Micron School of Materials Science and Engineering. Problem Set 7 Solutions

Transcription

1 Problem Set 7 Solutios 1. I class, we reviewed several dispersio relatios (i.e., E- diagrams or E-vs- diagrams) of electros i various semicoductors ad a metal. Fid a dispersio relatio that differs from the oes we covered i class. Label all the importat features icludig Eg, CB, VB, heavy ad light holes. Describe which, i geeral, is lighter ad faster carrier, the electro or hole, usig the effective mass ad group velocity. ω 1 E vg = = while the effective mass is give by: E meff == = 2 2. E 2 Solutio: The studets ca use ay dispersio relatio they fid off the iteret. It just eeds to be label with Eg, CB, VB, heavy ad light holes ad whether the electro mass is lighter or heavier tha the hole mass. The lighter carrier will lead to a greater group velocity. 2. Draw eergy bad diagrams i real space for the followig: a. Coductor b. Semicoductor c. Nocoductor Thoroughly label the diagrams at least icludig the valece bad top, coductio bad bottom, vacuum eergy, eergy bad gap, electro affiity, wor fuctio ad the eergy axis. Iclude the eergy bad gap coditios for semicoductor ad ocoductor. 1

2 3. For a itrisic semicoductor, we examied some of the ey steps i the derivatio of the cocetratio of electros i the coductio bad,. For this problem, fully derive the cocetratio of electros i the coductio bad,, showig all steps. I your derivatio, justify usig the Boltzma s approximatio. Solutio: See Lecture Note Hadout titled MSE 410 Ch5 Itrisic S-D g[e] f[e].pdf 4. For a itrisic semicoductor, we examied some of the ey steps i the derivatio of the cocetratio of electros i the coductio bad,. We also discussed the plot below (tae from Robert F. Pierret, Advaced Semicoductor Fudametals, Modular Series o Solid State Devices, 2d Ed., Volume VI (Pretice Hall, 2003) p ). a. Note that oe of the fuctios i the plot, Fermi-Dirac itegral of order j. For our case, j = ½. 1 x= x Fj ( η) = Γ ( j x= 0 ) j x e η dx, is ow as the i. Use a mathematical program to fid, ame, ad describe the fuctio that is used i the mathematical program to mathematically describe (i.e., solve) the Fermi-Dirac Itegral of order ½ (i.e., F ( ) 1/2 η ). Note that we did metio this fuctio i class relative to Mathematica ad both Mathematica ad Matlab has this fuctio. Be sure to correlate the variables i the fuctio used i your mathematical program with those i the above equatio, that is, x ad e x-η (i.e., traslate their variables to yours). ii. Plot F ( η ) usig your mathematical fuctio such that it mimics F ( ) 1/2 1/2 η i the plot below. Mae sure your plot is thoroughly labeled. Thoroughly documet your program. 2

3 Solutios i. Note that Mathematica solves the problem usig polylogarithms i the form of the PolyLog fuctio which is a summatio. From Wolfram s Documetatio ceter, we fid that Mathematica solves F 1/2 ( η ) by usig a PolyLog[, z] fuctio is the polylogarithm fuctio give by: Li z =. [1] ( z) = 1 I our case, = 3/2 ad z = -e ηη. Similarly, Matlab uses polylogarithms i the followig form: polylog(,x) returs the polylogarithm of the order ad the argumet x. Extra credit: taig it oe step further, we ca substitute ad z ito equatio 1, we have: Li ( z) = = = 1 = 1 e ( e ) η 3/2 3 [2]

4 Side Note: I our case, we are looig to solve the followig itegral: x= x 12 x= 0 x 1+ e η dx Which is related to F ( ) 1/2 η i the followig way: x= x= 0 12 x 3 dx = Γ F x 1 e η + 2 Where: 1/2 ( η) 12 3 π π = x Γ = F x = dx x e η x ; Hece: 1/2( v ) x= 0 ii. We the plot PolyLog[, z] which gives: Fig.: Both a liear ad a log plot of the Fermi-Dirac itegral of order ½, 3 which is a plot of PolyLog, e η 2 fuctio. The fuctio seems to be a expoetially growig fuctio as equatio 2 shows. 4

5 b. The Boltzma s approximatio states that F 1/2 ( η ) ca be approximated usig e η i Solutio: certai coditios of η. Mimic the plot below by plottig both F ( η ) ad 1/2 eη o the same plot with a rage of η from -10 to 2 o a Log Plot to mimic the plot below. Mae sure your plot is thoroughly labeled ad distiguishes betwee multiple curves o oe plot. Discuss your fidigs i terms of the defiitio for η. Also, discuss the results relative to Ef approachig Ec, ad the show ad explai uder what coditio should F 1/2 ( η ) ot be approximated usig e η. Fig.: The Fermi-Dirac itegral of order ½, which is a PolyLog fuctio, whe plotted o a log plot (red), is early the same as a expoetially growig fuctio that is plotted i blue. Commet: e η approximates F1/2(η) for egative η which is whe Ef is far from the coductio bad edge, Ec. However, as η approaches 0, that is, as Ef approaches the coductio bad edge, Ec, the two fuctios diverges. Hece, e η should ot be used to replace F1/2(η) as Ef approaches the coductio bad edge, Ec, that is, whe η ears 0 from egative values. The plot shows that e η should ot be used whe Ef is withi 2T of the coductio bad edge. That E E f c is, η = 2 or E E 2T or E E 2T or E E 2T f c c f f c T otherwise the Boltzma approximatio is ivalid thus F1/2(η) should be used istead of e η. Refereces: 5

6 [1] S.O. Kasap, Priciples of Electroic Materials ad Devices, 3 rd Ed. (McGraw Hill, 2006) p. 381 [2] R.R. Pierret, Advaced Semicoductor Fudametals, 2 d ed. (Pretice Hall, 2003) p. 93 [3] J.S. Blaemore, Semicoductor Statistics, (Dover, 1987), p Usig the relatioship of i = p i i that is a fuctio of Eg, recreate the plot show i Kasap figure 5.16 which is also show i your class otes. Note that the equatio adjacet to figure 5.16 is icomplete so do ot use it. Fully label your plot icludig, but ot limited to, labelig each plot relative to the semicoductor (e.g., leged) ad the room temperature itrisic carrier cocetratios of each semicoductor (e.g., vertical lie ad values). Do ot forget to iclude figure captios ad figure captio labels. Please commet o the fidigs revealed by your plot ad also compare it to figure 5.1. Hit: values for effective mass ca be foud i table 5.1 of Kasap. You may also use the effective desity of states (Nc & Nv i table 5.1, however, the values for Nc ad Nv do ot vary with temperature. So you will eed to modify Nc ad Nv to vary with temperature Solutio: 2T Usig the equatio from your slides: = NNe, ad values for Nc ad Nv i c v or meff for electros ad holes from Table 5.1 i Kasap, we ca the create the plots: E g 6

7 Fig. A log of itrisic electro carrier cocetratio versus iverse temperature (Arrheius plot) is plotted as a fuctio of the iverse of temperature for GaAs, Si, ad Ge. The dashed vertical lie idicates room temperature(300 K) ad the labeled cocetratio are itrisic carrier cocetratios at room temperature. Note that the steepest slope is GaAs ad the shallowest slope is Ge idicatig that Eg,Ge < Eg,Si < Eg,GaAs. Because of this relatio, ote that the itrisic carrier cocetratio, i, at room temperature follows: i,ge > i,si > i,gaas. The plot above is plotted with both Nc ad Nv are a fuctio of temperature ad are give as: N c 3 3 * 2 * 2 e b πmt h b N 2 v 2 2πmT 2 = 2 ad = 2. h h We use the effective masses that are specifically for the desity of states calculatios ad they ca be foud i table 5.1 i Kasap 3 rd editio o page 386. Also ote that the lies are ot quite liear. This is due to the T 3/2 depedece i the pre-expoetial. 7