DETERMINATION OF TRANSPORT PROPERTIES OF SILICON ELECTRON GAS

Transcription

1 MATHEMATICA MONTISNIGRI Vol XXXIX (2017) MATHEMATICAL MODELING DETERMINATION OF TRANSPORT PROPERTIES OF SILICON ELECTRON GAS O.N. KOROLEVA 1,2, A.V. MAZHUKIN 1,2 1 Keldysh Institute of Applied Mathematis of RAS 2 National Researh Nulear University MEPhI Abstrat. The paper disusses the results of mathematial modeling of harateristis of eletron gas of silion with intrinsi ondutivity. The properties of eletroni subsystem are determined within the framework of the ontinuum approah using the quantum statistis of eletron gas and Fermi-Dira integrals in an arbitrary degeneray range of the eletron gas with a temperature hange from 300K to 2000K. When modeling the properties of the eletroni subsystem, the effet of narrowing the band gap under the onditions of suffiiently strong heating of the intrinsi semiondutor and arrier degeneray is taken into aount. The properties of eletron gas, suh as arrier onentrations in the ondution and valene bands, mobility of arriers, eletrial ondutivity, and the oeffiient of ambipolar diffusion are determined. Numerial and graphial information on obtained properties and omparison results with experimental data are presented. 1 INTRODUCTION In the past few deades, tehnologial appliations related to the laser proessing of semiondutors with short pulses have been rapidly developing. The wide spetrum of tehnologies assoiated with this diretion inludes suh as the generation of nanopartiles and nanostrutures [1, 2], the reation of metamaterials [3], modifiation of the surfae of semiondutors by laser pulses [4]. The development of tehnologies related to laser ation on semiondutors, ativates researh on the fundamental mehanisms underlying suh proesses as ultrafast melting, ablation, whih are still the subjet of ative sientifi disussions [5]. In this onnetion, knowledge of the properties of semiondutors, in partiular silion, at temperatures lose to the melting and evaporation temperature is of deisive importane. Traditionally, the properties of semiondutors, like metals, are determined experimentally. In the literature, the experimental values of the equilibrium thermophysial properties of silion obtained for different temperature ranges (up to the melting point) are reported [6-9]. These data, mainly relating to thermal ondutivity and eletrial ondutivity, differ markedly from eah other. There are also methods for measuring the equilibrium thermophysial properties of semiondutors in a molten state [9]. The data obtained by measurements, in the ourse of experiments, are widely used for testing theoretial dependenies. Despite this, the experimental approah has a number of limitations, primarily on a range of measurement onditions, espeially in the field of melting. In the problems of laser ation on semiondutors the knowledge of equilibrium properties is insuffiient. Laser heating of semiondutors (silion) as well as of metals is nonequilibrium and proeeds with a large separation of the temperatures of urrent arriers 2010 Mathematis Subjet Classifiation: 82D37, 82B10, 82C70. Keywords and Phrases: eletron gas of silion, transport properties, mobility of arriers, ambipolar diffusion oeffiient, eletrial ondutivity, quantum statistis, Fermi-Dira integrals. 57

2 from the lattie; therefore, in problemsof laser ation on silion it an be onsidered as an objet onsisting of two interating subsystems, eletroni and phonon. At the same time, for eah of the subsystems it is neessary to determine thermophysial, optial and thermodynami harateristis that vary over a wide temperature range. In view of limited possibilities of the experimental approah in determining the properties of the eletron gas of silion, in this paper we propose to use the theoretial approah and the possibilities of mathematial modeling. For theoretial determination of the properties of eletroni subsystem, the present paper uses the quantum statistis of eletron gas, i.e. distribution funtions and Fermi-Dira integrals [10, 11]. Numerous experiments [5-8] have shown that in the proess of melting in silion ovalent bonds are destroyed, with a hange in the shortrange order aompanied by a sharp inrease in the onentration of ondution eletrons and leading to the transition of silion to the metalli state. The behavior of the eletroni subsystem of silion under phase transition onditions is deisive for laser impat problems. The most important harateristi of the phase transition, along with the temperature dependene of the arrier onentration, is the fundamental harateristi of silion - band gap. Under the onditions of temperature inrease, the band gap Eg(T) narrows, having a signifiant effet on the inrease in the arrier onentration reahing high values of N(T) m -3 and higher, whih is onfirmed by experimental studies [13,14]. The artile presents numerial and graphial information about some harateristis of the eletroni subsystem of silion with intrinsi ondutivity, determined by means of mathematial modeling. Suh eletron gas properties as eletron onentration Ne(T), holes onentration Nh(t), Fermi energy E F (T,N), band gap E g (T,N), arrier mobility μ(t,n), eletrial ondutivity σ(t,n), ambipolar diffusion oeffiient D a (T,N) are determined within the framework of quantum statistis in an arbitrary range of eletron gas degeneray with a temperature hange from 300K to 2000K. The results are ompared with the experimental data. 2 CALCULATION OF PROPERTIES OF ELECTRONIC SUBSYSTEM OF SILICON The alulation of the properties of eletroni subsystem of silion is based on the use of statistis of eletron gas of semiondutors. Central to this approah is the harge arrier distribution funtion for energy states. Eletrons in the ondution band and holes in the valene band of silion an be onsidered as an ideal Fermi gas. For an ideal Fermi gas, the probability of an eletron filling a state k with energy E at a temperature T is found using the Fermi-Dira distribution [10-12]: 1 f (E,T) = E EF 1+ exp kbt where E F is the Fermi energy, determined from the eletroneutrality ondition, k B is the Boltzmann onstant. For eletron gas, the value of Fermi energy oinides with the value of hemial potential at T = 0 K and is defined as the amount of energy needed to hange the number of partiles in the system per unit volume. (1) 58

3 An important harateristi of semiondutors, whih is neessary for determining the majority of the thermophysial properties of silion, is the onentration of harge arriers. In intrinsi semiondutors, unlike metals, the number of harge arriers and their mobility depend on temperature. The eletron N e (T) and holes N h (T) onentrations at the temperature T in ondution band under thermodynami equilibrium onditions are determined N T N f de e ( ) = (E) ( ) (E) E C C h E V N T = N f de (2) where E C is energy of the bottom of ondution band, E V - energy of the top of valene band, N C e B mhkbt N 2 V 2 m k T = 2, = 2 2π 2π valene band, ћ is the Plank onstant, m M 2 3 ( m m 2 ) 1 3 V effetive densities of states in ondution band and = effetive mass of the density of e l t states of eletrons in ondution band, taking into aount the number of equivalent energy minima in ondution band M (for silion M=6) [10-12], m l, m t the longitudinal and rosssetion masses, m h is the effetive mass of hole density of states in the valene band, and f(e) is Fermi-Dira distribution funtion (1). For an intrinsi semiondutor that does not ontain impurities, equality of onentrations holds The integrals in (2) an be represented in the form N e (T)=N h (T)=N(T). (3) ( ) = F ( ) N ( T ) N ( η ) N T N η e C 1 2 e = F (4) h V 1 2 h where F 1/2 (x) is Fermi-Dira integral of order j=1/2, a representative of the family of integrals that play an important role in determining the properties of semiondutors j 1 ε F j ( η ) = dε Γ (5) ε η ( j + 1) 1+ exp ( ) 0 where Γ(x) Gamma funtion, j is order of Fdermi-Dira integral, =e for eletrons and =h for holes, ε is redued eletron (hole) energy, redued Fermi energy for eletrons and holes η E E F C V F e = ηh. kbt kbt E E = (6) where E C is energy of the bottom of ondution band, E V - energy of the top of valene band. At low temperatures in semiondutors, the onentration of ondution eletrons is so small that they behave like a gas of noninterating partiles, the eletron gas is nondegenerate. In this ase, the Fermi level E F lies below the bottom of ondution band (E C -E F >0) in band gap E g and distribution funtion (1) easily redues to the lassial Maxwell-Boltzmann distribution funtion and the alulation of arrier onentration (4) redues to e E E ( ) exp F C N T N = C kbt h E E ( ) exp V F N T N = V kbt (7) 59

4 Taking into aount the intrinsi ondutivity N(T) 3/2 1 2kBT 3/4 Eg e ( ) = h ( ) = ( ) = 2 ( e h ) exp 4 π 2 B N T N T N T m m k T In the omputational ontext, determination of arrier onentrations will not be diffiult. As the temperature rises, the situation hanges. Hot eletrons give energy to the lattie, while the band gap dereases and onentration of free harge arriers in the ondution band inreases. The Fermi level penetrates either to the ondution band (E C -E F <0) or to the valene band (E F -E V <0), the eletron gas degenerates and the lassial statistis beome unfair, and (7) is not valid. Therefore, it beomes neessary to use quantum statistis and expressions (4) for arrier onentrations. This immediately leads to omputational diffiulties, sine the integral (5) with the exeption of an integral with order j=0 an not be alulated analytially. The omputational diffiulties assoiated with the use of Fermi-Dira integrals arise not only in determining the arrier onentrations, but also in determining the properties of eletron gas suh as eletrial ondutivity, arrier mobility, the ambipolar diffusion oeffiient and others, where were used Fermi-Dira integrals with integer and halfintegral orders, as a rule, not high 1/ 2 j 7 / 2 and 1 j 3. In [15,16] for Fermi-Dira integrals of orders j=-1/2, 1/2, 1, 3/2, 2, 5/2, 3 and 7/2 ontinuous analytial expressions have been obtained for eah single order in a wide range of degeneray m i F j ( η ) = exp aiη, = e, h, m = 5 7, (9) i= 0 Whih in this paper were used to alulate the properties of an eletron gas. To alulate integrals with order j=1/2, for example, an approximating funtion with m = 7 is used (9). As the temperature in the semiondutor inreases, proess of thermal exitation of eletrons from valene band to ondution band proeeds ontinuously, eletrons reombine from the ondution band to the valene band. In the intrinsi semiondutor these proesses are balaned, and eletron and hole onentrations are the same. From the eletroneutrality ondition, we an find the temperature dependene of Fermi energy E F, using (4) and (9) for j=1/2, we obtain 7 7 i i NC exp aiη e = NV exp aiη h i= 0 i= 0 (10) The band gap of silion Eg, like other semiondutors, with inreasing temperature and inreasing onentration of harge arriers, tends to narrow [10-14]. The effet of narrowing the band gap is due to three main mehanisms: thermal expansion of the lattie, eletron-lattie interation and olletive interations of arriers. Thermal expansion with inreasing temperature together with the enhanement of the eletron-lattie interation auses a shift in the relative positions of the ondution and valene bands. The total manifestation of the first two mehanisms of narrowing of the band gap is desribed by a semiempirial relationship [17] g 2 (, ) α ( β ) E T N = E T T +, (11) g,0 (8) 60

5 where E g,0 = ev is band gap at temperature 0 K, α and β are onstants, whose experimental estimates for silion: α = ev/t, β = 1108K. The third mehanism of narrowing of the band gap is assoiated with the effets of olletive interations of arriers, whih beome dominant at suffiiently high onentrations. The influene of quantum effets beomes notieable at a arrier onentration of N m -3 and is formulated in a omplex manner [18]. The most signifiant ontribution to the narrowing of the band gap is made by the exhange interation estimated by the empirial dependene of the form ΔE g (N)~γ N 1/3 (T) [19], where γ is the fitting parameter used for ombining theoretial estimates with the experimentally determined values of the narrowing of the band gap in various semiondutors. For silion at a temperature of T 300K and arrier onentration N= m -3, the value of the parameter γ is in the range of ( )10-8 ev m [18, 20]. An estimate of narrowing of Si band gap taking into aount all mehanisms at high temperatures and arrier onentrations N m -3 an be performed within a semiempirial relationship [21, 22] E g 2 3 ( T,N ) E α T ( T + β ) γ N 1 ( T ) =, (12) g,0 where the value γ = ev m was hosen from the ondition that the band gap should be zero at equilibrium melting point E g (T m )=0 [22]. Figure 1 shows temperature dependenes giving a lear piture of the shape and veloity of narrowing of the band gap E g (T,N) and the position of Fermi energy level E F (T) alulated with quantum statistis relative to the edges of valene E V (T) and ondution E C (T) bands and intrinsi Fermi level in the middle of the band gap. With inreasing temperature, Fermi energy E F (T) deviates from its own level toward the edge of valene band E V (T), whih is determined by lower effetive mass of density of states of the valene band. For silion, the ratio of effetive masses of states of eletron and hole is m de /m dh =1.89. Beause of this, the degeneray of hole gas (E V - E F <k b T, η h -4) ours earlier than the degeneray of eletron gas at a temperature T=1000K whih is muh lower than equilibrium melting point. Figure 1. Temperature dependenes of the edges of ondution E C (T) and valene E V (T) bands taking into aount: (1), (2) -thermal and quantum effets; (3), (4) - thermal effets. (5) - Fermi energy E F (T). Figure 2. The temperature dependene of the band gap with regard to: (1) - thermal and quantum effets; (2) - thermal effets. The markers show experimental data [23]. 61

6 Figure 2 shows the band gap omputed taking into aount quantum and thermal (12) and only thermal (11) effets. In the temperature range from 300 K to θ where influene of quantum mehanisms is weak the band gap is equally well approximated by both dependenes and ompletely oinide with experiment [23]. Above Debye temperature, the ontribution of olletive interation mehanisms to band gap beomes appreiable whih auses a stronger narrowing of band gap by means of the dependene (12) taking these effets into aount. Figure 3 shows temperature dependenes of the arrier onentration alulated with quantum statistis and band gap (12) and lassial Maxwell-Boltzmann statistis and the onstant value of band gap E g =1.12 ev at 300K. Figure 3. Temperature dependenes of arrier onentrations alulated using statistis: (1) - Fermi-Dira, (2) - Maxwell-Boltzmann, E g = 1.12 ev. 2.1 Mobility of arriers Mobility is an important transport harateristi of a semiondutor. Using quantum statistis the mobility of arriers is determined by following expression 0 where ( T ) μ 0 ( T, N ) = μ ( T ) F F ( η ) ( η ) = e, h μ is mobility of arriers for a nondegenerate semiondutor, F 1/2 (η ), F 0 (η ) (=e,h) are Fermi-Dira integrals of orders j=1/2 and j=0. In [24] was given method for alulating the mobility of nondegenerate harge arriers taking into aoount eletron-hole sattering whih was used to alulate μ. In this approah, it is assumed that the mobility of eletron or hole is a funtion of the following three omponents: mobility of lattie, mobility of impurity sattering and mobility of eletron-hole sattering. Figure 4 shows temperature dependenes of the mobilities of eletrons and holes. With inreasing temperature, the mobility of arriers dereases whih is due to the enhanement of eletron-lattie interation and effets of olletive interations ating at suffiiently high arrier onentrations and degeneray of eletron gas. Figure 4 also provides referene [25] and experimental [26, 27] data in temperature range T=300K 600K. 0 (13) 62

7 Figure 4. Temperature dependenes of mobilities (1) - eletrons, (2) - holes, (3) - total alulated with quantum statistis, markers: blue - referene data [25], light green - experimental data [26], dark green markers experimental data [27]. In the region of low temperatures T<600K as an be seen from Fig. 4, obtained temperature dependenes of arrier mobilities show good agreement with referene and experimental data. In the high-temperature region T > 600K (near the melting point), experimental values of arrier mobility are absent. 2.2 Ambipolar diffusion oeffiient As the mobility of arriers the ambipolar diffusion oeffiient is important transport semiondutor harateristi. When laser ation on a semiondutor, in the near-surfae layer the onentration of free arriers sharply inreases whih diffuse deep into the sample. As the temperature inreases, arrier onentration inreases reahing at equilibrium melting temperature T=Tm=1687K the value of m -3 (Fig. 3), with that diffusion rate dereasing due to an inrease in frequeny of eletron-eletron and eletron-phonon ollisions. This proess is haraterized by the ambipolar diffusion oeffiient D a (T,N) whih depends both on temperature and on arrier onentration. In arbitrary range of eletron gas degeneray using quantum statistis the ambipolar diffusion oeffiient is determined [28] D a ( T, N ) e ( T, N ) μh ( T, N ) (, ) (, ) ( ηe ) ( ) B = + ( ηh ) ( ) k T μ I I, (15) e μe T N + μh T N I 1 2 ηe I 1 2 η h where µ(t,n) is total mobility (14), F 1/2 (η ), F -1/2 (η ) (=e,h) are Fermi-Dira integrals of orders j=1/2 and j=-1/2. Approximating funtions for Fermi-Dira integrals from [15, 16] were used in the simulation. It is seen that D a (T,N) depends on arrier onentration and temperature, eletron μ e (T,N) and hole μ h (T,N) mobilities, degeneray of eletron gas. 63

8 Figure 5. Temperature dependene of ambipolar diffusion oeffiient Da(T,N) (solid line) alulated with quantum statistis. Blak markers denote referene data [25], blue - experimental [27]. Figure 5 shows temperature dependene of oeffiient D a (T,N) alulated from equation (15). Comparison of alulated data with experimental data in the temperature range 300K 600K [25, 27] and referene value at a temperature of 300K showed good agreement up to the temperature T=500K. In the region of higher temperatures T>600K the omparison was not arried out beause of the lak of experimental data. 2.3 Eletrial ondutivity In alulating the temperature dependene of speifi eletrial ondutivity taking into aount the degeneray of eletron gas we also used quantum statistis (Fermi-Dira integrals). Eletron and hole eletrial ondutivities are determined [10-12] σ where µ is arrier mobility (13). ( ) ( ) μ ( ) T, N = qn T T, N, = e, h (16) Figure 6. Temperature dependenes of eletrial ondutivity of eletrons (1), holes (2) alulated with quantum statistis. Experimental data are marked with blak markers [7], red marker is referene value [25]. 64

9 Figure 6 shows eletron and hole eletrial ondutivities of silion alulated using quantum statistis, experimental data and referene values are indiated by markers [7, 25]. Conentration of harge arriers Ne(T) and Nh(T) makes the main ontribution to temperature dependene of eletrial ondutivity of semiondutors. Therefore, temperature way of eletrial ondutivity in basi features repeats temperature dependene of onentration of harge arriers (Fig. 3) and qualitatively oinides with experimental dependene (Fig. 6). 4 CONCLUSION The properties of eletron subsystem of silion are alulated in the framework of ontinuum approah using quantum statistis and Fermi-Dira integrals in an arbitrary range of eletron gas degeneray with temperature hange from 300K to 2000K. Suh harateristis of eletron subsystem of silion with intrinsi ondutivity as arrier onentration and mobility, oeffiient of ambipolar diffusion, eletrial ondutivity are alulated. When modeling the properties of eletron subsystem, the influene of narrowing of band gap under onditions of suffiiently strong heating of intrinsi semiondutor and arrier degeneray is taken into aount. The results of alulations are ompared with the results of experiments. A omparison of alulated data with experimental ones showed aeptable quantitative agreement between band gap, eletron and hole mobilities and ambipolar diffusion oeffiient and qualitative oinidene of eletrial ondutivity. Numerial and graphial information on the properties obtained and omparison results with experimental data are presented. This work was supported by RSF (projet ). REFERENCES [1] U. Zywietz, A.B. Evlyukhin, C. Reinhardt, B.N. Chihkov, Laser printing of silion nanopartiles with resonant optial eletri and magneti responses, Nat. Commun., 5, (2014). [2] A. Mene ndez-manjo n, S. Barikowski, G.A. Shafeev, V.I. Mazhukin, B.N. Chihkov, Influene of beam intensity profile on the aerodynami partile size distributions generated by femtoseond laser ablation, Laser and Partile Beams, 28, 4552 (2010). [3] L. Shi, T.U. Tuzer, R. Fenollosa, F. Meseguer, A new dieletri metamaterial building blok with a strong magneti response in the sub-1.5-mirometer region: silion olloid nanoavities, Adv. Mater., 24, (2012). [4] A. Mathis, F. Courvoisier, L. Froehly, L. Furfaro, M. Jaquot, P. A. Laourt, and J.M. Dudley, Miromahining along a urve: Femtoseond laser miromahining of urved profiles in diamond and silion using aelerating beams, Appl. Phys. Lett., 101 (7), (2012). [5] K. Sokolowski-Tinten and D. von der Linde, Generation of dense eletron-hole plasmas in silion, Phys. Rev. B, 61 (4), (2000). [6] H. R. Shanks, P. D. Mayok, P. H. Shidles, and G. C, Danielson. Thermal Condutivity of Silion from 300 to 1400 K, Phys. Rev., 130 (5), (1963). [7] Ia.B. Magometov, G.G. Gadzhiev, Vysokotemperaturnaia teploprovodnost kremniia v tverdom i zhidkom sostoianiiakh, Teplofizika vysokikh temperatur, 46 (3), (2008). [8] C. J. Glassbrennert and Glen A. Slak, Thermal Condutivity of Silion and Germanium from 3K to the Melting Point, Phys. Rev., 134 (4a), A1058-A1069 (1964). 65

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