FYS4310. Problem Suggested solution

Transcription

1 FYS430. Problem Suggested solutio Problem: costruct semilog plot usig table 3. ad 3 cotributios to diffusivity of phosphorous as a fuctio of temperature from C assume C of phosphorous is e9 cm^-3 restart; Symbol defiitios Do = Diffusivity of P by eutral vacacy i itrisic Si at ifiite temperature i cm^/s Ea = activatio eergy for above i ev Dom = ( D zero mius) Diffusivity of P by egative vacacy i i-si at if. temperature Eam = activatio eergy for above Dodm =(D zero double mius ) Diffusivity of P by double egative vacacy diff i i-si at if. temperature Eadm = activatio eergy for above Bolzmas costat T absolute temp i K = electro cocetratio i cm^-3 i = itrisic carrier cocetratio io = costat givig the itrisic cocetratio Eg = Bagap of Si i ev Ego=Bagap at T=0 alpha, beta = costats deltaeg = decrease i Eg at high dopig ND=Door cocetratio #assume(t0,0,i0,nd0,ea0,eadm0,eam0,do0,dom0,dodm0, k0,io0,alpha0,beta0); The Diffusivity of P i Si ca be writte as eq := D = Do e eq := i = io T Ea eq Dom e i D = Do e Eam Ea Dodm e Dom e i Eam Eadm The itrisic carriercocetratio is give by (see 3.8 i textbook ) 3 Eg e eq i = io T 3 / e i Dodm e i Eadm The badgap is give by (exept for highdopig effects) is give by (text book 3.9) Eg eq3 := Eg = Ego T T eq3 Eg = Ego T T We discussed i the class wether to additioally iclude the badgap arrowig due to high

2 cocetratio. Some of that discussio ca be foud i aother worksheet. We decided to ot iclude heavy dopig effects o the badgap because a) tgf did ot believe the formulaes we have available yeld umerically reliable results. b) tgf thiks the uderlyig theoretical foudatio is lackig ad possibly quoted icorrect c) The high dopig badgap arrowig was ot used whe the diffusivities give i the table of the book was extracted from experimetal data ( agai from cocetratio profiles) (other correctios should also be applied for example the Fermi itegral would theoretically describe the electro cocetratio tha the Bolzma approximatio ) So we do ot here use equatio (.3.0) ad leared that 'life is ot simple' We have give the Phosphorous cocetratio, ND, we eed to fid the electrom cocetratio,. If we have o segregatio or complex formatio betwee poit defects the relatioship betwee them is give by charge eutrality yieldig eq4 := = ND i eq4 = ND Which came from requiremet of charge eutrality: ND+ - NA- + p - = 0 ad assume complite ioizatio of doors b settig ND+ = ND. We solve eq4 for eq5_:=solve(eq4,); eq5_ ND ND 4 i, i ND ND 4 i We defiitely have to pick the solutio with positive sig = ND ND 4 i, ND ND 4 i if (is(op(0,eq5_[])=`+`)) the eq7:==eq5_[] else eq7:== eq5_[]; fi; eq7 = ND ND 4 i The umerical values of parameters i the previous equatios are (as read from table 3., ad o page 4-43) i uits based o cm, ev, s ad K costats:={k= e-5, alpha= ,ego=.7,beta=636,io= 7.3e5, Do=3.9,Ea=3.66,Dom=4.4,Eam=4,Dodm=44,Eadm=4.37,ND=e9}; costats Do = 3.90, Dodm = 44, Dom = 4.40, Ea = 3.66, Eadm = 4.37, Eam = 4, Ego =.7, ND = , = , = 636, k = , io = Let us just estimate if we eed to use equatio eq7 or ca use =ND. We calculate i at 000K ad 400K at low dopig We thus use the followig equatio eq8:=subs(eq3,eq);

3 eq8 i = io T 3 / e Ego T T We calculate at a)t=000k, b)t=400k c)t=73k (=000 C) We just put i the values of the symbols, calc at T=000 ad 400, ad 000C=73 respectively eq9a:=evalf(subs(costats uio {T=000,ND=e9},eq8)); eq9b:=evalf(subs(costats uio {T=400,ND=e9},eq8)); eq9c:=evalf(subs(costats uio {T=73,ND=e9},eq8)); So we ca culculate eq9a i = eq9b i = eq9c i = eq0a:=subs(costats uio {eq9a},eq7); eq0b:=subs(costats uio {eq9b},eq7); eq0c:=subs(costats uio {eq9c},eq7); eq0a = eq0b = eq0c = We have preseted the equatios we use, from this we ca make the diffusivity a explicit fuctio of temperature We put eq8 ito eq7 to get ad put ad i (from eq8) ito eq, The lie below is doig exactly that DofT:=T-subs(costats,subs(subs(eq8,eq7),subs(eq8,eq))): We may also make each of the diffusio terms i eq a explicit fuctio of temperature I the followig lies first for eutral vacacy cotributio, egative vacacy ad double egative vacacy. Do_ofT:=T-subs(costats,subs(subs(eq8,eq7),subs(eq8,Do*exp(- Ea/k/T)))): Dm_ofT:=T-subs(costats,subs(subs(eq8,eq7),subs(eq8,Dom*exp(- Eam/k/T)*/i))): Ddm_ofT:=T-subs(costats,subs(subs(eq8,eq7),subs(eq8,Dodm*exp (-Eadm/k/T)*(/i)^))): We will calculate the diffusivity for a rage of temperatures ad plot We decide to plot log(d) versus /T, thus we will pick values of (/T) that are evely spaced We will do this i a procedure with(plots):

4 Makeplots:=proc() local j, ivt, delivt, TK, D, LogDarray, vd,vdo,vdm,vddm,logdoarray,logdmarray,logddmarray, ivtarray, dplot,doplot,dmplot,ddmplot: ivt:=.0/400:## ivt=/t, we start at the highest temparature delivt:=evalf((/000 -/400)/0.0):# we calculate the step size i /T for j from to 0 do TK:=/ivT: vdo:=do_oft(tk); vdm:=evalf(eval(dm_oft(t),t=tk)): vddm:=evalf(eval(ddm_oft(t),t=tk)): vd:=vdo+vdm+vddm: LogDarray[j]:=evalf(log0(vD)): LogDoarray[j]:=evalf(log0(vDo)): LogDmarray[j]:=evalf(log0(vDm)): LogDdmarray[j]:=evalf(log0(vDdm)): ivtarray[j]:=ivt: ivt:=ivt+delivt: od: dplot:=listplot([seq([ivtarray[i],logdarray[i]],i=..0)], color=red): doplot:=listplot([seq([ivtarray[i],logdoarray[i]],i=..0)], color=cya): dmplot:=listplot([seq([ivtarray[i],logdmarray[i]],i=..0)], color=gree): ddmplot:=listplot([seq([ivtarray[i],logddmarray[i]],i=..0)], color=blue): display({dplot,doplot,dmplot,ddmplot}); ed: Makeplots();

5 The Ed Curve Curve Curve 3 Curve 4 The above plot shows as red = Total diffusivity Cya = the cotributio from eutral vacacies Gree = the cotributio from egative vacacies Blue = the cotributio from double egative vacacies Axes horizotal /T i (K^-) vertical log0 to D We should compare the above figure with Data give i other books, But textbooks oly gives Do ad Ea values ad figures ad we dot kow what they use for ad i to extract the parameters.