The Temperature Dependence of the Density of States in Semiconductors

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1 World Joural of Codesed Matter Physics, 2013, 3, Published Olie November 2013 (htt:// htt://dx.doi.org/ /wjcm The Temerature Deedece of the Desity of States i Semicoductors Gafur Gulyamov 1, Nosir Yusujaovich Sharibaev 1,2, Ulugbek Ioyatillaevich Erkaboev 1 1 Namaga Egieerig Pedagogical Istitute, Namaga, Uzbekista; 2 Namaga Egieerig Istitute of Techology, Namaga, Uzbekista. gulyamov1949@mail.ru Received Setember 17 th, 2013; revised October 21 st, 2013; acceted November 10 th, 2013 Coyright 2013 Gafur Gulyamov et al. This is a oe access article distributed uder the Creative Commos Attributio Licese, which ermits urestricted use, distributio, ad reroductio i ay medium, rovided the origial work is roerly cited. ABSTRACT The temerature deedece of the desity of eergy states i semicoductors is cosidered. With the hel of mathematical modelig of the thermal broadeig of the eergy levels, the temerature deedece of the bad ga of semicoductors is studied. I view of the o-arabolic ad the temerature deedece of the effective mass of the desity of states i the allowed bads, grahs of temerature deedece of the bad ga are obtaied. The theoretical results of mathematical modelig are comared with exerimetal data for Si, IAs ad solid solutios of -Bi 2 x Sb x Te 3 y Se y. The theoretical results satisfactorily exlai the exerimetal results for Si ad IAs. The ew aroach is ivestigated by the temerature deedece of the bad ga of semicoductors. Keywords: Bad Ga; The Effective Mass Desity of States; The Eergy Sectrum; The Numerical Simulatio ad Exerimet 1. Itroductio Desity of states determies thermal, otical, magetic, electric ad other hysical roerties of semicoductors. The chage of the eergy gas with icreasig temerature ca be exlaied by the ifluece of lattice vibratios o the eergy levels i crystals. Statistical aalysis of the roblem is carried out by aalyzig the free eergy of the crystal as the sum of the eergies of the electro gas, lattice vibratios ad electro-hoo iteractios [1, 2]. O the other had, oe should take ito accout the thermal broadeig of the eergy levels of the radiative trasitios [1,2]. I [3-8] the temerature deedece of the desity of states is determied by relaxatio sectroscoy of eergy levels i semicoductors. It is show that the desity of surface states varies accordig to temerature. Due to the thermal broadeig of the levels, the discrete sectrum with hagig temerature becomes a cotiuous eergy sectrum. With the exasio of the eergy sectrum of the desity of states i the eergy derived from the robability required eergy level, it was show that the amout of eergy slits is deedet o temerature. The temerature deedece of the bad ga is determied by the desity of states of the coductio bad ad valece bad of the semicoductor. Due to the thermal broadeig of the desity of states ear the bottom of the coductio bad, valece bad reduces bad ga. I the calculatio of the temerature deedece of the forbidde assumed for simlicity, the desity of states i the areas of costat edge of the coductio bad ad valece bad is shar ad has a steed shae. I these works, the effective mass of the desity of states does ot deed o the temerature. However, as show i exerimets [9], the effective mass of the desity of states deeds o the temerature. This chage i the effective mass chages the temerature deedece of the bad ga. However, i the real state of the semicoductor, desity is a fuctio of seed ad eergy bad structure of the samle is determied. Moreover, the desity of states is so geeral that it ca be used eve whe there is o Brilloui zoe ad shar boudaries of ermitted ad rohibited zoes [10,11]. Thus, the aalysis of exerimetal results for comariso betwee theory ad exerimet is ecessary to cosider the secific form of the bad structure of the semicoductor ad the deedece of the effective mass of the charge carriers of the temerature. The aim of this work is to study the temerature deedece of the bad ga semicoductor with the bad Oe Access

2 The Temerature Deedece of the Desity of States i Semicoductors 217 structure ad temerature deedece of the effective mass of carriers ad comariso of theory with exerimet. 2. The Deedece of the Eergy Ga with Temerature As was show i [3-8], the desity of states ca be decomosed ito a series of GN fuctios. The temerature deedece of GN fuctio will determie the statistical thermal broadeig of the discrete levels. The resultig desity of states is determied by the exasio of discrete states by GN-fuctio takes ito accout the thermal broadeig of each discrete level. We assume that the desity of states at absolute zero arabolic N s (E). We defie a secific form of the electro disersio. For examle, the electro disersio arabola or accordig to the Kae model. Accordig to the rocedure [3-8] we exad the desity of states i a series of GN-fuctios. We will take ito accout oarabolicity zoes via the temerature deedece of the effective mass of the temerature: where, N E Т N E GN E, E, T (1) s si i i i1 GN E, E, T i ex Ei E ex Ei E kt kt kt N s (E) the desity of states at zero temerature [12, 13] where N0 EEc at E E c Ns E0at Ec E Ev N0 Ev E at Ev E N 2m , N 2m (2) ad E c, E v 2π 2π The value of the eergy of the coductio bad ad valece bad at T = 0. Substitutig (2) ito (1) we obtai the desity of states at temerature T. I determiig the width of the ga, we use the desity of states, which deeds o temerature. Whe modelig the rocess of measurig the width of the bad ga with icreasig temerature, use the coditio give i [8]. We assume that the desity of states corresodig to the eergy bad ga edges E c ad E v is N k. Eergy regio where the desity of states N s (T) is less tha the critical N k assume bad ga. The eergy rage where N s (T) > N k ermissio from the zoes. The values of the edges of the bad ga ositio of the bottom of the coductio bad E c (T) ad valece bads E v (T) is deter- mied by the solutio of the equatio trascedet Nsi EiGNЕi, Е, Т Nk (3) i1 trascedet solutio of Equatio (3) with resect to E at the secified temerature T, ad the critical value of the desity of states N k determies the ositio of the edges of the ga E c (N k, T) ad E v (N k, T). I Equatio (3) N k icluded as a arameter. N k value is determied by the coditio of the exerimet ad by the accuracy of measuremet techiques. The the bad ga is defied as the differece betwee the values of E c (T, N k ) ad E v (T, N k ): E T, N E T, N E T, N (4) g k c k v k It follows that the method of determiig the accuracy of the exerimet ad the imortat factors i determiig the width of the ga. Ideed the bad ga, determied by otical methods, otical width of the bad ga ca ot match the value of the bad ga, determied by the temerature deedece of the resistace of the semicoductor. Oe of the reasos is that differet values for N k otical ad electrical measurig techiques. 3. The Ifluece of the Effective Mass of the Desity of States at the Temerature Deedece of the Bad Ga i Solid -Bi 2 x Sb x Te 3 y Se y I [9] foud that i solid -Bi 2 x Sb x Te 3 y Se y effective desity of states i the valece bad is strogly deedet o temerature. Figure 1 shows the temerature deedece of the effective mass of the desity of states i solid -Bi 2 x Sb x Te 3 y Se y from [9]. Usig the data of Figure 1 calculated by the model badga variatio with temerature. Figure 2 shows lots of the desity of states at a temerature T = 100 K ad T = 300 K. As ca be see from Figure 2 ito accout the chage of the effective mass desity of states sigificatly affects the desity of states ear the valece bad. Figure 3 shows the temerature deedece of the grahics of the bad ga for the solid solutios - Bi 2 x Sb x Te 3 y Se y to chagig the effective mass desity of states take from Figure 3 [9]. For examle, for a solid solutio of -Bi 2 x Sb x Te 3 y Se y chage i the bad ga by chagig the effective mass at T = 100 K is Eg 100 Eg 100, m 0.93Eg 100, m cost 0.001эВ. By icreasig the temerature to T = 300 K, chagig the width of the bad ga due to chage i the 300 effective mass of the desity of states is E 300, m 1.35 E 300, m cost эВ g g This shows that the reductio of the bad ga by chagig the effective mass with icreasig temerature from E g Oe Access

3 218 The Temerature Deedece of the Desity of States i Semicoductors Figure 1. The temerature deedece of the effective mass of the desity of states m/m 0 i solid solutios -Bi 2 x Sb x Te 3 y Se y [14]. 1 x = 1, y = 0.06; 2 x = 1.1, y = 0.06; 3 x = 1.2, y = 0.06; 4 x = 1.2, y = 0.09; 5 x = 1.3, y = 0.09, 6 x = 1.3, y = 0.07; 7 x = 1.5, y = Figure 2. Grahic desity of states at T = 100 K ad 300 K. m mt ad m mt. m соst m mt. 100 K to 300 K ca be icreased more tha tefold. Figure 3 shows lots of the temerature deedece of the bad ga of the solid solutios -Bi 2 x Sb x Te 3 y Se y for the chages i the effective mass of the desity of states take from Figure 1. Thus, chages i the effective mass of the desity of states with temerature ca greatly affect the temerature deedece of the bad ga. 4. Comariso of Theory with Exerimet The temerature deedece of the width of the eergy ga deeds o the desity of states at the absolute temerature. I the model used i [3-8] the temerature deedece of the desity of states is determied by the temerature deedece of GN-fuctios ad bad structure of the allowed bads at the bottom of the coductio Figure 3. Grahic E g (T) temerature deedece of the bad ga. m соst и m m T. m соst ; ---- m mt, -Bi 0.7 Sb 1.3 Te 2.93 Se 0.07 ;... m mt -Bi 0.6 Sb 1.2 Te 2.91 Se 0.09 ; m mt, -Bi 0.5 Sb 1.5 Te 2.91 Se 0.09., bad ad at the to of the valece bad. Aalysis of the results of umerical modelig of chages i the desity of states at the temerature showed that the value of the desity of states ear the bad edges is determied by the umber of states of the bad edges, a few tes of kt or about 0.1 mev, the desity of states i the deths of the allowed zoes does ot affect the width of the ga. Sice the GN-fuctio of dee area of the zoe does ot eetrate ito the regio bad ga semicoductor, the mai cotributio to the shift of the bad edges give the states lyig close to the edges of the allowed bads. Accordig to this law of disersio ear the to of the valece bad ad the coductio bad edge is crucial i determiig the temerature deedece of the bad ga E g (T). Figures 4 ad 5 are grahs of temerature deedece of the bad ga of IAs [14] ad Si [15]. Usig mathematical modelig of the temerature deedece of E g (T) for a arabolic bad ad Kae model obtaied lots of the bad ga of the temerature. As ca be see i the ivestigated temerature rage of arabolic disersio ad a model for the use of Kae s model is i good agreemet with exerimetal data for IAs [14] ad Si [15]. Theoretical calculatios of the theoretical give for these materials is i good agreemet with the exerimetal data. It follows that the temerature deedece of the bad ga is satisfactorily described by a mathematical model of exasio of the desity of states i a series of GN-fuctios, which describes the temerature deedece of the thermal broadeig of idividual eergy levels i the zoes ad i the forbidde zoe. 5. Coclusio The temerature deedece of the eergy sectrum of the desity of states of solid solutios of Oe Access

4 The Temerature Deedece of the Desity of States i Semicoductors 219 Eg, ev Т, К Exerimet for Si [13] calculatios for the arabolic zoe calculatio for the of Kae Figure 4. The temerature deedece of the bad ga of Si. Eg, ev T, K Exerimet for IAs [14] calculatios for araboliс zoy calculatio for the model of Kae Figure 5. The temerature deedece of the bad ga of IAs. -Bi 2 x Sb x Te 3 y Se y takes ito accout of the temerature deedece of the effective mass of the desity of states i the valece bad. The temerature deedece of the bad ga for the chages i the effective mass of the desity of states is obtaied. The umerical exerimets show that at temeratures T > 120 K, chage i the effective mass of the desity of states by icreasig T has sigificat effect o the temerature deedece of the bad ga. I this temerature rage ([0, 300 K], [0, 500 K]), mathematical modelig of the temerature deedece of the bad ga is satisfactorily described by a arabolic disersio law ad the Kae model. The exerimetal results of chagig the bad ga of silico [13] ad IAs [14] withi the accuracy of measuremet is cosistet with the results of theoretical calculatios. Comariso of theory ad exerimet shows that the thermal broadeig of the eergy levels with the GN fuctio satisfactorily describes the rocess of the temerature deedece of the bad ga of Si ad IAs. REFERENCES [1] B. Ridley, Quatum Processes i Semicoductors, Claredo Press, Oxford, [2] R. Pässler, Semi-Emirical Descritios of Temerature Deedeces of Bad Gas i Semicoductors, Physica Status Solidi (b), Vol. 236, No. 3, 2003, htt://dx.doi.org/ /ssb [3] G. Gulyamov ad N. Yu. Sharibaev, Determiatio of the Desity of Surface States at the Semicoductor-Isulator Iterface i a Metal-Isulator-Semicoductor Structure, FTP (Fizika Techika Polurovodikov), Vol. 45, No. 2, 2011, [4] G. Gulyamov ad N. Yu. Sharibaev, Determiatio of the Discrete Sectrum of Surface State Desity i MOS- Structures Al-SiO 2 -Si Irradiated by Neutros, Poverkhost. Retgeovskie, Sikhrotroye i Neitroye Issledovaiya, No. 9, 2012, [5] G. Gulyamov, N. Yu. Sharibaev ad U. I. Erkaboev, Thermal Broadeig Desity of States ad Temerature Deedece of the Bad Ga Ge, FIP (Fyzicheckaya Ijeeriya Poverkhosti), Vol. 10, No. 4, 2012, [6] G. Gulyamov ad N. Yu. Sharibaev, Temerature Deedece of the Bad Ga of Si ad Due to the Thermal Broadeig of the Desity of States of, FIP (Fyzicheckaya Ijeeriya Poverkhosti), Vol. 10, No. 2, 2012, [7] G. Gulyamov, I. N. Karimov, N. Yu. Sharibaev ad U. I. Erkaboev, Determiatio of the Desity of Surface States at the Semicoductor-Isulator Structures i Al-SiO 2 -Si ad Al-SiO 2 --Si <Ni> at Low Temeratures, Uzbek Joural of Physic, Vol. 12, No. 3, 2010, [8] G. Gulyamov, N. Yu. Sharibaev ad U. I. Erkaboev, Ifluece Effective Mass Desity of States at Temerature Deedece of Bad Ga i Solid Solutios -Bi 2 x Sb x Te 3 y Se y, FIP (Fyzicheckaya Ijeeriya Poverkhosti), Vol. 11, No. 2, 2013, [9] L. N. Lukyaova, V. Kutasov ad P. Kostatiov, Effective Mass ad Mobility i Solid -Bi 2 x Sb x Te 3 y Se y for Temeratures <300K, Solid State, Vol. 47, No. 2, 2005, [10] N. Mott ad E. Devis, Electroic Processes i No-Crystallie Materials, 2d Editio, Claredo Press, Oxford, [11] V. B. Boch-Bruyevich, et al., Electroic Theory of No- Crystallie Semicoductors, Nauka, Moscow, [12] A. I. Aselm, Itroductio to Semicoductor, Nauka, Moscow, [13] K. V. Shalimova, Physics of Semicoductors, Eergoatomizdat, Moscow, [14] A. M. R. Godeir, Novel Dilute Nitride Semicoductor Oe Access

5 220 The Temerature Deedece of the Desity of States i Semicoductors Materials for Mid-Ifrared Alicatios, Uiversity of Lacaster, [15] A. Caiafa, X. Wag, J. L. Hudgis, E. Sati ad P. R. Palmer, Cryogeic Study ad Modelig of IGBTS, IEEE 34th Aual Power Electroics Secialist Coferece, Jue 2003, Oe Access