Hall effect: the role of nonequilibrium charge carriers

Transcription

1 INVESTIGACIÓN REVISTA MEXICANA DE FÍSICA 7 ( AGOSTO 20 Hall effect: the role of oequilibrium charge carriers S. Molia Valdovios ad Yu. G. Gurevich Cetro de Ivestigació y de Estudios Avazados del Istituto Politécico Nacioal, Deartameto de Física, Av. IPN 208, México, D.F., 07360, México, smolia@fis.civestav.mx Recibido el de abril de 20; acetado el 3 de juio de 20 A ew model of the Hall effect i the case of a biolar semicoductor is reset. Takig ito accout the oequilibrium carriers, thermal geeratio ad recombiatio rocesses assisted by tras (Shockley-Read model, the exressios for the electrochemical otetial of electros ad holes, Hall field ad Hall costat R H are obtaied. The deedece of these exressios of the distributio of the carriers alog the directio of the Hall field i the case of itrisic ad extrisic semicoductors is studied. Keywords: Galvaomagetic heomea; biolar semicoductors; recombiatio; Hall effect. Presetamos u uevo modelo del efecto Hall e el caso de semicoductores biolares. Se toma e cueta ortadores fuera de equilibrio, rocesos de geeració y recombiació asistidos or tramas (modelo de Shockley-Read. Se obtiee exresióes ara los oteciales electroquímicos de electroes y huecos, camo de Hall y costate de Hall R H. Estudiamos la deedecia de estás exresióes de la distribució de ortadores a lo largo de la direcció del camo Hall, e el caso de semicoductores itrísecos y extrísecos. Descritores: Feómeos galvaomageticos; semicoductores biolares; recombiació; efecto Hall. PACS: 72..Gd; My; i. Itroductio The Hall effect has bee used for a log time i the research of mechaisms of coductio. With the hel of the Hall coefficiet R H, the tye of carrier reset i the semicoductor samle ca be determied. The Hall voltage of a late may be regarded as a sigal carryig iformatio. If the material roerties ad the device geometry are kow, the Hall voltage ca give iformatio about the magetic field H ]. I this case the Hall device is alied as a magetic sesor 2]. For the Hall effect, a coductor material is subjected to a electric E = E x ad magetic H = H y fields. A electro that travel with velocity v exerimet a force give by e( v H/c, the Loretz force. This force roduce a deflectio of the electro i the directio eredicular to the velocity v ad the magetic field H. If the semicoductor samle has the shae of a aralleleied, some electros ca be deflected by the field H to accumulate o the bottom or o the to face of the material. This sace deviatio of the charges geerates the aearace of a electric field E. The rocess cotiues util the electric field balaces the Loretz force 3]. This rocess origiates a voltage that ca be measured through the surfaces of the samle. With the hel of the voltage, the coefficiet of Hall i the case of electros (R H = /e ca be obtaied 4]. The magetic field acts differetly o each of the carriers i the biolar material, because each carrier has differet eergies. This imlies that each of the electros ad holes will have differet mobilities. Hece, carriers with eergies higher tha the average (hot electros will accumulate i the regio of oe of the crystal surfaces, ad carriers with eergies less tha the average (cold electros will accumulate i the oosite surface. As a result, this redistributio geerates a trasverse temerature gradiet T. This heomeo is kow as the Ettigshause effect ]. I this case we will have a thermoelectric field. The total electric field i this directio will iclude the electrical ad temerature effects 6]. Very similar heomea is discussed i review 7]. The simlicity of the Hall effect has led us to make some aroximatio of the roblem. It geerates some subtle itfalls, which are reset i every techique, but are ot always studied or take ito accout. A simlificatio ofte made is to cosider that effects due to fiite breadth of the secime ca be eglected. These imly that the cocetratio of free charge carriers does ot vary throughout the directio of the Hall field, 0 = 0 = 0 ], the samle cotais o free sace charges 8]. As we will see this cosideratio is etirely correct, oly whe we have a very large recombiatio. O the other had, the deviatio of the equilibrium of the charge carriers (electros ad holes, give race to a cocetratio gradiet alog the z axis. This gradiet geerates a diffusio electric field. At this oit other simlificatio is take: 0 = 0, i.e., the deviatio of ad is cosidered of the same order thereby esures the sace charge eutrality 9, 0]. But, i geeral case it is wrog (see ]. These roblems are reset i others studies, for examle whe a flows of a costat electric curret through a aisotroic semicoductor, the carrier cocetratio deviates from the equilibrium value, roducig regios of erichmet ad deletio. I this case 2] cosiders oly itrisic semicoductor, i.e., =. Whe the system is i equilibrium we have the chemical otetials µ ad µ for electros ad holes resectively, which are related by the equatio µ + µ = ε g. Whe the system is far from equilibrium, the above relatioshi is o loger satisfied, i.e., µ + µ ε g. Two quasi Fermi levels

2 HALL EFFECT: THE ROLE OF NONEQUILIBRIUM CHARGE CARRIERS 369 are geerated, deedig o the distributio of carriers alog the z axis i the samle. I some refereces 2,4,7 9,,3], it was eglected this deedece. Whe the system is i a oequilibrium state, the authors 2, 4, 7 9,, 3] cosider that the chemical otetials for electros ad holes are equal 0, 2], ad that 0 = 0. I semicoductors, i geeral oe ca see that which obliges us to work with two quasi-fermi levels µ ad µ, which are differet. I this form we ca give to a ew quasieutrality coditio, that it takes ito accout the distributio of carriers i the semicoductor samles, ad the rocesses of volumetric ad suerficial recombiatio i the case of fiite samles. With hel of the oequilibrium distributio of carriers, we determied the Hall coefficiet, Hall field ad we aalyse the effects of the electrical otetial as well as the origi of this. We are ot cosider the coectio of the cotacts to measure the Hall otetial, but it is very imortat for makig the measuremets. This arises because usually the boudary coditios are formulated i the case of oe circuits ad free charge surfaces. Really it is ecessary to work with aroriate boudary coditios that describe the curret flow i the circuit, i.e, whe the circuit is closed 4, ]. 2. Model descritio Cosider a biolar semicoductor of aralleleied form, legth 2a alog the x axis ad 2b is the trasverse legth alog the z axis. Additioally, we assume that the semicoductor is ot degeerate ad the carrier eergy is give by ε = 2 /2m,, where is the quasimometum of articles, m β = m, m are the electro ad hole effective masses, ε is the carrier eergy. The relatios which gover the behaviour of electros ad holes i the resece of a logitudial electric ad trasverse magetic fields ca be formulated as follows. The semicoductor is subjected to a exteral electric field E x ad a magetic field H y i the x ad y directios resectively which do ot vary over time. The the hole ad electro currets are give by 6]: j βx = e 2 βi β 0 E x e 2 βi β 20 E z + e β I β 20 zµ β (z, ( j βz = e 2 βi β 20 E x + e 2 βi β 0 E z e β I β 0 zµ β (z, (2 where β =, I β ik = 4βωi Hβ 3 πm β θ/2 0 ε k+3/2 e ε/θ β 2 dε. (3 νβ ( i (ε + ( ωhβ ν β (ε Here, e β = e, e are the charge for electros ad holes, ω Hβ = eh/m β c are the cyclotro frequecy, θ β = T, T are the electro ad hole temerature, E=(E x, 0, E z =(E 0 x, 0, ϕ, where ϕ is the electric otetial, E 0 x is the fixed exteral electric field ad E z is the electric field (the Hall field variable i the z directio, c is the velocity of light i vacuum. We work i the rage of low electric field, i which the heatig of electros ad holes is very small (T T T 6]. This is achieved by assumig that the material is exosed at room temerature, ad quickly dissiate the eergy of the electros ad holes o the surface of the samle, so that the electros ad holes thermalize to the lattice temerature. I the resece of the uique relaxatio mechaism of the electroic (hole mometum, the frequecy ca be writte as ν β (ε = ν β0 (T (ε/t q β 7, 8], where q β are the characteristic of relaxatio mechaism. We will cosider that the magetic field is weak ω H /ν β, hece Eq. (3 ca be writte as: I β ik 4βωi Hβ 3 T k πm β νβ0 i (T Γ(iq β + k + /2, (4 where Γ(iq β + k + /2 is the gamma fuctio. Let us write the distributios of chemical otetials, electro ad hole cocetratios i the followig form, µ β (z = µ β0 (T + δµ β (z, β(z = β 0 (T + δβ(z, where δµ β (z deotes the oequilibrium chemical otetials for electros ad holes, δβ(z are electro ad hole oequilibrium cocetratio, µ β0 are the equilibrium electro ad hole chemical otetials, β 0 are the equilibrium electro ad hole cocetratios. The equilibrium chemical otetials are related by the kow exressio 9], µ 0 = µ 0 ε g. Electro ad hole cocetratio i biolar semicoductor are rereseted by the followig exressio 20], β(z = N β (T e µ β(t /T. ( Here N β (T = ( mβ T /2π 3 2 3/2 4 are the electro ad hole desity of states. Exadig the equatio we get β(z = N β (T e µ β0(t /T e δµ β(z/t β 0 + δµ ] β(z. T It follows that δβ(z = β 0 T δµ β(z. (6 The, the hole ad electro currets would be exressed as, (see Eqs. (-(2: j βx = σ β xxe x σ β xzhe z + D β xzh z δβ(z, (7 j βz = ±σ β zxhe x + σ β zze z D β zz z δβ(z, (8

3 370 S. MOLINA VALDOVINOS AND YU. G. GUREVICH with σxx β 4e 2 β = 3 πm β ν Γ(/2 + q β, β0 σ β xz = D β xz = 4e 3 β 3 πcm 2 Γ(/2 + 2q β, β ν2 β0 4e 2 T 3 πcm 2 Γ(/2 + 2q β, β ν2 β0 Dzz β 4eT = 3 πm β ν Γ(/2 + q β, β0 σ β zx = σ β xz, 3. Equatios σ β zz = σ β xx. The macroscoic descritio of the trasort of oequilibrium charge carriers is erformed usig the cotiuity equatios for the electro ad hole curret desities ad the Poisso Equatio 2]. We cosider the static case ad the absece of the exteral geeratio of carriers (by light or other mechaism: j β = ±er β, (9 E = 4π ɛ ρ. (0 where ρ is the bulk charge desity, ɛ is the material ermittivity, R β are the electro ad hole recombiatio rates. Subtractig Eqs. (9, we obtai that ( j + j e(r R = 0. ( The cotiuity equatio for the total curret j = j + j from the Maxwell s equatios is Comarig Eqs. ( ad (2, we obtai j = 0. (2 R R = 0. (3 As it follows, the deviatio of the cocetratio of the electros traed i the imurity level δ t from the equilibrium value 0 t deeds o the deviatios of the electro ad hole cocetratios from their equilibrium values (δ= 0, δ = 0 through R ad R. The recombiatios rates R ad R are actually defied as the differece betwee the rates of cature of electros ad holes ad the thermal geeratio 22]. It follows from Eq. (3 that R = R = R, the cotiuity equatios for electros ad holes ad Poisso equatio take the form, j β = ±er β = ±er, (4 δ E = 4π ɛ δρ = 4πe ɛ (δ δ δ t. ( where E = E 0 + δ E, ρ = ρ 0 + δρ. I the case of a semicoductor that cotais a cocetratio N t of imurity, geeratio ad recombiatio rocesses are assisted by tras (Shockley Read model. The recombiatio rates R ad R ca be calculated by (we are assumig that electro, hole ad hoo have the same temerature 23], R = α (T (N t t α (T N t, (6 R = α (T t α (T (N t t. (7 Here, α (T ad α (T are the electro ad hole cature coefficiets, ( the electro (hole cocetratio whe the Fermi level matches the activatio eergy of the imurity. From R = R, we obtai the followig exressio for t, t = N t (α + α α ( + + α ( +, (8 By substitutio of Eq. (8 i Eq. (6 we obtai R = R = R = α α N t ( α ( + + α ( +. (9 Cosider that the alied field is weak. I this case, the excess of the electro cocetratio (δ t = t 0 t o the level of imurities is reduced to, δ t = α (N t 0 t δ + α 0 t δ α ( α ( + 0, (20 where 0 t, 0, ad 0 are the equilibrium values of the resective cocetratios. From Eq. (6 or (7 we ca obtai the recombiatio rate through the tras as follow 24]: ( Nt 0 ( ] t 0 t with R = τ + 0 δ δ, (2 τ = α α ( ( α ( α ( + 0 ( Quasieutrality aroximatio The cocet of quasieutrality is basic i semicoductor device aalysis ad widely used i the literature o trasort heomea 22]. Traditioally it defied as r 2 D l2 D, d2 ; where r D is the Debye radius, l D the diffusio legth, d the thickess of the samle. It ca be readily see from Poisso s equatio, 2 ϕ = 4πρ/ɛ, (23 that δρ/ρ 0 = (r D /l D 2. If (r D /l D 2, the δρ 0. Uder this coditio the Poisso equatio becomes a algebraic equatio (δρ 0, that does ot eed boudary coditios. This algebraic equatio established a relatioshi betwee the excess of carriers (electros ad holes ad make the Poisso equatio redudat. From the coditio δρ 0, the o-equilibrium charge carrier cocetratio δ ad δ, are related by, δρ = e(δ δ δ t = 0. (24

4 HALL EFFECT: THE ROLE OF NONEQUILIBRIUM CHARGE CARRIERS 37 It follows from the Eqs. (20 ad (24 that, where δ = ζδ, (2 ζ = α ( N t 0 t + α ( α ( α ( (26 0 t I this case, where ad R = R = R = δ τ, (27 ( Nt 0 t Λ = + 0. Boudary coditios τ = τ Λ, (28 ( 0 + t + 0 ζ. (29 The cotiuity equatios for electros ad holes have to be sulemeted by aroriate boudary coditios 4]. Alog the z directio, it follow that, (j z + j z z=±b = 0. (30 Similarly to the case of volume recombiatio, the exressio for surface recombiatio, ca be rewritte as follow: R s = R s = R s = S δ + S δ. (3 From the quasieutrality aroximatio (2, with R s = S δ, (32 S = S + S ζ. (33 I this case, S takes the meaig of surface recombiatio velocity. The surfaces of the semicoductor samle, chage their roerties at distaces of the order of Debye radius ad becomes ihomogeeous, which exlais the deedece of the rate of recombiatio R as a fuctio of the coordiate z, thus j z z=±b = ± es,2 δ, (34 where S ad S 2 are the surface recombiatio velocity i z = b ad z = b. 6. Solutio of the equatios The cotiuity equatios j = ( j + j = 0 show, that j z + j z = j z = cte. From the geometry of the Hall exerimet ad from the Eq. (30, we ca see that o curret flow i the z directio, the the costat is zero at all oits of the samle, j z + j z = j z = 0. With this coditio, we obtai the Hall electric field E H = E z, ad cosiderig the quasieutrality coditio (δ = ζδ, we get E z = xz σxz σxx+σ xx HE x + zz ζd zz σ xx+σ xx δ z. (3 Substitutig this exressio ito Eq. (8, we obtai: j z = xx σxz + σxxσ xz σxx + σxx HE x xx Dzzζ + σxxd zz δ σxx + σxx z, (36 j z = j z. (37 We ca see that uder the quasieutrality coditio ad the Eq. (37, we work oly with oe of the cotiuity equatios for electros or holes. By substitutio of the Eq. (36 i Eq. (9, it ca be rewritte as follows Thus 2 δ z 2 l 2 D δ l 2 D = 0. (38 = τ D, (39 ad D = e(σxx + σxx σxxd zzζ + σxxd zz. (40 The solutio of the Eq. (38 is: δ(z = γ e z/ld + γ 2 e z/ld. (4 The costats γ ad γ 2 should be determied from boudary coditios (34. Assumig that surface recombiatio velocity o the surfaces of the semicoductor samle is the same, S = S 2 = S, we obtai: γ = HE x γ, γ 2 = γ = HE x γ, σ γ = xxσxz + σxxσ xz (. 2 (σxx + σxx es sih( b l D + ( ed l D cosh( b l D Thus, the variatio of electro cocetratio alog the samle (z directio is: δ(z = 2HE x γ sih(z/l D. (42 Now we ca fid the Hall field E z E z = xz σxz σxx + σxx HE x + zz ζdzz σxx + σxx HE x (2 γ/l D cosh(z/l D. (43 From this result we obtai the value of the otetial δϕ(z, which ca be calculated from the equatio E z = z ϕ(z, itegratig from ( b, z ad assumig that ϕ( b = 0, δϕ(z = xz σxz (z + bhe x σ xx + σ xx zz ζdzz σxx + σxx 2HE x γ(sih(b/l D + sih(z/l D.

5 372 S. MOLINA VALDOVINOS AND YU. G. GUREVICH The Hall coefficiet R H = E z /Hj x is give by: R H = σ xz σxz (σxx + σxx 2 + zz ζdzz (σxx + σxx 2 (2 γ/l D cosh(z/l D, (44 Notice that R H = R H (z, thus it is ot ossible to defie R H as a costat. Aother asect to cosider is the deedece of R H o the amout ζ = ζ( 0, 0, N t, 0 t, 0, 0, which coect with the equilibrium electros ad holes cocetratios, as well as the imurity cetres (tras resets i the samle. 7. Results ad Discussio With the results reseted above we ca see i detail some secial cases, whe we have a fiite ad ifiite lifetime of recombiatio, ad study the itrisic ad extrisic semicoductors: i Strog Recombiatio: This case is obtaied whe the volumetric recombiatio rocesses are strog. This haes whe the lifetime is small τ 0 (d = 2b l D r D 0 or the surface recombiatio velocity is large, S. I this case the cocetratio of electros δ(z 0. The electrochemical otetial is give by, δ ϕ β (z = ξ(z + bhe x, (4 where e b 2 0 Γ( 2 ξ = + 2q 0 Γ( 2 + 2q ] cm ν 0 b 0 Γ( 2 + q + 0 Γ( 2 + q, here b = m ν 0 /m ν 0. Notice that the electrochemical otetial is equal to the otetial differece, i.e., δ ϕ (z=δ ϕ (z=δϕ(z. The electrochemical otetial is ot affected by the resece of recombiatio rocesses because that rocesses are very fast. The electric otetial deeds solely o the accumulatio of charge o the surface of the samle, which causes the olarizatio of the samle. O the other had the Hall coefficiet R H, takes the form: R H = σ xz σxz (σxx + σxx 2 = 3 π 0 Γ( 2 + 2q b 2 0 Γ( 2 + 2q ] 4ec 0 Γ( 2 + q + b 0 Γ( 2 + q ] 2. (46 We see that δ = δ = 0, ad recover the classic case for the Hall coefficiet. If the domiat scatterig mechaism is by acoustic vibratios for electros ad holes (q = q = /2, we get R H = (3π/8ec(( 0 b 2 0 /( 0 + b 0 2. (47 From this result we ca aly two cases: extrisic ad itrisic semicoductors. i.a Itrisic Semicoductors ( 0 0. I this case there are o imurities cocetratio (N t 0 ad the oly viable recombiatio mechaism is bad-bad, the quasieutrality coditio (2, chage by δ δ. I this case ζ ad ( 0 = 0 = i, we obtaied: R H = 3 π 4ec i Γ( 2 + 2q b 2 Γ( 2 + 2q ] Γ( 2 + q + bγ( 2 + q ] 2. (48 If the domiat scatterig mechaism is by acoustic vibratios for electros ad holes (q = q = /2, we get the classic result, R H = 3π ] b. (49 8ec i + b i.b Extrisic Semicoductors ( 0 0. The otetial differece (Hall otetial is give by: e Γ( 2 δϕ β (z = + 2q ] cm ν 0 Γ( 2 + q (z + bhe x. I this case the Hall coefficiet is: R H = 3 π 4ec 0 Γ(/2 + 2q Γ 2 (/2 + q. (0 Deedig o the domiat disersio rocess we obtai the classical results. If the scatterig mechaism does ot deed o eergy (q = 0, the R H = / 0 ec. If the domiat scatterig mechaism is by acoustic vibratios (q = /2 we obtaied R H = 3π/8ec 0. ii Weak Recombiatio: This case is obtaied whe the volumetric recombiatio rocesses are weak (this haes whe the lifetime is great τ, l D d = 2b r D, ad the surface recombiatio is weak, S 0. The cocetratio of o-equilibrium electros is: σ δ(z = xx σxz + σxxσ xz ] σxxd zz + ζσxxd zz HE x z, ( I the oit z = 0, the electroic cocetratio is zero ad icrease reachig its maximum value at the surface of the semicoductor samle z = ±b. Similar situatio we have with the cocetratio of holes. This behaviour will be reflected i the form of the otetial differece, σ δϕ(z = xz σxz ] (z + bhe x σxx + σxx (D zz ζdzz(σ xxσ xz + σxxσ xz (σxx + σxx(σ xxd zz + ζσxxd zz ] (z + bhe x, Notice that i this case the otetial differece is affected by the variatio of the electro cocetratio δ(z, ad beig differet from the classical results. This exressio icludes exlicitly the term ζ, which takes ito accout the hysical material characteristics, i.e., cocetratio of tras reset i the material, the activatio eergy, as well as the recombiatio coefficiets α ad α. This is a big differece with all results reseted i the literature, which excludes this deedece 9,,3]. As we have a variatio i cocetratio two

6 HALL EFFECT: THE ROLE OF NONEQUILIBRIUM CHARGE CARRIERS 373 quasi-fermi levels exist, which reflected i the electrochemical otetials for electros ad holes (δ ϕ (z δ ϕ (z. The Hall coefficiet deedece o the term ζ, as we see below R H = σ xz σxz (σxx + σxx 2 (D + zz ζdzz (σxxσ xz + σxzσ xx ] (σxx + σxx 2 (σxxd zz + ζσxxd zz (2 It is imortat to ote that whe the recombiatio rocesses are weak ad oly whe ζ = it ca be reroduce of the result resets i 9, 3] for the Hall effect. If the domiat scatterig mechaism is by acoustic vibratios for electros ad holes (q = q = /2, we obtai: R H = 3π 8ec 0 b 0 (b ( ]. (3 ii.a Itrisic Semicoductors ( 0 0. I this case there are o imurities cocetratio (N t 0 ad the oly viable recombiatio mechaism is bad-bad. The quasieutrality coditio (2, chage to δ δ, with ζ, ad Γ( 2 δ(z = + 2q Γ( 2 + q + Γ( 2 + 2q ] e 2 0 HE x z b Γ( 2 + q, 2cT m ν 0 The Hall costat i the case whe the domiat scatterig mechaism is by acoustic vibratios for electros ad holes (q = q = /2, is: R H = 3π ] b. (4 6ec i + b ii.b Extrisic Semicoductors ( 0 0. The cocetratio of o-equilibrium electros is: bγ( 2 δ(z = + 2q Γ( 2 + q + Γ( 2 + 2q ] e 2 0 HE x z Γ( 2 + q 2cT m ν 0 ζ, I this case we have ζ = ζ( 0, 0, N t, 0 t, 0, 0 thus we cosider two cases, i low temerature, where N t 0 t , cosequetly ζ, ii high temerature, where 0 N t t 0, thus ζ too. The similarity of the limits is due to the followig: cosiderig that the recombiatio coefficiets for electros ad holes are of the same order, i the first case the cocetratio of traig cetres domiate over other material cocetratios ad the low temerature reduce the trasitio of electros from the valece bad to the coductio bad. I the secod case the cocetratio of electros i the coductio bad domiates i the semicoductors materials, additioally the hight temerature stimulate the geeratio of electros. The electric otetial is give by δϕ(z = ad the Hall coefficiet is e(z + b Γ( cm ν q Γ( 2 + q ] HE x, ( R H = 3 π 4ec 0 Γ(/2 + 2q Γ 2 (/2 + q. (6 From the above results we see that the otetial differece ad the Hall coefficiet are idetical to the case of strog recombiatio for extrisic semicoductors. This ca be exlaied by sayig due to the high cocetratio of electros i the samle, the effects of a redistributio of carriers i the otetial are egligible, oly it is imortat the redistributio of charge o the surface of the samle. 8. Coclusio Whe the recombiatio is strog the Hall otetial, Hall field ad Hall coefficiet do ot deed o the distributio of the oequilibrium carriers, ad the traig cetres. We ca say that ideedet of the amout of imurities resets i the samle they do ot affect the measuremet of the Hall otetial. The source of the Hall otetial is due to accumulatio of carriers o the surface of the samle. If the recombiatio is weak, the Hall coefficiet, the Hall field, as well as the otetial is modified by the resece of the distributio of the oequilibrium carriers δ(z, that icludes exlicitly the deedece of the cocetratio of imurities. I the case of extrisic semicoductors, the Hall costat does ot deed o the distributio of the oequilibrium carriers i cases of strog ad weak recombiatio. The measured otetial does ot deed o the resece of the redistributio of the oequilibrium carriers alog the z axis, it oly deed o the cocetratio of carriers i the surface of the semicoductor samle. Ackowledgmets The authors thaks to CONACyT, México, for hel ad suort to make this work.. It is ractice i the literature to write H i a Loretz force ( F = e v H/c, wheever H B. For those cases where H B, it would be ecessary to relace H with B i the Loretz force ad i the esuig theory. 2. Edward Ramsde, Hall-Effect Sesors (Newes-Elsevier, USA, If we take a electro, the Loretz force will ot be balaced across the Hall field, evh/c ee z, because the velocity of

7 374 S. MOLINA VALDOVINOS AND YU. G. GUREVICH electros deeds o the eergy of the carrier v(ε. I this case it is ecessary to work with the average velocity of the electros < v(ε >, which give equality i the Lorez force e < v(ε > H/c = ee z. 4. R.S. Poovic, Hall Efect Devices (Istitute of Physics Publishig, Bristol ad Philadelhia, L.S. Stilbas, Physics of Semicoductors (Nauka, Moscow, F.G. Bass, V.S. Bochkov ad Yu. G. Gurevich, Soviet Physics- Semicoductors 7 ( Z.S. Gribikov, K. Hess, ad G.A. Kosiovsky, J. of Al. Phys. 77 ( D.C. Look, Electrical Characterizatio of GaAs Materials ad Devices (Joh Wiley ad Sos, New York, R. Ladauer ad J. Swaso, Physical Review 9 ( A. Koi ad R. Raguotis, J. Phys.: Codes. Matter 2 ( P.C. Babury, H.K. Heisch ad A. May, Proc. Phys. Soc. 66A ( E.I. Rashba, Soviet Physics-Solid State 6 ( G.E. Pikus, Techical Physics 26 ( O. Yu. Titov, J. Giraldo ad Yu. G. Gurevich, Al. Phys. Letters 80 ( I.N. Volovichev, J.E. Velázquez-Pérez ad Yu. G. Gurevich, Solid State Electroics 2 ( Yu. G. Gurevich ad F. Pérez Rodríguez, Feómeos de Trasorte e Semicoductores (Fodo de Cultura Ecoomica, México, V.F. Gatmakher ad Y.B. Leviso, Carrier Scatterig i Metals ad Semicoductors (North Hollad, Amsterdam, M. Ludstrom, Fudametal Carrier Trasort (Cambridge Uiversity Press, Cambridge, I.N. Volovichev ad Yu. G. Gurevich, Semicoductors 3 ( A.I. Aselm, Itroductio to Semicoductor Theory (Pretice Hall, Eglewood Cliffs, NJ, K. Seeger, Semicoductors Physics (Sriger, Berli, Yu. G. Gurevich, J.E. Velazques Pérez, I.N. Volovichev, ad O. Yu. Titov, J. Al. Physics 0 ( P.T. Ladsberg, Recombiatio i Semicoductors (Cambridge Uiversity, Eglad, I.N. Volovichev, G.N. Logviov, O. Yu. Titov ad Yu. G. Gurevich, Joural of Alied Physics 9 (