A Model of the Growth of Copper Selenide Thin Films Controlled by Diffusion and Chemical Reaction

Pro põe-se um mo de lo de cres ci men to de fil mes fi nos de se le ne to de co bre. É apre sen ta da uma equa ção ma te má ti ca que des cre ve a ci né ti ca de cres ci men to do fil me. São apre sen ta dos re sul ta dos si mu la dos bem como dis cus são so bre os re sul ta dos do mo de lo. Fez-se um pro ce di men to de ajus te dos da dos de ri va dos do mo de lo com re sul ta dos ex pe ri men ta is. Foi es ti ma do o co e fi ci en te de di fu são dos íons Cu(I) em se le ne to de co bre.


Introduction
This pa per de als with a the o re ti cal des crip ti on of the growth of thin films of cop per se le ni de on cop per subs tra te.Thin films of se mi con duc tor ma te ri als like Cu 2 Se, ZnS, GaAs, CdS, Cu 2 S, CuIn Se 2 , amongst ot hers, are po ten ti ally al ter na ti ve ma te ri als for subs ti tu ting bulk si li con in elec tronic de vi ces and par ti cu larly in pho to vol ta ic systems.The draw back of si li con is its high pro duc ti on cost with the neces sary pu rity.Use of thin films (1-10 µm thick) re pre sents con si de ra ble cost sa ving be ca u se of re du ced ma te ri al use (ca.20 g/m 2 ).
The se films can be pre pa red in se ve ral ways.Low pressu re va pour de po si ti on is pro bably the tech ni que which yields the best re sults, but re qui res sop his ti ca ted equip ment.Elec tro che mi cal met hods are sup po sed to be che a per and they can be more ea sily adap ted from la bo ra tory to in dustri al sca le.A very sim ple met hod is the che mi cal bath de posi ti on.It is ba sed on dip ping a so lid subs tra te in a con ve ni ently for mu la ted so lu ti on.More than one me chanism is pos si ble in this case.The se mi con duc tor can be formed at the so lid/so lu ti on in ter fa ce and then de po si ted onto the subs tra te.Cad mi um sulp hi de was pre pa red by dip ping a glass she et in a so lu ti on con ta i ning Cd(NH 3 ) 4 2+ and thi ou -rea 1 .The lat ter slowly de com po ses and ge ne ra tes sulp hi de ions whi le the for mer acts as a sour ce of Cd(II).Both ions com bi ne and ge ne ra te CdS, which de po sits on the glass.ZnS was pre pa red si mi larly 2 .
Anot her me cha nism ope ra tes when the subs tra te it self acts as sour ce of me tal ions of the se mi con duc tor to be prepa red.In this case, once the first mo no la yer is for med, it is ne ces sary that me tal lic ions move through the film, sin ce mo ve ment of se le ni um ions in cop per se le ni des prac ti cally does not oc cur.Elec trons also have to move to wards the exter nal in ter fa ce and com bi ne with some oxi di zing agent in the ne igh bour ho od.Alu mi ni um, zir co ni um, cop per and nic kel oxi des 3 and cop per se le ni de 4 are for med in this way.This work de als with a growth mo del of ber ze li a ni te, Cu 2-x Se (x re pre sen ting a small de vi a ti on of the sto i chi ometry).Cu 2-x Se is a se mi con duc tor with a band gap bet ween 1.1 and 1.7 eV.The elec tro nic trans port pro per ti es are cha rac te ri zed by po si ti ve signs of the ther mal emf and the Hall cons tant and by the ne ga ti ve tem pe ra tu re co ef fi ci ent of elec tri cal re sis ti vity in cer ta in tem pe ra tu re ran ges.Appa rently, the se mi con duc tor type of pro per ti es re sults from the de fect struc tu re of Cu 2-x Se ari sing from the oc curren ce of va can ci es in the Cu site, who se for ma ti on energy Ar ti cle is cha rac te ris ti cally low.The cop per va can ci es sup po sedly cre a te low-lying ac cep tor le vels in the band gap that are occu pi ed at or di nary tem pe ra tu res, and the re sul tant ho les in the va len ce band ac count for the hole con duc ti on 5 .
Thin films of Cu 2-x Se com pound can be pre pa red by dip ping cop per me tal in a so di um se le no sul fa te so lu ti on 4 .
(2-x)Cu(so lid) + SeSO 3 2-(aq.)→ Cu 2-x Se(so lid) + SO 3 2-(aq.) The growth rate de pends on the mo bi lity of the me tal lic ion through the so lid film.The less the dif fu si on co ef fi ci ent in the so lid, the lo wer the growth rate.As far as we know, the dif fu si on co ef fi ci ent of Cu(I) in cop per se le ni de has not yet been de ter mi ned.The elec tri cal con duc ti vity is also impor tant, for the elec trons also have to move in or der to main ta in the elec tri cal ne u tra lity.Ho we ver, this pa ra me ter is more im por tant in a gal va nos ta tic pro cess (like the ano dic for ma ti on of Cu 2 S 6 and CdS 7 ) than in che mi cal bath de posi ti on.
Con trol of the growth ki ne tics of se mi con duc tor is prere qui si te to the pre pa ra ti on of thin films with tech no lo gi cal ap pli ca ti ons.With re gard to Cu 2-x Se, the film grows ac cording to the pa ra bo lic law 4 , δ = At 1/2 , whe re δ is the film thick ness, t is the im mer si on time of the subs tra te in the solu ti on and A is a cons tant.Wag ner 8,9 de du ced this law, assu ming the exis ten ce of con cen tra ti on dif fe ren ces in each pha se.
For the growth of cop per se le ni de in aque ous se le no sulfa te, the cons tant A is a func ti on of the se le no sul fa te concen tra ti on 4 .This de pen den ce is not pre dic ted by the mat he ma ti cal ex pres si on that de fi nes this pa ra me ter.In order to ex pla in the de pen den ce, a mo del re fi ne ment is ne cessary.The ba sis for this can be the fact that tra ces of cru ta i te, CuSe 2 , were found in the film to get her with Cu 2-x Se, depen ding on the ex pe ri men tal con di ti ons 4 .This re sult led to the hypot he sis that a su bla yer of CuSe 2 is for med be fo re the for ma ti on of Cu 2-x Se.The gra di ent of che mi cal po ten ti al of Cu(I) at the CuSe 2 /so lu ti on and the Cu/Cu 2-x Se in ter fa ces would be the dri ving for ce le a ding the re ac ti on.Des crip tion of this pro cess as well as its for mu la ti on es ta blis hed a new mo del for cop per se le ni de growth.The aim of this paper is to pre sent such a mo del.The pa ra bo lic law will still be con si de red va lid, but the ki ne tics of the se le no sul fa te reduc ti on and the for ma ti on of the su bla yer will be in clu ded.

The Model
Fi gu re 1 re pre sents the steps sup po sed to oc cur du ring the growth of Cu 2-x Se.First, be fo re any re ac ti on oc curs, the subs tra te (cop per me tal) is put in con tact with aque ous se leno sul fa te.As the subs tra te is ho mo ge ne ous, the Cu(I) chemi cal po ten ti al pro fi le is cons tant at some va lue at the me tal and zero at the so lu ti on if it does not con ta in any cop per ion.
Then, a first la yer of Cu 2-x Se is for med by the at tack of se leno sul fa te: The so lu bi lity pro duct of cop per se le ni de is qui te low (10 -60.8 mol 3 /L 3 , ac cor ding to Ref. 10) and fa ci li ta tes the oc cur ren ce of this re ac ti on.The ac ti on of se le no sul fa te goes on and ma kes the sur fa ce ri cher in se le ni um.This leads to the for ma ti on of CuSe 2 .The for ma ti on of po li se le nides is known 11 .The che mi cal po ten ti al of Cu(I) de cre a ses in the se two pha ses in re la ti on to the one at the me tal.Fi gure 1 re pre sents the de cre a se as li ne ar in the CuSe 2 be ca u se this la yer is al ways very thin, whi le the Cu 2-x Se la yer grows.This hypot he sis is re a so na ble be ca u se just tra ces of CuSe 2 are found ex pe ri men tally and no CuSe 2 could be detec ted if the ex pe ri ment is car ri ed out fas ter 4 .The gra di ent of che mi cal po ten ti al le ads to the dif fu si on of Cu(I) through both pha ses.When Cu(I) goes through the su bla yer, the CuSe 2 is con ver ted into Cu 2-x Se: CuSe 2 is ri cher in se le ni um than Cu 2-x Se and ac cordingly CuSe 2 is for med near the film/so lu ti on in ter fa ce.The se le no sul fa te up ta kes the elec tron ge ne ra ted by the Cu(I) for ma ti on.Then, it is as su med that the se pro ces ses re pe at n ti mes.

Mathematical Description
The time for for ming CuSe 2 in each cycle is al ways the same.On the ot her hand, the dif fu si on time of Cu(I) through the la yer and su bla yer in cre a ses in each cycle, since the ir thick nes ses in cre a se gra du ally.
It is as su med that the rate of the che mi cal re ac ti on is pro por ti o nal to the so di um se le no sul fa te con cen tra ti on, C. The time re qui red by the che mi cal re ac ti on is gi ven by: where δ 1 ' is the thickness of the sublayer, k is the kinetic constant and the subscript indicates the number of the cycle.
The time re qui red by Cu(I) to dif fu se, t", may be ex pected to obey the pa ra bo lic law.Thus t" is gi ven by t" = ( δ/A) 2 , whe re δ is the thick ness of the film to be crossed.So, af ter for ma ti on of the first Cu 2-x Se la yer, with thick ness δ 1 , one has: Each cycle con sists of the se two steps.The time of each step in each cycle is na med t 2 ', t 2 ", t 3 ', t 3 ", , t n ', t n ", so that: The fi nal thick ness is the sum of the thick nes ses in each cycle.The thick ness over the first mo no la yer is: The time re qui red for for ming the film is the sum of the re qui red time in each of the n steps: Subs ti tu ting the terms t' and t" yi elds: Fi gu re 1. Pro po sed mo del of for ma ti on of Cu 2-x Se film on cop per from aque ous so di um se le no sul fa te.Once a first mo no la yer of Cu 2-x Se is formed, it is at tac ked by se le no sul fa te ions, yi el ding a su bla yer of CuSe 2. Ions Cu(I) dif fu se from the subs tra te to wards the in ter fa ce film/so lu ti on and con vert CuSe 2 to Cu 2-x Se, in cre a sing the thick ness of the film.The pro cess re pe ats n ti mes.
Using the iden ti ti es 12 i n n 2 the following equation can be obtained: Subs ti tu ti on of ( 1) in ( 2) re sults:

Results and Discussion
The growth of cop per se le ni des was si mu la ted as signing va lu es for the pa ra me ters kC, A, δ 1 and δ 1 ' in Eq. ( 3).Fi gu re 2 shows some cur ves ge ne ra ted in this way.The value used both for δ1 and for δ1 ' was 6 x 10 -8 cm, which corres ponds ap pro xi ma tely to the di men si on of a mo no la yer of cop per se le ni de.The size of a unit cell in the crystal lat ti ce of cop per se le ni de is 13 ca .5.8 x 10 -8 cm.Va lu es of A 2 and of the pro duct kC have di men si ons of cm 2 /s and cm/s, respec ti vely.The se pa ra me ters va ri ed from 10 -8 to 10 -10 cm 2 /s and from in fi nity (1/kC = 0) to 10 -8 cm/s.
The growth cur ve in cre a ses mo no to ni cally, is con ca ve down wards and has no asympto te.The con ca vity is a conse quen ce of the terms δ 2 and δ 3 in Eq. ( 3) and of the fact that the ir co ef fi ci ents are al ways po si ti ve.The re is a li ne ar re gion, who se ex ten si on de pends on kC and t.If kC is small com pa red to δ1 ' A 2 /δ1 2 , A 2 /δ1 and A 2 /δ1 ' , the first term of the se cond mem ber in Eq. ( 3) pre do mi na tes over the ot her terms.The Eq. ( 3) sim pli fi es and be co mes li ne ar.This li near re gi on me ans that the che mi cal re ac ti on is the rate de termi ning step of the growth.The film growth is also li ne ar at the ini ti al sta ges of for ma ti on, i.e. , if t tends to zero.In this case, the film is so thin, that the trans port of Cu(I) oc curs ea sily and does not cons ti tu te hin dran ce to the growth.Obvi ously, the smal ler is kC, the smal ler is the li ne ar in terval.
If Eq. (3) can not be sim pli fi ed, then dif fu si on of Cu(I) is the rate de ter mi ning step.Thus, ki ne tic in for ma ti on is predo mi nantly con ta i ned at the ini ti al part of the growth cur ve, whi le in for ma ti on about ion trans port is con ta i ned at the non li ne ar re gi on.
Equa ti on (3) was fit ted to the ex pe ri men tal re sult of Ref. 4, var ying the pa ra me ters A 2 and kC.Ini ti ally both para me ters were fi xed at low va lu es, and A 2 was in cre a sed.
The low va lu es of kC yi eld li ne ar plots.Then the con ca vity was cor rec ted by in cre a sing kC.De vi a ti ons ca u sed by this last in cre a se were com pen sa ted by var ying A 2 .Fi gu re 3 shows the re sults ob ta i ned with A 2 = 10 -8 cm 2 /s and k = 7.9 x 10 -6 cm s -1 L mol -1 .
The agre e ment bet we en cal cu la ti ons and ex pe ri men tal data by it self does not va li da te the mo del but per mits es tima ti on of physi co che mi cal pa ra me ters.The dif fu si on coef fi ci ent of Cu(I) in Cu 2-x Se can be cal cu la ted from A 2 if the dif fe ren ce of Cu(I) con cen tra ti on in cop per se le ni de and in cop per me tal is known.(see ap pen dix): This con cen tra ti on is re la ted to the num ber of de fects per unit vo lu me of the crystal.The re a der is re fer red to Fi gu re 2. Hypo tet hi cal growth cur ves of cop per se le ni de film ac cording to the pro po sed mo del.Used pa ra me ters: δ1 = δ1 ' = 6 x 10 -8 cm.For cla rity, the cur ves were se pa ra ted in three se ri es, shif ted 2000 s from each ot her: (1) A 2 = 10 -10 cm 2 /s; (2) A 2 = 10 -9 cm 2 /s and (3) A 2 = 10 -8 cm 2 /s.In each group the pa ra me ters kC were: ◊ ( i.e. , 1/kC=0), Fi gu re 3. Com pa ri son of ex pe ri men tal and the o re ti cal cur ves.Pa rame ters used in Eq. ( 3): A 2 = 10 -8 cm/s; δ1 = δ1 ' = 6 x 10 -8 cm; the pa rame ters kC at the fi gu re cor res pond to the same va lue of k = 7.9 x 10 -6 cm s -1 M -1 , C be ing the gi ven so di um se le no sul fa te con cen tra ti on.
Mor ri son 14 for more de ta ils.Typi cally, the num ber of defects may be of the or der 10 16 cm -3 , ac cor ding to Bam ford and Tip per 15 .Assu ming the num ber of de fects in cop per me tal to be much lo wer than the one in cop per se le ni de, ∆CCu(I) can be as su med to be 10 16 cm -3 = 1.66 x 10 -8 mol cm -3 .By using this va lue with the va lue of A ob ta i ned ex peri men tally and the den sity and mo lar mass of the Cu 2Se equal to 6.749 g/cm 3 and 206.0 g/mol, res pec ti vely, one obta ins 2 x 10 -10 cm 2 s -1 as an es ti ma te for the co ef fi ci ent of dif fu si on of cop per.This va lue can be com pa red with the dif fu si on co ef fi ci ent of Cu(I) in a si mi lar com pound, copper sulp hi de.Accor ding to Eti en ne 16 , the dif fu si on co ef fici ent of Cu(I) in chal co ci te is D Cu(I) = 8.1 x 10 -3 e (-5870/T) , and in di ge ti te is D Cu(I) = 3.6 x 10 -2 e (-6100/T) , whe re T is the ther mody na mic tem pe ra tu re.At room tem pe ra tu re (300 K) the va lu es are res pec ti vely 2.6 x 10 -11 and 5.3 x 10 -11 cm 2 /s.On the ba sis of the se cal cu la ti ons, Cu(I) may have a mo bility four to eight ti mes gre a ter in cop per se le ni de than in cop per sulp hi de.
Sin ce the re is in ot her mo dels no ot her equi va lent pa rame ter to k, it is dif fi cult to com pa re its va lue.The ra tio A 2 /k has di men si ons length ti mes mo la rity (of so di um <%-2>sele no sul fa te).Its ex pe ri men tal va lue, 1.26 x 10 -3 cm mol L -1 re ve als that the dif fu si on li mits the growth much more than the rate of su bla yer for ma ti on.
Fi nally, a re mark about the mag ni tu de of δ1 and δ1 ' must be made.As men ti o ned, the va lue 6 x 10 -8 cm used for the se pa ra me ters is ap pro xi ma tely the thick ness of a cop per se leni de mo no la yer.The ap pro pri a te δ1 and δ1 ' va lu es might be ex pec ted to be gre a ter if se le ni um could dif fu se into the film to wards the me tal con co mi tantly with Cu(I) mo vement.Then the film would grow at the film/so lu ti on in terfa ce as well as at the film/subs tra te in ter fa ce.Ho we ver this se ems to be im pro ba ble, be ca u se se le ni um has prac ti cally no mo bi lity in cop per se le ni de 13 .

Conclusion
A mo del for the growth of cop per se le ni de on cop per from se le no sul fa te so lu ti on was pre sen ted.The mo del assu mes that the en rich ment with se le ni um of the la yer ad jacent to so lu ti on le ads to for ma ti on of a su bla yer of CuSe 2 .Dif fu si on of Cu(I) from the subs tra te le ads to film growth.The time ne ces sary for the growth va ri es ac cor ding to a poly no mi al func ti on of third de gree of the thick ness.The terms of su pe ri or de gree are a con se quen ce of the dif fu si on hin dran ce.The prin ci pal pa ra me ters that de ter mi ne the pro per ti es of the growth cur ve are dif fu si on co ef fi ci ent, rate cons tant, thick ness of the su bla yer and ac ti vity of Cu(I) in the film.By com pa ring ex pe ri men tal and si mu la ted growth cur ves, the dif fu si on co ef fi ci ent of Cu(I) in cop per se le ni de was es ti ma ted as 2 x 10 -10 cm 2 s -1 .The pre ci si on of this va lue de pends on know led ge of the che mi cal po ten ti al of Cu(I) in cop per se le ni de re la ti ve to the one in me tal lic cop per. t