9.1 Direct ad idirect semicoductors Valece bad (VB) ad coductio bad (CB) of a semicoductor are searated by a eergy ga E G = 0.1... 4 ev. Direct semicoductor (e.g. GaAs): Miimum of the CB ad maximum of the VB are at the same ositio i the Brilloui zoe. Idirect semicoductor (e.g. ): Miimum of the CB ad maximum of the VB lie at differet ositios i the Brilloui zoe. (a) 3 GaAS (b) 4 Eergy (ev) 2 1 0 0.29 ev 0.38 ev E =1.42 ev G Eergy (ev) 2 0 2 3.4 ev E =1.1 ev G 1 L Γ Wavevector X 4 L Γ Wavevector X Figure 9.1: Scheme of the badstructure of (a) GaAs ad (b). Adoted from Wikiedia. I the viciity of the extrema of the CB ad VB, resectively, the bad structure E(k) ca be exressed i the arabolic aroximatio (oly terms u to oder k 2 are ket). The surfaces of costat eergy are ellisoids. 91
Withi the arabolic aroximatio the disersio of the coductio bad ca be writte as: E CB (k) = 2 [ k 2 x k 2 y 2m t k2 z 2m l ]. (9.1.1) Here, m t ad m l are the socalled trasverse ad logitudial effective mass, resectively. is the eergy of the coductio bad miimum. For, the ratios of the effective masses ad the free electro mass m are give by m t m = 0.19, m l m = 0.98. (9.1.2) (9.1.3) [001] [010] [100] E (k)=cost CB Figure 9.2: Scheme of the surfaces of costat eergy for the coductio bad of. The badstructure i the viciity of the VB maximum is usually more comlicated tha suggested by Fig. 9.1. Two bads with differet curvatures meet at the VB maximum. These bads are called the heavy hole bad ad the light hole bad. The corresodig effective masses are give by m hh ad m lh, resectively (see Fig. 9.3). A third bad, the socalled slitoff bad, with effective mass m soh is lowered by the slitoff eergy from the other two bads due to siorbit iteractio. The values for ad GaAs are reseted i table 9.1. 92
9.2 Charge carrier desity i itrisic semicoductors E(k) k Heavy * holes (m ) hh Light * holes (m ) lh slitoff * holes (m ) soh Figure 9.3: Scheme of the badstructure i the viciity of the VB maximum. m hh/m m lh/m m soh/m (ev ) 0.49 0.16 0.23 0.044 GaAs 0.45 0.082 0.17 0.34 Table 9.1: Effective masses ad slitoff eergy of ad GaAs. Source: Hukliger, Festkörerhysik. 9.2 Charge carrier desity i itrisic semicoductors I what follows, we assume that the disersio relatios i the viciity of the CB miium ad the VB maximum, resectively, are give by: E CB (k) = 2 k 2 2m c (9.2.1) E V B (k) = E v 2 k 2, 2m v (9.2.2) with : Eergy of a electro at the CB miimum, E v : Eergy of a electro at the VB maximum, m c : Effective mass of a electro ear the CB miimum, m v : Effective mass of a electro ear the VB maximum. 93
The corresodig desities of states are D c = (2m c) 3/2 E E 2π 2 3 c with E, (9.2.3) D v = (2m v) 3/2 2π 2 3 E v E with E E v. (9.2.4) The desity of states withi the eergy ga is zero. I thermal equilibrium, the occuatio of the electroic states is govered by the Fermi Dirac distributio: f(e) = 1 ex [(E E F ) /] 1. (9.2.5) For udoed semicoductors, the Fermi eergy E F lies withi the eergy ga (see below). For E E F 2, the FermiDirac distributio ca be aroximated by the Boltzma distributio: f(e) = ( 1 ex [(E E F ) /] 1 ex E E ) F 1. (9.2.6) The desity of electros i the coductio bad ca be writte as: = f(e)d c (E)dE. (9.2.7) With equatio (9.2.3) ad (9.2.6), the electro desity becomes = (2m c) 3/2 2π 2 3 e EF E e E k BT de. (9.2.8) Substitutig X = (E )/, we obtai: = (2m c) 3/2 2π 2 3 = 2 ( mc 2π 2 () 3/2 e Ec E F X e X dx, (9.2.9) 0 }{{} π 2 ) 3/2 e Ec E F k BT. (9.2.10) 94
9.2 Charge carrier desity i itrisic semicoductors The corresodig desity of holes i the valece bad is give by: = Ev D v (E) [1 f(e)] de. (9.2.11) After a short calculatio, we fid: ( ) 3/2 mv = 2 e Ev E F k BT. (9.2.12) 2π 2 Next, we defie the socalled effective desities of states for the electros ad holes, resectively: N c ( ) 3/2 mc = 2, 2π 2 (9.2.13) N v ( ) 3/2 mv = 2. 2π 2 (9.2.14) The roduct of the electro ad hole cocetratios is give by the socalled law of mass actio: ( ) = 2 i = N c N v e Eg ( ) 3 ( ) kb T = 4 (m 2π 2 c m v ) 3/2 e Eg. (9.2.15) For a itrisic semicoductor, charge eutrality demads =. (9.2.16) The itrisic charge carrier cocetratio is give by: ( ) i = i = N c N v e Eg ( ) 3/2 ( ) 2 kb T = 2 (m 2π 2 c m v ) 3/4 e Eg 2. (9.2.17) For at 300 K, the itrisic carrier cocetratio is i = 1.5 10 10 cm 3. The FermiLevel E F (T) takes the value for a give temerature T such that the charge eutrality coditio is fulfilled. With we obtai = = N c e Ec E F E F (T) = E v 2 = N v e Ev E F k BT, (9.2.18) k ( ) BT 2 l Nv = E v N c 2 3k BT 4 l ( mv m c ). (9.2.19) 95
9.3 Doig of semicoductors The free carrier cocetratio i semicoductors ca be i icreased by doig, i.e, by the additio of electrically active imurities to the semicoductor. I this coectio, oe distiguishes betwee two tyes of imurities: Imurities which icrease the umber of electros i the coductio bad are called doors. Imurities which icrease the umber of holes i the valece bad are called accetors. P B doed silico doed silico Figure 9.4: Doig of silico. I what follows, we will discuss doig usig the examle of a crystal. Here, each atom is covaletly boud to four eighborig atoms. A door is created by relacig a atom by a valecefive atom such as P. Four of the five valece electros of the P atom are required to bid the P atom i the crystal. The fifth electro, however, fids o arter ad is oly weakly boud to the ositively charged door core. Because of its small bidig eergy, the fifth electro ca be easily romoted to the coductio bad. To estimate the bidig eergy E d of the fifth electro, we use a simle hydrogeatom model i which we relace the free electro mass m by the effective mass m ad iclude the screeig effect of the surroudig via the dielectric costat ǫ r : E ν = 1 m e 4 1 2 (4πǫ r ǫ 0 ) 2 2 ν = E 2 H,ν m mǫ 2 r. (9.3.1) Here, ν is the ricial quatum umber ad E H,ν is the corresodig eergy level of the hydroge atom. 96
9.4 Carrier desities i doed semicoductors The ioizatio eergy of the ν = 1 level of a hydroge atom is 13.6 ev. For the P atom i silico, we fid with m = 0.3m ad ǫ r = 11.7, that the ioizatio eergy is i the order of 30 mev. The modified Bohr radius of the electro i the crystal is give by r = 4πǫ 0ǫ r 2 m e 2 = a 0 ǫ r m m, (9.3.2) where, a 0 is the Bohr radius of the hydroge atom. For the values metioed above, we fid r 2 m. This value is cosiderably larger tha the iteratomic searatio betwee two atoms i the crystal (0.23 m). This is a a osteriori justificatio of the use of the dielectric costat to describe screeig. Relacig a atom by a valecethree atom such as B creates a accetor. Here, oe bod with a eighborig atom is usatisfied. This usatisfied bod ca easily accet a electro from the valece bad. Formally, this rocess ca be iterreted as the romotio of a hole from the accetor level to the valece bad. The bidig eergy E a of the hole ca be estimated i a calculatio aalogous to the case of a door. Here, however, the effective mass is that of a hole i the valece bad. doed silico doed silico Eergy Door level E d E D Eergy Accetor level E a E A E v E v Positio Positio Figure 9.5: Eergy levels of doors ad accetors. 9.4 Carrier desities i doed semicoductors The law of mass actio is also valid for doed semicoductors: ( ) = N c N v e Eg. (9.4.1) 97
The charge eutrality coditio i the resece of accetors ad doors reads: N A = N D. (9.4.2) The total door cocetratio N D cosists of the sum of the cocetratios of the eutral doors N 0 D ad the ioized doors N D: N D = N 0 D N D. (9.4.3) The electrooccuatio robability of the door level ca be calculated with the hel of the FermiDirac distributio: N 0 D = N D ex [(E D E F ) /] 1. (9.4.4) At this oit, we eglect a slight comlicatio that arises from the fact that the door level ca be usually oly occuied by a sigle electro. The total accetor cocetratio is give by N A = N 0 A N A. (9.4.5) Here, N 0 A is the cocetratio of the eutral accetors ad N A is the cocetratio of the ioized accetors. The holeoccuatio robability of the accetor level is give by N 0 A = N A ex [(E F E A ) /] 1. (9.4.6) I what follows, we cosider a tye semicoductor (N A = 0). I that case, we have = N c e Ec E F k BT, N D = N 0 D N D, (9.4.7) (9.4.8) N 0 D = N D ex [(E D E F ) /] 1. (9.4.9) Electros i the coductio bad origiate either from the valece bad or from doors: = N D. (9.4.10) 98
9.4 Carrier desities i doed semicoductors As a further simlificatio, we assume that the effect of doig domiates over the itrisic carrier cocetratio: N D = N D ND 0 ) 1 = N D (1. (9.4.11) ex [(E D E F ) /] 1 The Fermi level E F follows from equatio (9.4.7): e E F k BT = N c e Ec k BT. (9.4.12) Substitutig equatio (9.4.12) i (9.4.11), we obtai: N D 1 e E d k BT /N C (9.4.13) with E d = E D. (9.4.14) This is a quadratic equatio for 2 e Ed N c N D, (9.4.15) which has the followig hysically meaigful solutio: 2N D (1 1 4 N D N c e Ed ) 1. (9.4.16) Next, we will discuss the temerature behavior: Freezeout rage (low temeratures with 4 N D Nc e Ed k BT 1) : N D N C e 2E d k BT, (9.4.17) E F (T) E C E d 2 k BT 2 l ( ) Nc (T). (9.4.18) N D 99
Saturatio rage (itermediate temerature regime with 4 N D Nc e Ed k BT 1): N D = cost. (9.4.19) ( ) Nc (T) E F (T) E C l. (9.4.20) N D Itrisic rage (high temeratures): i N D. (9.4.21) Gradiet E g/2kb l Gradiet E d/2kb T 1 Eergy Fermi Eergy E (T) F E D Itrisic rage Saturatio rage Freezeout rage E v T 1 Figure 9.6: Qualitative temerature deedece of the carrier cocetratio ad the Fermi eergy E F of a tye semicoductor. 9.5 Mobility of semicoductors I thermal equilibrium, the average kietic eergy of a coductio electro i a semicoductor ca be estimated with the hel of the theorem for equiartitio of eergy: 1 2 m v 2 th = 3 2 k BT. (9.5.1) The thermal velocity v th for at room temerature is i the order of 10 7 cm/s. The radom thermal motio leads to a vaishig average dislacemet of a electro. 910
9.5 Mobility of semicoductors If we aly a small electric DCfield E, the electros will acquire a additioal velocity comoet, the so called driftvelocity v d, which is suerimosed o the thermal velocity. This drift velocity is resosible for the et curret desity. To calculate the drift velocity, we start with the classical equatio of motio: m v m v = qe. (9.5.2) τ I steady state ( v = 0), the average electro velocity is the drift velocity: v d = qeτ m. (9.5.3) The electro mobility b is defied by: with v d = be b = qτ m. It is related to the coductivity via: σ = qb. (9.5.4) (9.5.5) (9.5.6) So far, we have oly cosidered electros. The total curret desity results from the motio of both electros ad holes. It is give by: j = q (v d,e v d,h ) = q (b e b h ) E. (9.5.7) I what follows, we cosider the temerature deedece of the electro mobility. The hole mobility follows a similar tred ad will ot be exlicitly discussed. The scatterig time τ is roortioal to the average electro velocity < v > ad the scatterig cross sectio Σ of the corresodig scatterig rocess: 1 τ < v > Σ. (9.5.8) ce the drift velocity is usually much smaller tha the thermal velocity for small field stregth, we fid < v > T. (9.5.9) 911
First, we will cosider scatterig of electros from acoustic hoos. For this urose, we assume that the scatterig cross sectio is roortioal to the average vibratioal amlitude < s 2 > of a hoo. For temeratures above the Debye temeratue, oe fids Mω 2 < s 2 >=. (9.5.10) Hece, we exect that the relaxatio time due to scatterig from acoustic hoos varies as 1 τ h < v > Σ h T 3/2. (9.5.11) For the mobility, we thus obtai the estimate b h T 3/2. (9.5.12) Aother imortat source of scatterig i semicoductors is scatterig from charged defects (ioized doors or accetors). Here, we assume that this rocess ca be described aalogous to Rutherford scatterig. The corresodig scatterig cross sectio is roortioal to the iverse fourth ower of the velocity: Σ def < v > 4. (9.5.13) The iverse scatterig time varies as 1 τ def T 3/2 (9.5.14) ad the mobility is roortioal to b def T 3/2. (9.5.15) log (b) ~T 3/2 due to ~T 3/2 due to charged hoos defects log (T) Figure 9.7: Qualitative temerature deedece of the electro mobility of a semicoductor. 912
9.6 The juctio 9.6 The juctio 9.6.1 Ubiased juctio I what follows, we cosider a juctio, i.e., a semicoductor crystal which is doed o the left half with accetors (semicoductor) ad o the right half with doors (semicoductor). I a gedakeexerimet, we start with the two searate halves. The roerties of these halves, e.g., the ositios of the corresodig FermiLevels, have bee discussed i the revious sectios. If we combie the two halves ad aly o voltage, the large carrier cocetratio gradiets causes carrier diffusio. The electros from the side diffuse i the side ad recombie there with holes. Likewise, holes from the side diffuse i the side ad recombie with electros. As a result, a double layer of ucomesated egative accetors ad ositive doors forms ear the juctio. This sacecharge zoe creates a electric field that couteracts the diffusio. I thermal equilibrium, the Fermi level, i.e. the electrochemical otetial, takes a costat value withi the whole crystal. The bedig of the badstructure ear the juctio ca be described by a macrootetial V (x). This macrootetial is related to the sace charge ρ(x) via the Poisso equatio: 2 V (x) x 2 = ρ(x) ǫǫ 0. (9.6.1) Far away from the juctio, the cocetratio of majority carries (electros i the regio, holes i the regio) is give by: = N c e E c E F, = N v e E F E v k BT. (9.6.2) (9.6.3) The cocetratio of the corresodig miority carriers (holes i the regio, electros i the regio) ca be calculated from 2 i = = N v N c e E c E v k BT. (9.6.4) The diffusio voltage is related to the carrier cocetratios: ( ) ( ) ev D = (Ev Ev) NA N D = l = l. (9.6.5) 2 i 2 i 913
Here, we assume that we are i the saturatio rage. I thermal equilibrium, the et curret flow of carriers (electros ad holes) across the juctio vaishes. For each tye of carrier, the drift curret caused by V D is exactly comesated by the diffusio curret due to the carrier cocetratio gradiet: j = j diff j = j diff j drift j drift = e = e ( ) D x b E ( ) D x b E = 0, (9.6.6) = 0. (9.6.7) Here, D ad D are the diffusio costats for electros ad holes, resectively. With the hel of equatio (9.6.6), we fid where D x = b V x, E = V x has bee used. (9.6.8) (9.6.9) I the sacecharge regio, the electro cocetratio is ositio deedet with ( ) (x) = N c ex E c ev (x) E F. (9.6.10) The gradiet of the electro cocetratio ca be calculated as x = e V x. (9.6.11) A comariso with equatio (9.6.8) yields the socalled Eistei relatio: D = k BT e b. (9.6.12) I what follows, we use the socalled Schottky model of the sace charge zoe: ρ(x) = 0 for x < d en A for d < x < 0 en D for 0 < x < d 0 for x > d (9.6.13) 914
9.6 The juctio With a iecewise costat sacecharge desity, the Poisso equatio ca be easily itegrated. We fid for the regio (0 < x < d ) of the sacecharge zoe: ad E = en D ǫǫ 0 (d x) (9.6.14) V (x) = V ( ) en D 2ǫǫ 0 (d x) 2. (9.6.15) For the regio (d < x < 0), we get ad E = en A ǫǫ 0 (x d ) (9.6.16) V (x) = V ( ) en A 2ǫǫ 0 (x d ) 2. (9.6.17) Charge eutrality requires, that N D d = N A d (9.6.18) ad the cotiuity of V (x) at x = 0 demads: e ( ) ND d 2 N A d 2 2 = V ( ) V ( ) = V D. (9.6.19) 2ǫǫ 0 With the hel of the last two equatios, we ca calculate the satial exted of the sacecharge zoe: d = d = 2ǫǫ0 V D e 2ǫǫ0 V D e N A /N D N A N D, (9.6.20) N D /N A N A N D. (9.6.21) 915
N semicoductor semicoductor N D N A x ρ x E x E Drift ev(x) Diffusio ev D E v Diffusio l 0 l Sacecharge zoe Drift E F E v x Figure 9.8: Ubiased juctio. 916
9.6 The juctio 9.6.2 Biased juctio I this sectio, we cosider the effect of a exteral voltage U o the juctio. Because of the deletio of free carriers, the sacecharge zoe has a cosiderable larger resistace tha the rest of the semicoductor crystal. Hece, we ca assume that the otetial dro across the sacecharge zoe is equal to the exterally alied voltage. Outside of the sacecharge zoe, (x), E v (x), ad V (x) are costat withi the resective regios. The total otetial dro across the sacecharge regio is thus give by: V ( ) V ( ) = V D U. (9.6.22) Here, we defie that the otetial U is ositive whe the otetial of the side is icreased with resect to the side. As a effect of the alied voltage, the extet of the sacecharge zoe becomes: d (U) = d (U = 0) d (U) = d (U = 0) 1 U V D (9.6.23) 1 U V D (9.6.24) I thermal equilibrium ad without alied voltage, the drift currets of electros ad holes are comesated by the corresodig diffusio currets. The electro drift curret (hole drift curret) results from electros (holes) which have bee thermally geerated withi the side (side) of the sacecharge zoe ad which move uder the ifluece of V D to the side (side). For that reaso, the drift currets are ofte called geeratio currets ( I ge ad I ge, resectively). They are largely ideedet of the exterally alied voltage. However, they ca be icreased if we icrease the geeratio rates of the miority carriers by illumiatio of the juctio. The situatio is differet for the diffusio currets which stem from the majority carriers. ce the majority carriers have to move agaist the otetial, oly the fractio ex e(v d U)/ (Boltzmafactor) of the majority carriers ca overcome the otetial ad reach the other side. There the electros ad holes each are miority carriers which recombie with the corresodig majority carriers (holes ad electros, resectively). The diffusio curret is thus ofte called the recombiatio curret ( I rec, resectively). I rec Combiig these effects, we obtai for the electro currets: I rec (U = 0) = I ge = cost, (9.6.25) I rec Vd U e k (U) e BT. (9.6.26) ad 917
Reverse bias Forward bias U U E F E v Drift Diffusio eu Diffusio E f E F E v Drift Diffusio eu Diffusio Drift E F E v Drift E v Figure 9.9: Biased juctio. Together, we thus have I rec (U) = I ge e eu k BT. (9.6.27) The total electro curret is give by I = I rec I ge ( ) = I ge e eu k BT 1. (9.6.28) For the hole curret, we fid after the aalogous aalysis a corresodig exressio. The total curret through the juctio is thus give by: I(U) = (I ge ( ) I ge ) e eu k BT 1. (9.6.29) 918
9.6 The juctio U I U I I ge ge (I I ) U Figure 9.10: Schematic reresetatio of the curret voltage characteristic of a juctio. 919