Page 1 Quantum free electron theory(Sommerfield) • Particles of micro dimension like the electrons are studied under quantum physics • moving electrons inside a solid material can be associated with waves with a wave function ?(x) in one dimension (?(r) in 3D) • Hence its behaviors can be studied with the Schrödinger's equation (1D) For a free particle V=0, hence the equation reduces to E With 0 ) ( ) ( 2 ) ( 2 2 2 x V E m x x ? , 0 ) ( ) ( 2 2 2 x k x x 2 , 2 , 2 2 2 2 2 h m k E or mE k ? ? ? K E f Page 2 Quantum free electron theory(Sommerfield) • Particles of micro dimension like the electrons are studied under quantum physics • moving electrons inside a solid material can be associated with waves with a wave function ?(x) in one dimension (?(r) in 3D) • Hence its behaviors can be studied with the Schrödinger's equation (1D) For a free particle V=0, hence the equation reduces to E With 0 ) ( ) ( 2 ) ( 2 2 2 x V E m x x ? , 0 ) ( ) ( 2 2 2 x k x x 2 , 2 , 2 2 2 2 2 h m k E or mE k ? ? ? K E f Sommerfield’s model • As the freely moving electrons can not escape the surface of the material, they may be treated as particles confined (trapped)in a box • Hence, V(x) =0, for 0<x<L . e- = 8, for x=0 & x=L V(x) 0 L x Sch’s equation , Solution of the equation can be obtained as ?(x) = A sin kx + B cos kx From boundary conditions , at x=o & x=L , ?(x)=0, we can get B=0 and k= ± np / L , Putting the normalization condition we get A= 0 ) ( 2 ) ( 2 2 2 x mE x x ? P= 0 ?= 0 P= 0 ?= 0 L 2 Page 3 Quantum free electron theory(Sommerfield) • Particles of micro dimension like the electrons are studied under quantum physics • moving electrons inside a solid material can be associated with waves with a wave function ?(x) in one dimension (?(r) in 3D) • Hence its behaviors can be studied with the Schrödinger's equation (1D) For a free particle V=0, hence the equation reduces to E With 0 ) ( ) ( 2 ) ( 2 2 2 x V E m x x ? , 0 ) ( ) ( 2 2 2 x k x x 2 , 2 , 2 2 2 2 2 h m k E or mE k ? ? ? K E f Sommerfield’s model • As the freely moving electrons can not escape the surface of the material, they may be treated as particles confined (trapped)in a box • Hence, V(x) =0, for 0<x<L . e- = 8, for x=0 & x=L V(x) 0 L x Sch’s equation , Solution of the equation can be obtained as ?(x) = A sin kx + B cos kx From boundary conditions , at x=o & x=L , ?(x)=0, we can get B=0 and k= ± np / L , Putting the normalization condition we get A= 0 ) ( 2 ) ( 2 2 2 x mE x x ? P= 0 ?= 0 P= 0 ?= 0 L 2 • Substituting all the values • ? n (x) = (v 2/L) sin npx /L • & E n (x) = h 2 k 2 / 2m = h 2 p 2 n 2 / 2m L 2 = h 2 n 2 / 8mL 2 This shows that energy of the electrons inside the material is quantized and hence is discrete ?3 n=3 E3 ?2 n=2 E2 ?1 n=1 E1 In 3D, ? n (r) = (v 8/L 3 ) sin n x px /L sin n y py /L sin n z pz /L & E n (r) = h 2 k 2 / 2m = (n x 2 +n y 2 +n z 2 ) h 2 p 2 / 2m L 2 Page 4 Quantum free electron theory(Sommerfield) • Particles of micro dimension like the electrons are studied under quantum physics • moving electrons inside a solid material can be associated with waves with a wave function ?(x) in one dimension (?(r) in 3D) • Hence its behaviors can be studied with the Schrödinger's equation (1D) For a free particle V=0, hence the equation reduces to E With 0 ) ( ) ( 2 ) ( 2 2 2 x V E m x x ? , 0 ) ( ) ( 2 2 2 x k x x 2 , 2 , 2 2 2 2 2 h m k E or mE k ? ? ? K E f Sommerfield’s model • As the freely moving electrons can not escape the surface of the material, they may be treated as particles confined (trapped)in a box • Hence, V(x) =0, for 0<x<L . e- = 8, for x=0 & x=L V(x) 0 L x Sch’s equation , Solution of the equation can be obtained as ?(x) = A sin kx + B cos kx From boundary conditions , at x=o & x=L , ?(x)=0, we can get B=0 and k= ± np / L , Putting the normalization condition we get A= 0 ) ( 2 ) ( 2 2 2 x mE x x ? P= 0 ?= 0 P= 0 ?= 0 L 2 • Substituting all the values • ? n (x) = (v 2/L) sin npx /L • & E n (x) = h 2 k 2 / 2m = h 2 p 2 n 2 / 2m L 2 = h 2 n 2 / 8mL 2 This shows that energy of the electrons inside the material is quantized and hence is discrete ?3 n=3 E3 ?2 n=2 E2 ?1 n=1 E1 In 3D, ? n (r) = (v 8/L 3 ) sin n x px /L sin n y py /L sin n z pz /L & E n (r) = h 2 k 2 / 2m = (n x 2 +n y 2 +n z 2 ) h 2 p 2 / 2m L 2 Fermi Level and Fermi Energy: • Electrons are fermions or Fermi particles, which obey Pauli’s exclusion principle • At 0K temperature the highest filled energy level is called the “Fermi Level” & the energy possessed by the electrons in that level is “Fermi Energy” • E f = h 2 k f 2 / 2m or k f = (2mE f / h 2 ) 1/2 • In 3D electrons will fill up the k-space with one state accommodating 2 electrons each ( ??) • If N is the total no. of electrons & is large , electrons will occupy a sphere of radius k f , then highest occupied state n f = N/2 From uncertainty principle electron states ?x ?p =h or, ?x h?k = h Or, L (h/2p) ?k = h or, ?k = 2p / L k x k y k f Page 5 Quantum free electron theory(Sommerfield) • Particles of micro dimension like the electrons are studied under quantum physics • moving electrons inside a solid material can be associated with waves with a wave function ?(x) in one dimension (?(r) in 3D) • Hence its behaviors can be studied with the Schrödinger's equation (1D) For a free particle V=0, hence the equation reduces to E With 0 ) ( ) ( 2 ) ( 2 2 2 x V E m x x ? , 0 ) ( ) ( 2 2 2 x k x x 2 , 2 , 2 2 2 2 2 h m k E or mE k ? ? ? K E f Sommerfield’s model • As the freely moving electrons can not escape the surface of the material, they may be treated as particles confined (trapped)in a box • Hence, V(x) =0, for 0<x<L . e- = 8, for x=0 & x=L V(x) 0 L x Sch’s equation , Solution of the equation can be obtained as ?(x) = A sin kx + B cos kx From boundary conditions , at x=o & x=L , ?(x)=0, we can get B=0 and k= ± np / L , Putting the normalization condition we get A= 0 ) ( 2 ) ( 2 2 2 x mE x x ? P= 0 ?= 0 P= 0 ?= 0 L 2 • Substituting all the values • ? n (x) = (v 2/L) sin npx /L • & E n (x) = h 2 k 2 / 2m = h 2 p 2 n 2 / 2m L 2 = h 2 n 2 / 8mL 2 This shows that energy of the electrons inside the material is quantized and hence is discrete ?3 n=3 E3 ?2 n=2 E2 ?1 n=1 E1 In 3D, ? n (r) = (v 8/L 3 ) sin n x px /L sin n y py /L sin n z pz /L & E n (r) = h 2 k 2 / 2m = (n x 2 +n y 2 +n z 2 ) h 2 p 2 / 2m L 2 Fermi Level and Fermi Energy: • Electrons are fermions or Fermi particles, which obey Pauli’s exclusion principle • At 0K temperature the highest filled energy level is called the “Fermi Level” & the energy possessed by the electrons in that level is “Fermi Energy” • E f = h 2 k f 2 / 2m or k f = (2mE f / h 2 ) 1/2 • In 3D electrons will fill up the k-space with one state accommodating 2 electrons each ( ??) • If N is the total no. of electrons & is large , electrons will occupy a sphere of radius k f , then highest occupied state n f = N/2 From uncertainty principle electron states ?x ?p =h or, ?x h?k = h Or, L (h/2p) ?k = h or, ?k = 2p / L k x k y k f • state in k-space of radius k f = • = K f 3 L 3 / 6p 2 • Hence no. of electrons ‘N’ = 2 X ( K f 3 L 3 / 6p 2 ) • • So, k f = {3p 2 (N/ V) } 1/3 • where n = electron density = N/V = 1/ V (k f 3 V / 3p 2 ) • Or, n= 3 3 2 3 4 L k f N = k f 3 V / 3p 2 k f = (3p 2 n) 1/3 2 3 2 3 2 2 2 3 1 f E m ?Read More

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