Gains in Activation Energy from Quasi Fermi Splitting, In Selectively Doped MQW Solar Cells

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1 Gains in Ativatin Energy frm Quasi ermi Splitting, In Seletively Dped MQW Slar Cells ARGYRIOS C. VARONIDES, ROBERT A. SPALLETTA ANDREW W. BERGER Department f Physis and Eletrial Engineering, University f Srantn, A Jesuit University, 8 Linden Street, Srantn, PA 1851, USA. Abstrat: - Illuminatin f a Multiple Quantum Well (MQW slar ell, affets the ermi level psitin in the gap, diretly under the ndutin band f the narrw gap layer. In this mmuniatin, we explre ermi level shifting and splitting in these strutures. The neutrality nditin is examined in terms f (a availability f arriers (b arrier nfinement ( esaping arriers (d dping levels (e layer width and (f ermi level dependene n dping and illuminatin. We nlude that shrt iruit urrents and pen iruit vltage are seriusly affeted. Key Wrds: - Quantum wells, ermi level, slar ells, pen iruit vltage, shrt iruit urrent. 1 Intrdutin Lw dimensinal ptial devies are ideal prbes fr the study f exess arriers in quantum wells. The latter are regins where quantum size phenmena beme imprtant, beause they are essentially traps f exess eletrns. Optial devies r ptial heterstrutures ffer tw advantages ver their bulk unterparts: (a they prvide wider ptial gaps, expanding the spetrum t shrter wavelengths and (b they redue rembinatin thrugh mass separatin: pre-existing eletr-stati separate eletrn-hle pairs (in the ndutin band and the valene band respetively This latter phenmenn is mre aentuated in redued dimensinality strutures. Typial examples are GaAs and its Allys: GaAs-AlGaAs (at several Al perentages. In slar ells; they prvide higher shrt iruit urrents (at least the III-V nes. Nmenlature: ΔΕ = ndutin band differene E 2 -E d2 = dnr shallw levels d 1, d 2 = lw and wide gap width N d2 = dnr nentratin N = number f perids γ,1 = miniband width ΔΕ = minibands (frm the bttm f the QW ΔΕ C = E E (ativatin 2 The neutrality nditin The nentratin f eletrns bund in dnr atms an be estimated by ignring eletrn-eletrn interatins. The mean number f eletrns is [1,2,3]: N d nd = 1 + (1/ 2 exp( Ed! E (1 Where, E d is the dnr level f energy in a sample with N d (m -3 dnr atms and E is the ermi energy r hemial ptiental. r an n-type sample (N d >N a, eletrns distribute themselves in suh a way that verall harge neutrality is sustained. n + n d = (N d " N a + p v + p a (2 The number f ndutin and dnr eletrns at sme temperature T inreases ver the T= state by the number f hles in the valene band (ρ v. This is the neutrality nditin, whih allws ne t late the ermi level at any temperature. In a multi-layered struture nsisting f N repeatable perids, eah f width (d equal t the sum f the ptential barrier width (d 1 and the quantum well width (d 2. Cndutin arriers me frm the dnr sites in the gap f the layer dped with dnr impurities. The rle f suh dnr atms is t raise the ermi level

2 lser t the ndutin band f the lwer gap material. Mre free eletrns spill int neighbring quantum well sites inreasing the eletrn ppulatin and reduing the gap between the ndutin band and ermi energy, and sine ndutivity σ = σ ο exp(-(e -E /kt the transprt prperties f slar ells are enhaned. While ndutin is pssible beause f exess arriers frm the dnr sites, neutrality requires that the eletrns (per vlume ntributed by dnr atms be equal in numbers with the arriers left at the dnr levels, plus the arriers that remain in the quantum wells, plus the ndutin band free arriers. This is summarized in the fllwing expressin (2 refers t the wide gap and 1 t lw gap layer: (NV 2 + n w (NV 1 + # V T "E deg 3d (E f (E (3 (NV 2 Where n d2 is the eletrn nentratin due t dnr atms, V 1 and V 2 are the vlumes f layers 1 and 2, V T is the ttal vlume f the sample (the slar ell itself, N is the number f perids f the superlattie, and N d2 is the dnr nentratin in layer 2 (AlGaAs. ΔE is the ndutin band disntinuity, g 3d is the bulk density f states (DOS, and f(e is the prbability distributin f the eletrns (ermi-dira r Maxwell-Bltzmann. Sine V i /V T = d i /(Nd, expressin (3 bemes: m * g 2d = (! #( E " E( k z 2 h k z (6 Where Theta is the step funtin, and k z is the wavevetr f the III-V ally in the grwth diretin arding t the tight binding thery. The energy dispersin relatin is then: E( k = E + 2 s( k d (7 z n! n z Assuming a Maxwell-Bltzmann distributin funtin fr the f(e prbability, and integrating (5 ver the widths f the minibands in the quantum wells, we btain: n w (T = 4" exp # exp, #8" exp # exp #, +1" exp # exp, [ ] +K Terms with the 2 nd mini - band (8 2!3 Where " = m * ( kt /# h ( m, and "E is the distane f the miniband frm the ermi level plus the ativatin energy: n 2 d 2 ' + n 1 # d w ' + # d = N 2 d 2 ' # d * (E def (Eg 3d (E (4 E " E = #E = #E + #E C All quantities in (8 are knwn parameters based n the gemetry f the superlattie, and the expnential fatrs inlude the ermi energy. 3 Carriers in QW s Carriers in the quantum wells are expressed via similar integrals as in (3: n w (T = Eg 2d (E f (E (5 Where the DOS in (5 is 2-dimensinal: 4 Carriers due t dnrs and arriers abve E 2 The number f arriers dnated by shallw dnr impurities is [4, 5, 6, 7]: # exp " E 2 " E ( (9 kt ' Replaing the differene in (9 in terms f the band disntinuity:

3 exp " #E ' C exp " #E ' C (1 kt ( kt ( On the ther hand, free arriers abve the ndutin band f the wide gap layer, are simple arriers in the energy ntinuum in the bulk f bth layers in a mqw-slar ell. Namely, n = N exp " #E ' exp " #E ' C (11 kt ( kt ( Where N is the effetive DOS f the wide gap material. 5 Ativatin energy ΔΕ C We slve fr the ativatin energy frm (4 via (5, (8, (1 and (11: + exp # "E ' / - kt ( - "E C = kt ln, sinh "E * '.- 2kT ( 1 - With = 1 A " N ' + d " ' exp ( E ' # 2d 2 N d 2 # 2N d 2 d 2 # kt And where (12 A = 4" exp # E ( ' * sinh + (, ' * 3 sinh + ( ' * # sh + (/. ' * 1 kt kt - kt kt, +5 exp # E + 2+ (/. ' * 1 - kt 6 QL s and illuminatin We nsider a single quantum well between tw layers f wide gap material (ptential wells. Typially, this means a GaAs layer in ntat with tw layers f AlGaAs frm left and right. An peniruited slar ell, under illuminatin, develps a nn-zer vltage in its utput. This alled the pen iruit vltage, whih is the mst imprtant slar ell parameter. It is beause f this vltage, that the devie will eventually sustain a urrent thrugh a lad, when applied fr prdutin f eletriity. The questin bemes: what is the effet n the ermi levels n bth sides f the slar p-n juntin? Under zer illuminatin the ermi levels inide. This is beause the number f arriers frm left t right equals the number f arriers frm right t left. Nte that if f 1 (E, f 2 (E are the ermi funtins in the tw regins, (E, g 2 (E the DOS f the tw regins, then: ( E ( 1" f 1 ( E g 2 E ( E f 1 ( Eg 2 E and therefre f 1 = f 2. ( f 2 ( E = ( ( 1" f 2 ( E (13 Als, under illuminatin, pen iruit vltage develps, s that: ( E ( 1" f 1 ( E g 2 E + qv ( E f 1 ( Eg 2 E + qv rm (14 it fllws that f 1 ( E = f 2 E + qv Or that ( f 2 ( E + qv = ( 1" f 2 ( E + qv ( ( (14 E 2 " E 1 = qv (15 Expressin (15 establishes the fat that under peniruit nditin and under illuminatin, the slar ell (any slar p-n juntin reats within neutrality ardingly. irst f all, due t pen-iruit vltage, the devie adjusts t zer urrents by splitting the ermi level int tw quasi-ermi level in regins 1 and 2. This exatly is the meaning f expressin (15. On the ther hand, illuminatin auses a further split in the quasi ermi level f medium 1 (lw gap medium. This is due t the fat that as arriers in the quantum wells inrease in numbers, the mere quantum event f arrier nfinement ause a readjustment f the quasi-ermi level in medium ne. Thus, under illuminatin (a the ermi levels split by qv due t neutrality nditins and (b the quasi ermi level E 1 shifts upwards by an amunt Δµ f the rder f 12 mev. Thus the verall pen iruit

4 vltage will inrease. igure 1 belw depits the situatin: a quantum well and ermi level behavir. igure 1: Three quasi-ermi levels in the viinity f a quantum well with its eigen-energy. The ttal pen iruit vltage and the exess ermi level splitting are shwn expliitly. d 2 (A E -E 1 (ev N split E -E 1 (ev With split Table 2: Ativatin energy dependene n dping. E 2 -E 1 =qv E 2 -E 1 Δµ N d2 (m -3 E 1 -E 1 (ev Withut Q-split E 1 -E 1 (ev With Q- split 8.65 x x x x x x As seen frm ig. 1 abve, the expeted pen iruit vltage inreases by an amunt Dm, due t quantum size effets. Thus an verall gain is expeted as far as pen iruit nditins are nerned. 7 Results The gal in this mmuniatin is t shw lear imprvements in the ativatin energy f a slar ell with quantum wells in its intrinsi regin. Ativatin energy is the differene in energy between the bttm f the quantum well (hene the ndutin band E 1 and the atual ermi energy right beneath it. As seen in ig. 1, the ermi energy shifts and splits under illuminatin. Table 1 belw depits the gain n ativatin energy. Table 1: Ativatin energy withut and with ermi level split due t quantum size effets, in the quantum wells. As seen frm Table 2, high dping f the barrier regin auses narrwer ativatin energy. The third lumn f table 2 depits the final psitin after the ermi level split rretin. We ntie that the ativatin energy redues by a net amunt f the rder f 12 mev. Suh redutins ause an inrease t ndutivity by a fatr exp (Δµ/kT. Thus gains in the ativatin energy and in the pen iruit vltage are expeted when superlatties are inrprated in heterstruture slar ells. 8 Cnlusins When superlatties [8,9] are used in slar ells, the ativatin energy is the differene in energy between the bttm f the ndutin band and the ermi level (see ig. 1. This study has shwn that the ermi level depends n (a the gemetry f the devie (this means the layer harateristis and (b n the dping levels, as seen frm Table 2. In bth ases (as in Table 1, 2 the ativatin energy redues. Our general nlusin is that exessive dping f the wide gap layer and gradual inrease f the f the width f the ptential barrier prdue a ermi level

5 split whih is equal t the pen iruit vltage. urthermre, the pen iruit vltage will inrease by the amunt f ~12 mev, whih prdues an even higher pen iruit vltage. Lsses are fund frm the fat that temperature inreases ause a redutin in the ativatin energies. We attribute this fat t the rle f high temperatures: the higher the temperatures the mre sattering expeted, and hene mre lsses. 9 Referenes [1] Asft and Mermin, Slid State Physis, Saundres Cllege, 1976 [2] I. Chen, Phys Rev B, 32 ( [3]. Capass, IEEE, J. Quantum Eletrnis-22 ( [4] H.L. Strmer et al, Appl. Phys. Lett, 44 ( [5] AC Gssard, IEEE, J. Quantum Eletrnis-22, N 9 ( [6] SK Ly, Phys Rev B, 35(15 (1987 II 335 [7] A Rthwarf and AC Varnides, 21 st IEEE PVSC, lrida ( [8] L. Esaki, IEEE J. Quantum Eletrnis 22, N 9 ( [9] V Milanvi et al, Phys Rev B, 37 (

Content 1. Introduction 2. The Field s Configuration 3. The Lorentz Force 4. The Ampere Force 5. Discussion References

Content 1. Introduction 2. The Field s Configuration 3. The Lorentz Force 4. The Ampere Force 5. Discussion References Khmelnik S. I. Lrentz Fre, Ampere Fre and Mmentum Cnservatin Law Quantitative. Analysis and Crllaries. Abstrat It is knwn that Lrentz Fre and Ampere fre ntradits the Third Newtn Law, but it des nt ntradit