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Construction of the exact solution of the stationary Boatman equation for the semiconductor with isotropic dispersion law MALYK O.P. Semiconductor Electronics Department “Lviv Polytechnic” National University Bandera Street. 12, 79013, Lviv, UKRAINE Abstract: - The way of construction of totality of the exact solutions of a stationary Boltzmann equation for the homogeneous semiconductor with the isotropic dispersion law is proposed in case of Hall effect and electrical conductivity. The criterion of selection of the physical solutions from totality of the mathematical solutions of Boltzmann equation is determined. Key-Words: - Transport Phenomena, Boltzmann Equation, Exact Solution 1 Introduction The description of transport phenomena in semiconductors is usually carried out on the base of Boltzmann equations with the use of relaxation time approximation or variational method [1,2]. However, these methods are approximate and therefore do not allow to answer the question: as far as the selected quantum mechanical models of charge carriers scattering describe correctly the physical reality? The aim of the present work is the precise solution of the stationary Boltzmann equation that will allow directly to determine the adequacy of charge carriers scattering model to physical reality. 2 Solution of stationary Boltzmann equation Let's consider the stationary Boltzmann equation for the homogeneous semiconductor in isothermal conditions when the magnetic field is directed along Z-axis B={0,0,B} and the electric field has components E={E1, E2, 0} : q E q vB f ( p ) W ( p, p ) f ( p ) p 1 f ( p ) - W ( p , p ) f ( p ) 1 f ( p )dp , (1) where f ( p ) - distribution function; v p charge carrier velocity; q – charge of the carrier. Suppose that the charge transport is carried out by two types of carriers (electrons and holes) with isotropic dispersion law. Let’s find the solution of equation (1) in the form: f ( p) (2) f a ( p ) f0 a ( p ) - 0 a a( p ) , where a 1, 2 - respectively for electrons and holes; f 0 a ( p ) - equilibrium Fermi-Dirac function; a ( p ) - unknown functions. Substituting (2) in (1), using a principle of detail equilibrium and relations: f0 1( p ) 1 f 0 1 ( p ) 1 f 0 1 ( p ) ; kB T f0 2 ( p ) 1 f 0 2 ( p ) 1 f 0 2 ( p ) (3) kB T we shall receive a set of equations for functions a ( p ) ( a 1, 2 ) : f ( p ) a ( p ) f 0 a ( p ) qa E v a 0 a q a v a B p qa E f 0 a ( p ) a ( p ) 2 f0 a q a E v a a ( p ) p 2 2 1 b 1 B k T W ab ( p , p ) f 0 a ( p ) 1 f 0 a ( p ) a ( p ) - a ( p )dp Wab ( p , p ) kB T 2 f 0 a ( p ) 1 f 0 a ( p ) f 0 a ( p ) f 0 a ( p ) a ( p ) a ( p ) dp , (4) where Wab ( p , p ) - the charge carrier transition probability from initial state “p” in band “a” to final state “p’ “ in band “b”; the upper sign before the first integral in a right side of the equation relates to electrons and the lower sign - to holes. It should be noticed that equation (4) differs from the known Bloch equation, obtained by a linearization of Boltzmann equation, by the availability of two last items in the left side of equation and by the availability of a second item in the right side of the equation which is absent in the case of elastic carriers scattering . 1 2 Let’s find the unknown functions a ( p ) in such a form: a( p ) C na p ( ) n , (5) n, where C n a - unknown coefficients; n = 0,1,2,3.. , a = 1,2 respectively for electrons and holes; = 1,2,3 ; p ( ) - quasi momentum component. Let's substitute (5) in the Boltzmann equation (4), multiply this equation by p ( ) m (m = 0,1,2,3 …, = 1,2,3) and integrate its both parts over the quasimomentum. At an integration of the left side of the equation (4) we transfer to a spherical coordinate system and then obtain: f0 a ( p ) 2V 4 qEv p ( ) m dp 2 3 3 f 0 a ( ) k ( ε ) 3 ε m dε B ma ; ε a ( p ) f0 a ( p ) q v B p ( ) m dp p 2V 4 ( 1 2 2 1 ) C nα a qB 3 3 2 n , 1 E1 2 E 2 q a f 0 a ( ) k ( ε ) 3 ε m n dε C nα a M βmαna ; ε n , f 0 a ( p ) a ( p ) p ( ) m dp p a qa E q a n, , a f 0 a ( p ) E C na n n-1 k k k m n dp 0 ; p ( ) k q a E v a a ( p ) q a E n, , a f0 a ( p ) 2 2 Wab ( p , p ) Wab ( p, p ) ( ) , (8) where Wab ( p , p ) - some function; - some energy equal to null at the elastic scattering mechanism. For the calculation of the first integral in the right side of the equation let’s transfer to a spherical coordinate system concerning a vector p having guided an axis OZ along a vector p . Then the integral over a state p’ in a band “b” becomes: W ab ( p , p ) f 0 a ( p ) 1 f 0 a ( p ) a ( p ) - b a ( p ) dp V ( 2 ) 3 C W ( p, p ) na n, ab b f 0 a ( p ) 1 f 0 a ( p ) p ( ) n p ( ) n dp ( ) p 2 ( ) sin ( ) d d d d C F ( ) C na 1 na F2 ( ) p ( ) , (9) n, where F1 ( ) and F2 ( ) - known functions of an energy. At a repeated integration of a relation (9) over the state of the vector p we transfer to a spherical coordinate system and then the first item at integration over angles gives null. The integral for a second item in the view of a manifestative aspect of function F2 ( ) and relation (7b) becomes: W ab ( p , p ) f 0 a ( p ) 1 f 0 a ( p ) a ( p ) - - a ( p )p ( ) m dp dp 4V 2 ( 2 )6 C na n, 4 3 ( ) f 0 a ( ) 1 f 0 a ( ) n m p( ) 4 Wab p ( ) dp m f0 a ( p ) k C k p ( ) na ma 2 2 m n dp 0 , where k( ) - module of a wave vector; V – crystal volume; - Kronecker deltas; ma - charge carrier effective mass and at an integration the relations are taken into account: a( p ) k k C na n n-1 (7a) ; p k k n , 4 p ( ) p ( ) sin d d 3 | p | 2 a p( ) p( ) d , (10) ( ) is determined by relation : where Wab p( ) 2 ( ) Wab ( p, p ) sin d d . Wab (10a) Passing from integration over a quasimomentum to integration over a wave vector we discover final expression obtained at an integration of the first item in the right side of the Boltzmann equation: 1 Wab ( p , p ) f0 a ( p ) 1 f0 a ( p ) k BT a ( p ) - a ( p ) p ( ) m dp dp . (7b) At an integration of the right side of the equation (4) we take into account that the carrier transition probability Wab ( p , p ) is proportional to a deltafunction: C n, na 4 2 3 k BT W ( ) f ab a 0 a( 4V 2 ( 2 )6 ) 1 f0 a ( ) n m k( )4 k( )2 C na K n m b a . f 0 a ( ) 1 f 0 a ( ) f 0 a ( ) - f 0 a ( ) k( ) k( ) d (10b) ( ) n m k ( )3 n, We do similarly at an evaluation of the second integral in the right side of the Boltzmann equation . In this case the integral over a state p in a band "b" becomes: Wab ( p , p ) f 0 a ( p ) 1 f 0 a ( p ) f 0 a ( p ) - b f 0 a ( p ) a ( p ) a ( p ) dp V ( 2 ) 3 C C W a na n , , , ab ( p , p ) f 0 a ( p ) 1 - b f 0 a ( p ) f 0 a ( p ) f 0 a ( p ) p ( ) p ( ) n p 2 ( ) sin ( ) C 3 a Cn a L m n b a . k( ) k( ) k( )4 d (14) n,, It should be noticed that for given dispersion laws k( ) and scattering probabilities Wab ( p , p ) the values K, L, M, B can be always defined with any given degree of accuracy. Thus as a result of integration the Boltzmann equations are reduced to the system of the nonlinear algebraic equations concerning values C na for electrons and holes: C na K mn b a C 3a C na L mn b a n , , b dp ( ) d d d Bma d n , , , b C na M mn a (15) n, ) f 0 a ( )1 f 0 a ( ) For the further analysis let's rewrite (15) in the other form separating electronic and hole components and n , , also - components of the equations ): f 0 a ( ) - f 0 a ( ) p ( )( ) n p( ) 3 mn mn 3 mn 3 mn C n1 1 ( K11 11 K 11 12 C 1 L11 11 C 2 L11 12 ) p( ) , (11) 2 nm m C n 1 M 12 1 B11 2 mn mn 3 mn 3 mn ( ) is determined by a relation: Where Wab C n 1 ( K 22 11 K 22 12 C 1 L22 11 C 2 L22 12 ) V ( 2 ) 3 C C na 3 ( aW ab ( ) Wab ( p, p ) cos sin d d , Wab (12) and it is also taken into account that the scattering probability depends only on an angle . Repeatedly integrating a relation (11) over the states of a vector p we shall receive: 1 Wab ( p , p ) f 0 a ( p ) 1 f 0 a ( p ) 2 2 kB T f 0 a ( p ) - f 0 a ( p ) a ( p ) a ( p ) p ( ) m dp dp 4 V 2 4 ( 2 )6 3k B T C C W ( ) na n , , 3 a ab a f 0 a ( ) 1 f 0 a ( ) f 0 a ( ) - f 0 a ( ) ( ) n m p( ) 3 p( ) p( ) 4 p( ) d . (13) Passing to integration over a quasimomentum we receive for the second integral in the right side of the Boltzmann equation: 1 Wab ( p , p ) f0 a ( p ) 1 f0 a ( p ) 2 2 kB T f0 a ( p ) - f0 a ( p ) a ( p ) a ( p ) p ( ) m dpdp 3 4 V 2 4 ( 2 )6 3 k B 2T 2 C C W ( ) na n , , 3 a C 1 M n m B m 21 n 1 21 1 3 m n mn 3 mn C n 1 ( K 33 11 K 33 12 C 1 L33 11 3 mn (15a) C 2 L33 12 ) 0 3 mn mn 3 mn C n 2 ( K 33 22 K 33 21 C 1 L33 21 3 mn C 2 L33 22 ) 0 C 2 ( K m n K m n C 3 Lm n C 3 Lm n ) 22 21 1 22 21 2 22 22 n 2 22 22 nm m C n1 2 M 21 2 B 22 1 mn mn 3 mn 3 mn C n 2 ( K 11 22 K 11 21 C 1 L12 21 C 2 L12 22 ) nm m C n2 2 M 12 1 B12 The system (15a) can be solved in two steps: 1) for the fixed value m = 0,1,2 .. one can obtain С3n1 та С3n2 ( n =0,1,2 ..) from 3 rd and 4 th system equations; 2) one can obtain the values С 1n1 , C 2n1 and C 1n2 , C 2n2 (n =0,1,2...) respectively from 1 st and 2 nd , 5 th and 6 th system equations, which form two independent linear systems equations. It should be noticed that values С αn b (α=1,2; b=1,2; n=0,1,2..) are the linear functions from the components of the electric field intensity vector Е. The system of nonlinear algebraic equations for values С 3n1 and С 3n2 ( n=0,1,2 ..) may have such types of solutions: 1) С 3n1 = С 3n2 = 0 ; 2) С 3n1 = 0, С 3n2 0; 3) С3n1 0 , С3n2 = 0; 4) С3n1 0, С 3n2 0 . That’s why the necessity of choice of ab a In relation (15a) the rule of summarizing over the indexes which are repeated twice is used. physical solutions between the totality of mathematical solutions arises. To formulate the choice criterion let's calculate beforehand the components of a current density vector: 1 f 0 a J 3 3 q a v a a dp 4 a f 0 a (16) k ( )3 n d 3 a , n Let’s formulate the next criterion of choice: Jz = Jz n + Jzp = 0 , where Jz n , Jz p are the electron and hole contributions into the Z – component current density vector. Only the solution of type 1) satisfies this criterion. Other solution types under the value of the magnetic field induction B = 0 transform determinants of the linear system equations for values С 1n1 and С 1n2 ( 1 st and 6 th equations of system (2a) ) into zero , thus they are physically senseless and that’s why must be rejected. For taking off the physical solutions among solutions of type 1) it is necessary to determine at fixed value n = 0,1,2 … the components of a conductivity tensor and compare them with experiment. For evaluation of conductivity tensor 1 2 q C a na components let’s use that the values C n1 a (a =1,2) are the linear functions of E1 and E2 then Jx will also linearly depend upon them and the values attached to them will be equal to respective conductivity tensor components 11 and 12 for electrons and holes. References: [1] A.I. Anselm. Introduction to the Theory of Semiconductors, Nauka, Moscow, 1978 (in Russian). [2] I.M. Dykman , P.M. Tomchuk, Transport Phenomena and Fluctuations in Semiconductors, Naukova Dumka, Kiev, 1981 (in Russian).