Hole Drift Mobility, Hall Coefficient and Coefficient of Transverse Magnetoresistance in Heavily Doped p-type Silicon

Transcription

1 Iteratioal Joural of Pure ad Alied Physics ISSN Volume 6 Number (). 9 Research Idia Publicatios htt:// Hole Drift Mobility Hall Coefficiet ad Coefficiet of rasverse Magetoresistace i Heavily Doed -tye Silico esfaye Getiet Deartmet of Physics College of Educatio Addis Ababa Uiversity P.O. ox 76 Addis Ababa Ethioia Abstract I this commuicatio the ormalied Fermi-eergy the ormalied hole drift mobility the ormalied Hall coefficiet ad the ormalied coefficiet of trasverse mageto resistace are umerically calculated for differet doig cocetratios of accetor imurities at a temerature 3 K i a heavily doed -tye silico. he aalysis of the results show that the formatio of valece bad tails ad its effect o trasort roerties become sigificat for accetor doig cocetratio 9 /cm 3 ad it is exected that results may be articularly of iterestig above 9 /cm 3. he result of this article might be very imortat i characteriig semicoductor devices. Itroductio Recetly differet worers have studied the effect of bad tails due to heavy doig o bad ga arrowig ad some trasort heomea i heavily doed -tye silico. Lee ad Fossum [ have calculated the bad ga arrowig i heavily doed -tye silico. Poortmas et. al. [ have reorted the bad ga arrowig i heavily doed -tye silico. Sharma [3 have calculated the diffusio mobility ratio i heavily doed -tye silico by taig i to accout the effect of bad tails. Y. Elfgad et.al [4 have calculated some galvaomagetic trasort coefficiets i heavily doed -tye silico by taig ito accout the effect of bad tails.. Getiet [ has calculated hall coefficiet ad coefficiet of trasverse magetoresistace of itermmediately doed -tye silico by taig i to accout the effect of bad tails with mixed scatterig of electros both by ioied iurities ad acoustic hoos. I this aer we umerically calculate some galvaomagetic trasort coefficiets vi. hole drift mobility Hall coefficiet ad coefficiet of trasverse magetoresistace i heavily doed -tye silico. o carry out the umerical

2 esfaye Getiet calculatios we first derive exlicit exressios of these trasort coefficiets by taig ito accout the effect of valece bad tails which exted above the valece bad edge dee ito the forbidde bad ga. o derive the exlicit exressios of the trasort coefficiets we used the followig model: () comlete ioiatio of accetor imurities () Fermi - Dirac statistics to describe the equilibrium distributio of holes i the valece bad (3) ioied imurities as a domiat scatterig mechaism ad (4) the collisio frequecy due to ioied imurity scatterig derived by Cowell-Weissoft [6. ecause we are iterested oly studyig the effect of valece bad tails o galvaomagetic trasort coefficiets i heavily doed -tye silico we further assume a sherical costat eergy surface i mometum sace. Hole drift mobility Hall coefficiet ad coefficiet of trasverse magetoresistace i heavily doed -tye silico Exlicit exressios of the trasort coefficiets i heavily doed -tye silico ca be obtaied if we ow the cocetratio of holes i the valece bad. he radom distributio of imurity atoms i heavily doed -tye silico gives rise to aeriodicity of the crystal. his crystal aeriodicity causes a fluctuatio i the local electrostatic otetial which i tur results i a satially deedet valece bad edges ad hece satially deedet desity of states. I our aalysis we use the desity of states fuctio derived by Kae [7 ad Slotboom's [8 aroximatio of the Kae's desity of states fuctio. I derivig the exressios of the trasort coefficiets we use the oltma trasort equatio i the resece of the crossed d.c. electric ad magetic fields. We further use the collisio frequecy due to ioied accetor imurities derived by Cowell-Weissoft. o obtai the exlicit exressios of the trasort coefficiets i heavily doed -tye silico we first derive the exressios by assumig arabolic desity of states ad the we modify the exressios whe the desity of states fuctio is o loger arabolic usig the cocet suggested by Elfagd et. al. for heavily doed -tye silico. For the sae of umerical calculatios we further ormalie the hole cocetratio i the valece bad ad the trasort coefficiets. hus the ormalied hole cocetratio the ormalied hole drift mobilityμ the ormalied Hall coefficietr ad the ormalied coefficiet of trasverse magetorésistace are derived to be b * ( m ) o π ħ () μ 3 o () 3 o 4 r (3)

3 Hole Drift Mobility Hall Coefficiet ad Coefficiet 4 3 b (4) where is the stadard deviatio of the Guassia distributio of imurities cosidered by Kae ad it is give by 8 s o πε λ () he screeig legth λ is exressed as. ) ( F V o s E E e ε λ (6) I these exressios o is the uiform hole desity that revails i the absece of io coulomb iteractio s ε is the dielectric costat of silico * m is the hole desity of states effective mass ( * m.8m o at 3 K) that accouts for the slit off subbads as well as the two degeerate sub-bads with m o beig the free electro mass ad 6 )}.96( {..6.6 d d π (7) 6 )}.96( {..6 ) (.6 d d m m m π (8)

4 esfaye Getiet EF where is the ormalied ( or dimesioless) Fermi-eergy is defied i terms of the hole eergy E as E ad i the itegral exressio m m / 4 /. Numerical calculatio of the trasort coefficiets o umerically calculate the trasort coefficiets we first calculate the umerical value of the ormalied (or dimesioless) Fermi-eergy for a give hole cocetratio from equatio () by iteratio. o do this for a give ormalied hole cocetratio we arbitrarily assume a value for ad the we comute RHS of equatio () ad comare the result with the LHS i.e. with the ormalied (or dimesioless) hole cocetratio. We the try some other value of ad the cotiue to chage it util we get a umerical value for which maes equatio () self-cosistet. his value of is the ormalied value of Fermi-eergy for the give hole cocetratio. After calculatig for various hole cocetratios we comute the defiite itegrals occurrig i the exressios of the trasort coefficiets i.e. equatios (7) to (). We the use these values of the defiite itegrals i equatios () (3) ad (4) to umerically calculate the ormalied hole drift mobility Hall coefficiet ad coefficiet of trasverse magetoresistace resectively i heavily doed -tye silico. Fially we lotted the ormalied Fermi-eergy hole drift mobility Hall coefficiet ad coefficiet of trasverse magetoresistace versus the ormalied doig cocetratio of accetor imurities. Discussio Figures () () (3) ad (4) show the lot of the ormalied Fermi-eergy hole drift mobility Hall coefficiet ad coefficiet of trasverse magetoresistace versus the ormalied doig cocetratio of accetor imurities resectively. Fig. () shows the deedece of the dimesioless Fermi-eergy o dimesioless doig cocetratio. We see from this figure that the ormalied Fermi-eergy icreases with icreasig doig cocetratio. Fig. () shows the variatio of the ormalied hole drift mobility with doig cocetratio i heavily doed -tye silico. It shows that the drift mobility icreases slowly with icreasig doig cocetratio of accetor imurities for ad raidly with icreasig doig cocetratio for >. Fig. (3) idicates the deedece of the ormalied Hall coefficiet o the ormalied cocetratio of accetor imurities i heavily doed -tye silico. We ote from this figure that the ormalied Hall coefficiet decreases mootoically with icreasig ormalied doig cocetratio. I cotrast to this we see from fig. (4) that the ormalied coefficiet of trasverse magetoresistace icreases with icreasig doig cocetratio. As we ca see from all grahs the results of the aalysis are very iterestig aroud i.e. whe the doig cocetratio is

5 Hole Drift Mobility Hall Coefficiet ad Coefficiet 3 aroud 9 /cm 3. Aroud this doig cocetratio some hase chage occurs i the system. Doig cocetratio aroud 9 /cm 3 may be thought as the oit where the radomess of the imurity distributio which give rise to the aeriodicity of the crystal which i tur causes the fluctuatio of the electrostatic otetial resultig i satially deedet desity of states fuctio starts to become sigificat. his meas that the formatio of the valece bad tails ad its effect o trasort roerties become sigificat for accetor doig cocetratio 9 /cm 3 ad it is exected that results may be articularly of iterestig above 9 /cm 3. We could ot comare our results with that of exerimet for lac of exerimetal data o -tye silico i which our theory is alicable. However it will be iterestig to comare them with the exerimetal data wheever it becomes available. he error icurred i the umerical calculatio is oe i 6. he limitatios i these calculatios are () the logarithmically varyig eergy i the exressio of the collisio frequecy has bee assumed as a costat ad () oly ioied imurity scatterig as a domiat scatterig mechaism has bee cosidered while of course oe could also iclude acoustic hoo scatterig i the aalysis of the trasort coefficiets. hese trasort coefficiets i heavily doed -tye silico could also be calculated as fuctios of temerature. Calculatios of these trasort coefficiets as a fuctio of doig cocetratio ad temerature simultaeously i heavily doed - tye silico have bee erformed ad the results will be reorted soo. I all calculatios I used 3 K ad the dielectric costat of silico ε s.8. 6 Normalied Fermi-eergy Normalied hole cocetratio Figure : Deedece of the ormalied Fermi-eergy o the ormalied doig cocetratio i heavily doed -tye silico.

6 4 esfaye Getiet 4 Normalied hole drift mobility Normalied hole cocetratio Figure : Variatio of the drift mobility with doig cocetratio i heavily doed -tye silico.4. Normalied Hall coefficiet Normalied hole cocetratio Figure 3: Variatio of the ormalied Hall coefficiet with doig cocetratio i heavily doed -tye silico.

7 Hole Drift Mobility Hall Coefficiet ad Coefficiet 4 Normalied coefficiet of trasverse magetoresistace Normalied hole cocetratio Figure 4: Deedece of the ormalied coefficiet of trasverse mageto resistace o doig cocetratio i heavily doed -tye silico. Refereces [ D. S. Lee ad J. G. Fossum IEEE ras Electro Devices. 3 66(983) [ J. Poortmas S. C. Jia D. H. oller M. Camax J. F. Nijis R. P. Mertes ad R. Va Overstraete Solid State Electro (993) [3 S. K. Sharma IEEE ras Electro Devices (989) [4 Y. Elfagd. Woralemahu ad S. K. Sharma Phys. Stat. Sol. (b) 77 49() [. Getiet Idia Joural of Pure ad Alied Physics Vol () [6 E. M. Cowell ad V. F. Weissoft Phys. Rev. Vol (9) [7 E. O. Kae Phys. Rev. 3 79(963) [8 J. W. Slotboom Solid State Electro. 79(977)

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