ANEXO AL TEMA 4 FÍSICA DEL ESTADO SÓLIDO II
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1 ANEXO AL TEMA FÍSICA DEL ESTADO SÓLIDO II
2 ESTRUCTURA DE BANDAS E INTERVALO DE ENERGÍA PROHIBIDA PARA ALGUNOS SEMICONDUCTORES ENERGY (ev) ENERGY (ev) ENERGY (ev) ENERGY (ev) ENERGY (ev) Silicon E g =.7.37 x T (ev) T k E g =.ev at 3K E g T = Temperature in K ENERGY (ev) Gallium Arsenide E g =.3eV.3.58 E g at 3 K [] [] k E g = x T T + T = Temperature in K (ev) Germanium.3 E g E g =.66eV at 3K.9 L [] Γ [] X Indium Arsenide E g =.35eV at 3K E g L [] Γ [] X k ) ) Aluminum Arsenide.75 Eg =.5eV at 3K E g L [] Γ [] X Indium Phosphide E g =.3eV at 3K E g.9 L [] Γ [] X k
3 EFFECTIVE MASS ( ) c MASA EFECTIVA EN BANDA DE CONDUCCIÓN PARA SEMICONDUCTORES MÁS TÍPICOS m * m o. CdSe CdS AlAs AlSb ZnTe GaP ZnSe. GaSb CdTe InP GaAs InAs InSb.. 3. BANDGAP E g (ev) I Banda de huecos ligeros II Banda huecos pesados III Split-Off banda de Valencia Δ Split-Off energía E c (k) = E c () + h k m* c
4 Material Bandgap Relative (ev) Dielectric Constant C 5.5, I 5.57 Si., I.9 Ge.66, I 6. SiC.6, I 9.7 GaAs., D 3.8 AlAs.53, I.6 InAs.35, D 5.5 GaP.7, I. InP.3, D.56 InSb.3, D 6.8 CdTe.75, D. AlN 6., D 9. GaN 3., D. ZnSe.8, D 9. ZnTe.39, D 8.7 Material Electron Hole Mass Mass (m ) (m ) AlAs. AlSb. m dos * =.98 GaN.9 m dos * =.6 GaP.8 m dos * =.6 GaAs.67 m* lh =.8 m* hh =.5 GaSb. m dos * =. Ge m l =.6 m lh * =. m t =.8 m* hh =.8 m dos =.56 InP.73 m dos * =.6 InAs.7 m dos * =. InSb.3 m* dos =. Si m l =.98 m* lh =.6 m t =.9 m hh * =.9 m dos =.8 Propiedades de algunos semiconductores. D e I indican transición directa e indirecta respectivamente. Los datos mostrados corresponden con una temperatura de 3K.
5 ELECTRONES Y HUECOS EN UN SEMICONDUCTOR INTRÍNSECO E E c E F E v f F (E) g c (E) g v (E) g c (E)f F (E) n(e) Area n electron concentration g v (E)( f F (E)) p(e) Area p hole concentration E E c E v E f F (E) g v (E) g c (E) [ ff (E)] Densidad de estados en banda de valencia y conducción y función de Fermi. Las áreas representan la concentración de electrones y huecos en el caso de que el nivel de Fermi esté en el centro del intervalo de energía prohibida. Mostramos ampliación de las zonas cerca de los bordes de banda de valencia y banda de conducción. f F (E) f F (E)
6 ELECTRONES Y HUECOS EN UN SEMICONDUCTOR INTRÍNSECO Aproximación de Maxwell-Boltzman: FERMI - DIRAC. Fermi Dirac function f # E = + exp E E # Boltzmann approximation E E F f / E exp (E E #) MAXWELL - BOLTZMANN
7 ELECTRONES Y HUECOS EN UN SEMICONDUCTOR INTRÍNSECO En un semiconductor intrínseco y bajo la aproximación de Maxwell-Boltzman: 9 9 n = 5 g 7 E f # E de 5 g 7 E f / E de πm? : ; h B : ; C B exp (E D E # ) : H : H p = 5 g G E ( f # E )de 5 g 7 E ( f / E )de πm J I9 p = N G exp (E # E G ) I9 h B = N 7 exp (E D E # ) C B exp (E # E G ) = En todo semiconductor intrínseco n = p
8 ELECTRONES Y HUECOS EN UN SEMICONDUCTOR INTRÍNSECO Consecuencias: n = p N 7 exp (E D E # ) = N G exp (E # E G ) operando E # = E G + E D + Ln N G = N D E G + E D + 3 Ln m S m? E G + E D = E TUJ n p = n W B = N D N G exp E TUJ = cte. Material n i (cm -3 ) n W = p W =.86 x B_ m? m S m? B C/a T C/B exp E TUJ m IC Si,5 x GaAs,8 x 6 Ge, x
9 ELECTRONES Y HUECOS EN UN SEMICONDUCTOR INTRÍNSECO 5 T( C) σ total = neμ? + peμ S = Intrinsic carrier density n i (cm 3 ) Ge Si GaAs =.86 x B_ m? m S m? B = 773. m? m S m? B C/a C/a T C/B e μ? + μ S T C/B μ? + μ S σ total = σ j exp E TUJ exp E TUJ exp E TUJ = T(K ) Concentración de portadores intrínsecos en diferentes semiconductores en función dela temperatura.
10 ELECTRONES Y HUECOS EN UN SEMICONDUCTOR INTRÍNSECO σ total = σ j exp E TUJ El intervalo de energía prohibida depende del parámetro de red del material
11 MOVILIDAD EN UN SEMICONDUCTOR 5 N D 6 N D n 5 T N D cm 3 5 T (K) N A N A 7 N A 6 μ = + μ k μ m n (cm /V-s) N D 7 5 N D 8 N D T ( C) (a) p (cm /V-s) N A 8 N A 9 N A cm 3 T 5 T (K) T ( C) p 5 Variación de la movilidad de electrones y huecos en silicio en función de la temperatura y en función de la concentración de dopantes. Los insets muestran la dependencia con la temperatura parauna muestra intrínseca. Dependencia de la movilidad con la temperatura: La movilidad de ve sólo afectada por la red cristalina: μ k T IC/B La movilidad de ve sólo afectada por impurezas ionizadas: μ m nop/q r s, donde N I es el número de impurezas ionizadas
12 MOVILIDAD EN UN SEMICONDUCTOR n T 3 K 3 p Ge Mobility (cm /V-s) 3 Si n p Variación de la movilidad de electrones y huecos en silicio y germanio en función de la concentración deimpurezas. n 3 p GaAs 5 Impurity concentration (cm 3 )
13 DEPENDENCIA DE LA RESISTIVIDAD CON LA TEMPERATURA EN UN SEMICONDUCTOR T (K) 75 Electron concentration (cm 3 ) 6 5 n i n.. Conductivity ( cm) ρ = σ = e μ u n + μ J p Relación entre la concentración de electrones y la conductividad en función del inverso de la temperatura en silicio T (K )
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