Chater our (II) 6

63 Iue letri iel i Grae Imurity Distributio We hae thus far osiere semioutors beig oe uiformly. We will osier i the followig semioutors that are oe o-uiformly. Cosier a semioutor with the oor oetratio ereasig with ireasig x. The majority arriers (eletros) will iffuse i the +x iretio, leaig behi ositiely harge ios. The searatio of ositie a egatie harges iues a eletri fiel that will reets the eletros from further iffusio. The ermi eergy will be the same throughout the semioutor. Whe equilibrium is establishe, the mobile arrier oetratio is ot exatly equal to the fixe imurity oetratio. But i most ases, the mobile arrier oetratio is ot too ifferet from the oat esity. PHYS53 Chater our (II) 63

64 Iue letri iel i Grae Imurity Distributio The eletri otetial is relate to the ifferee betwee the ermi eergy a the itrisi ermi eergy: 1 i e 1 i The iue eletri fiel is x x e x If we assume that the eletro oetratio is almost equal to the oor imurity oetratio, the we a write: i i ex x x i l i 1 x i x x e x x x x x PHYS53 Chater our (II) 64

65 The istei Relatio Cosier agai the o-uiformly oe semioutor. If there are o eletrial oetios, the the semioutor is i thermal equilibrium a the eletro a hole urret will be zero: J ex ed x Assume (x), the: Substitute the exressio for the eletri fiel, we obtai: 1 x x e x ed e x x x D e x J e x ed We a also obtai the exressio for holes: x x PHYS53 Chater our (II) 65 D istei relatio e

66 The istei Relatio xamle: assume the mobility for a artiular arrier is 1 m /(Vs) at T = 3 K. Determie the iffusio oeffiiet of this arrier. D e 5.9m /s.59v1m / V s PHYS53 Chater our (II) 66

67 Reaig Materials D. A. eame, Semioutor Physis a Deies: Basi Priiles, Irwi, Bosto, MA 116, 199, Chater 5, Carrier Trasort Pheomea. PHYS53 Chater our (II) 67