´ lido Estado Estado Solido o Bolet´ ın de prob problem lemas as
Dpto. Dpto. F´ ısica Aplicada. Aplicada.
Andr´ Andr´es es Canta Ca ntare rero ro Ana Cros Julio Pellicer Alfredo Segura Septiembre 2010
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2
Estructuras Estructuras cristalinas. cristalinas.
1. Estructuras cristalinas. cristalinas. 1. 1. Considera Considera la red bidimensional bidimensional rectangular de la figura. En ella se han representado representado los vectores base de cuatro celdas unidad diferentes.
�
� �
a) b) c) d)
¿Cu´antos antos puntos de la red contiene en total cada una? ¿Cu´ ales ales son primitivas? ¿Cu´ ales ales son no primitivas? Determina Determina la celda de Wigner-Seitz Wigner-Seitz correspondiente. correspondiente.
1. 2. Demostr Demostrar ar que la relaci´ relacion c o´n c/a /a en en una estructura hexagonal compacta ideal es
8 3
.
1. 3. Demostrar que la fracci´ on de empaquetamiento de las siguientes estructuras es la on que se indica: a) b) c) d)
C´ ubica ubica simple: π/6 π/ 6 ≈ 52%. √ C´ ubica centrada en el cuerpo: π 3/8 ≈ 68%. ubica √ C´ ubica centrada en las caras: π 2/6 ≈ 74%. ubica √ Hexagonal Hexagonal compacta: compacta: π 2/6 ≈ 74%.
1. 4. El sodio se transforma de bcc bcc en en hcp a una temperatura de 23 K. Supongamos que durante esta transformaci´ on la densidad del cristal permanece constante, y que la on relaci´on on c/a se c/a se corresponde con la ideal de la estructura hcp. Calcula el par´ametro ametro de red a red ahcp de la celda hcp sabiendo que el par´ametro ametro de red de la celda convencional ˚ para la fase c´ ubica ubica es abcc = 4.23A. A. 1. 5. En la siguiente red bidimensional, determina la celda unidad de Bravais y su base.
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Estado S´ olido. Curso 2010-2011.
3
1. 6. El siguiente siguiente mosaico ha sido realizado realizado por el artista japon´es es Makoto Makoto Nakamu Nakamura. ra. Identifica la celda unidad de Bravais y la base.
1. 7. Descri Describir bir la estruct estructura ura de la figura adjunta adjunta como red m´ as una base at´ omica, omica, y dar su f´ormula ormula estequiom´etrica. etrica. ¿Es compatible con el compuesto BaTiO3 ?
1. 8. Demuestra que los unicos ´unicos ejes de rotaci´ on on compatib com patibles les con co n la simetr sim etr´´ıa de translaci´ tran slaci´ on on son los de orden 1, 2, 3, 4 y 6. 1. 9. Determina Determina las matrices matrices que determinan determinan las suguientes suguientes operacion op eraciones es de simetr´ simetr´ıa: a) Inversi´ Inversi´ on. on. b) Reflexi Reflexi´ o´n por un plano determinado por un vector unitario on unitario n perpendicular al plano. Particulariza la expresi´ on para un plano que contenga al eje Z on Z y que forme 45 grados con el eje X . Ayuda: Conviene descomponer el vector que se refleja en sus componentes paralela y perpendicular al plano. c) Rotaci´ Rotaci´ on o n de angulo a´ngulo φ alrededor de un eje que contenga el vector unitario n. Determina en concreto la matriz de rotaci´ on correspondiente a un angulo on a´ngulo φ alrededor del eje Z Z as´ı como la rotaci´ o n de 180 grados alrededor de un eje on contenido en el plano X Y y Y y que forma 45 grados con el eje X . 1. 10. Considera la red cristalina bidimensional de la figura:
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4
Estructuras Estructuras cristalinas. cristalinas.
b) Analiza los elementos de simetr´ simetr´ıa. Busca en las tablas t ablas cristalogr´ aficas aficas el grupo espacial (bidimensional) apropiado para describir la estructura. c) ¿ Cu´ales ales son las posiciones de Wyckoff necesarias para describir la estructura? Comprueba que tanto la multiplicidad como co mo la simetr´ simetr´ıa puntual de las posiciones de Wyckoff son las adecuadas. 1. 11. A la hora de describir la estructura de un s´ olido oli do crista cri stalino lino con simetr´ sime tr´ıa ıa rombo´ ro mbo´edrica edr ica los cristal´ografos ografos prefieren a menudo describir la estructura con una celda unidad hexagonal. Sin embargo, en los c´alculos alculos de primeros principios es m´ as as util u ´til utilizar la celda cel da rombo´ romb o´edrica edr ica,, m´as as peque˜ na, para ahorrar tiempo de c´ na, alculo. alculo. La relaci´ on on entre ambas celdas (siguiendo las convenciones de las tablas cristalogr´ aficas) aficas) aparece en la Fig. a).
a) Encuentra la relaci´ on on entre los par´ ametros de red de ambas celdas. ametros b) ¿ Cu´ antas antas veces es m´ as as grande la celda c elda unidad hexagonal que la rombo´edrica? edrica? c) Expres Expresaa las posi p osicion ciones es at´ omicas omicas de la celda rombo´ rombo´edrica edrica en funci´ on de las posiciones referidas a la celda hexagonal. d) El ´oxido oxido transparente conductor CuAlO conductor CuAlO2 (Fig. b) cristaliza en el grupo espacial R¯3m. Las posiciones de Wyckoff de la base cristalina, referidas a la celda hexagonal, son: C son: C u 3(a 3( a) (0, (0, 0, 0), Al 0), Al 3(b 3(b) (0, (0, 0, 1/2), O 2), O 6( 6(cc) (0, (0, 0, u). Calcula las posiciones de Wyckoff referidas a la celda rombo´ r ombo´edrica. edrica. ¿ Cu´ anto anto vale ahora la multiplicidad de las posiciones at´ omicas?. omicas?. 1. 12. En el problema anterior, acerca del compuesto CuAlO compuesto CuAlO2, calcula: a) las distancias y Cu a´ngulo Al − O − C u. Considera que C u − C u, Al − Al, Al, Cu C u − O , Al − O y C u − Al, Al, b) el angulo Al los par´ametros ametros de la celda hexagonal son a son a = = 2.858 ˚ A, c = c = 16.958 ˚ A y el par´ametro ametro interno que describe la posici´ on on del ox´ ox´ıgeno ıge no vale u = 0.1099.
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12
Propiedades t´ ermicas ermicas de los s´olidos olidos ´ ´ lidos 4. Propiedades Propiedades termi e rmica cas s de los los solidos o
4. 1. Considerar una cadena monoat´ omica lineal de constante de red a, cuya relaci´ omica on on de 2 2 dispersi´on on viene dada por ω por ω (k) = M C [1 [1 − cos(ka cos(ka)]. )]. Calcular: a) La velocidad del sonido en ese medio. b) La densida densidad d de estados estados de fon´ on. on. c) La densidad densidad de estados en la aproximaci´ aproximaci´ on de Debye. d) Compara Compararr gr´ aficamente las densidades de estados de los apartados b) y c). aficamente
4. 2. Considera un s´olido olido tridimensional is´otropo otropo a temperatura T temperatura T .. Utilizando la aproximaci´on on de Debye, calcula la probabilidad de encontrar un fon´ on on con frecuencia ω , sabiendo que viene dada por el producto de la probabilidad de ocupaci´ on < n > a esa frecuencia y el n´ umero de estados con esa frecuencia. ¿Cu´ umero al al ser´ a la frecuencia m´as as probable de los fonones? 4. 3. Calcula Calcula la energ´ energ´ıa de vibraci´ vibraci´ on o n por atomo a´tomo del punto cero (T (T → 0) de un cristal tridimensional en la aproximaci´ on de Debye. Estima dicho valor para el helio s´olido on olido (ΘD =24 K) y para el diamante (ΘD =2250 K) 4. 4. Determina la densidad de estados de un cristal tridimensional, uno bidimensional y otro unidimensional en la aproximaci´ on de Debye. Utilizando estos resultados, on encuentra encuentra la dependencia dependencia con la temperatura del calor espec´ espec´ıfico de los tres tipos de cristales c ristales en el l´ımite de d e muy bajas baj as temperaturas. temp eraturas. 4. 5. Conside Considerem remos os un s´ olido cristalino monoat´ olido omico omico con red c´ ubica ubica simple de constante constante de red a = 5 ˚ A. Supondremos que la velocidad de propagaci´ A. on on del sonido es is´otropa, otropa, igual a 7000 m/s a una temperatura de 300 K.
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13
Estado S´ olido. Curso 2010-2011. olido.
Debye y considerando que la fusi´ on se produce cuando la amplitud de vibraci´ on on on alcanza alcanza el 9 % de la distanci distanciaa a primeros primeros vecinos vecinos.. Calcula la temperatura de fusi´ on on del C u utilizando el criterio de Lindemann. Con˚. Su sidera que el C u adopta una estructura f cc con cc con par´ ametro ametro de red de 3.615 A temperatura de Debye es de 343 K 343 K .. La masa de un ´atomo atomo de C u es de 63.546 u 63.546 u.. 4. 9. La expansi´on on t´ermica ermica se puede comprender desde un punto punto de vista cl´ asico considerando la asimetr´ asimetr´ıa que introduce la parte par te anarm´ a narm´ onica del potencial en el desplazamiento de los atomos a´tomos alrededor de la posici´ on de equilibrio. Como ejemplo on cualitativo, considera que la energ´ energ´ıa potencial p otencial de un atomo ´ cuando se le desplaza una distancia x de su posici´on on de equilibrio viene dada por: U (x) = cx 2 − gx 3 − f x4 con c con c,, g y f positivos. Utiliza la estad esta d´ıstica de Maxwell-Boltzmann para calcular el valor medio del desplazamiento < x >. Sabiendo que el valor del coeficiente de dilataci´ on on t´ermica ermica del C u es de 1.7 × 5 1 10 K , estima el orden de magnitud del t´ermino ermino c´ ubico respecto del cuadr´ ubico atico atico para el C el C u a temperatura ambiente y a la temperatura de fusi´ on (1360 K (1360 K ). ). Utiliza Utiliza los datos necesarios del problema 4.8. −
−
4. 10. Conside Considera ra una cadena lineal lineal de atomos, a´tomos, de longitud L longitud L y par´ametro ametro de red a red a,, sometida a un potencial con un t´ermino ermino anarm´ onico: 1 U (x) = kx2 − gx 3 2 donde x = d = d − a, siendo d la distancia entre dos atomos a´tomos vecinos.
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14
Propiedades t´ ermicas ermicas de los s´olidos olidos
4. 12. a) Mues Muestra tra que que la la ecu ecuaci aci´ o´n de movimiento de un oscilador anarm´ on onico: onico: mx¨ + 2cx 2cx − 3gx 2 = 0 admite una soluci´ on on del tipo: ∞
x =
an einω
0t
n=1
con ω0 = 2c/m. c/m. Discute Discute el resultado resultado en t´erminos erminos creaci´ on, destrucci´ on on y dispersi´ disp ersi´on on de fonones. fono nes. b) En la gr´ gr´ afica se muestran las curvas de dispersi´ afica on de los fonones de la delafosita on CuAlO2 . Γ indica el centro de la zona de Brillouin. T , T , X y L son puntos de borde de zona, seg´ un se ha representado en la figura de la esquina superior un izquierda. Sugiere posibles decaimientos del modo E modo E g de centro de zona.
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19
Estado S´ olido. Curso 2010-2011.
6. Metales 6. 1. Considera la estad est ad´´ıstica de Fermi: f ( f (E, T ) T ) =
1 e
E −E F KT
+1
.
a) Prueba que la probabilidad de que un estado est ado con co n la energ´ energ´ıa de Fermi est´e ocupado es 1/2, independientemente de la temperatura. b) Determi Determina na la recta recta tangente tangente a f para una energ´ energ´ıa igual a la de Fermi. c) Prueba Prueba que la funci´ funci´ on f on f ,, a temperatura constante, tiene un punto de inflexi´ on para la energ´ıa ıa de Fermi. d) Determina Determina la energ´ energ´ıa para la cual la recta corta a f = 1 y f = = 0. Evalua f en estos puntos y compara el valor con los l´ımites f = 1 y f = 0. e) Demuestra que la probabilidad de ocupaci´ o n de los estados por debajo del on nivel de Fermi disminuye en la misma medida que aumenta la probabilidad de ocupaci´ on on por encima de esta energ´ energ´ıa. f ) Teniendo en cuenta los resultados anteriores, aproxima aproxima la distribucion de Fermi Fermi para T para T << T F F por tres tramos rectos de forma adecuada. g) Utilizando esta aproximaci´ on, on, estima la fracci´on o n de electrones de un gas de electrones libres con energ´ energ´ıa mayor mayor que la de Fermi. h) Calcul Calculaa de forma forma simpli simplificad ficadaa la contrib contribuci uci´ o´n del gas de electrones libres al on calor espec esp ec´´ıfico de los metales. metal es. 6. 2. La aleaci´ on on binaria CuZn binaria CuZn sufre sufre cambios estructurales a medida que se aumenta la concentraci´ o n de Zn. La estructura del cobre puro es f cc. on cc. Esta fase se mantiene hasta una cierta concentraci´ on de Zn, a partir de la cual aparece la fase bcc. on bcc. Para entender esta transformaci´ transformaci´ on conviene calcular el n´ umero umero medio n de electrones libres por atomo. a´tomo. A partir de la estructura electr´ onica onica del C u y el Z n se sabe que el C el C u aporta un electr´ on on libre, mientras que el Z el Z n aporta dos. Se encuentra que o a unos 600 C la C la fase bcc comienza bcc comienza a aparecer cuando n = 1.36, y cuando n ≥ 1.50 la fase bcc se bcc se desestabiliza. Para interpretar estos cambios, determina: a) El valor de n para que la esfera de Fermi toque la frontera de la primera zona de Brillouin de un cristal f cc. cc. b) El valor de n para el que la esfera de Fermi toque la frontera de la primera zona de Brillouin en un cristal bcc. bcc. Comenta tus resultados para explicar el cambio de fase de la aleaci´ on on teniendo en cuenta el modelo de electrones cuasilibres. 6. 3. Consid´ Con sid´erense ere nse tres tre s hipot´ hip ot´eticos eti cos s´ solidos o´lidos bidimensionales con concentraciones electr´ onioni−2 −2 −2 14 15 15 cas n1 = 1.30 × 10 cm , n2 = 1.20 × 10 cm y n3 = 2.49 × 10 cm . Su estructura cristalina es una red rectangular de par´ ametros a ametros a = = 0.625 nm y b y b = = a/ a/2. 2. Suponiendo que las superficies de Fermi son circulares:
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20
Metales
6. 4. Para estudiar las propiedades del cobre se le aplica una presion hidrost´ atica, de forma que el material se comprime. a) Encuentra Encuentra la variaci´ variaci´ on on de la energ´ energ´ıa de d e Fermi con el volumen del cristal. cr istal. b) Determina el m´odulo odulo de compresibilidad del gas de electrones de conducci´ on on del cobre cuando la temperatura tiende al cero absoluto. c) Sabiendo que el m´ odulo de compresibilidad del cobre, medido experimentalodulo mente, vale 134 GP 134 GP a, discute la importancia de la contribuci´ on on de los electrones electrones libres. Datos: peso
at´ omico omico 63.5 g/mol 63.5 g/mol,, densidad 8.9×103 kg/m3 .
6. 5. Un gas bidimensional de electrones es un gas de electrones libres de moverse en un plano, pero fuertemente confinados en la direcci´ on perpendicular perp endicular a ´el. el. Este tipo de comportamiento se encuentra habitualmente en transistores y pozos cu´ anticos. anticos. Considera un gas bidimensional de electrones que se encuentra confinado en un plano de espesor d por un potencial infinito en la superficie del plano y 0 en su interior. a) Calcul Calculaa los nivele niveless de energ energ´´ıa del gas de electro electrones nes incluy incluyendo endo el efecto efecto del confinamiento cu´ antico. antico. b) Dibuja un esquema esquema de la estructura de bandas resultante. resultante. Para ello considera ˚ y calcula d = 20 A calcula la energ´ energ´ıa de las tres primeras primeras subbandas en centro de zona. Determina adem´ as as la energ´ energ´ıa de la primera subbanda en borde de zona, teniendo en cuenta que la periodicidad de la red bidimensional est´ a descrita ˚. por una red cuadrada con par´ ametro ametro de red a red a = = 4 A c) Obt´ en en la densidad de estados del gas bidimensional bidimensional de electrones. electrones. d) Determi Determina na la densidad densidad de electro electrones nes necesaria necesaria para que la segund segundaa subbanda empieze a llenarse a T = 0. e) ¿Cu´ anto anto valdr´ a la energ´ energ´ıa media por electr´on on si el material tiene la densidad de electrones del apartado anterior? 6. 6. En ciertos materiales con dimensiones del orden del nan´ ometro, como nanotubos ometro, de carbono, hilos cu´anticos anticos o nanocolumnas semiconductoras, los electrones son libres de moverse en una direccion, pero se encuentran fuertemente confinados en la direccion radial. Podemos asimilar el comportamiento de estos electrones al de un gas de unidimensional de electrones libres de moverse a lo largo del eje z eje z y y cuya energ´ıa ıa viene dada por: por : h ¯ 2 k2 E n (k) = + εn , 2m donde n es un entero y εn representa repr esenta la energ´ıa ıa del n−´esim es imoo nivel ni vel.. a) En un modelo sencillo, en el que el gas de electrones est´ a confinado en una caja de paredes infinitas, los niveles de energ´ energ´ıa εn vienen dados por: h ¯ 2π2
2
2
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Estado S´ olido. Curso 2010-2011.
b) c) d) e)
21
a T = (temp eratura ura del helio l´ıquido), ıquido) , T =77 ogen o l´ıquido) y a T = 4.2 K K (temperat T =77 K (nitr´ogeno T =300 T =300 K (temperatura ambiente). ˚. Determina la energ´ energ´ıa de los dos primeros niveles de energ´ energ´ıa si L = 50 A Calcul Calculaa la densid densidad ad de estados estados por unidad unidad de energ energ´´ıa del gas de electro electrones nes unidimensional. Se estudia estudia un nanotubo con una concentraci´ concentraci´ on electr´ onica onica de 107 cm−1. ¿ D´onde onde est´a situado el nivel de Fermi a 4.2 K ? ¿Cu´al al ser´ a la contribuci´ on on electr´ onica onica al calor ca lor espec´ espec´ıfico de una nanoestructura nanoestruct ura en las condiciones del apartado anterior?
6. 7. El aluminio es un metal trivalente cuyos electrones de valencia pueden describirse mediante el modelo de gas de electrones libres. La densidad del aluminio es de 2700 kg/m3 , su peso at´ omico 27 y su temperatura de Debye 428 K. Encuentra: omico a) La densidad de electrones libres. b) La temperatura de Fermi. Fermi. c) El cocient cocientee entre entre la capacid capacidad ad calor calor´´ıfica ıfica fon´ onica y la capacidad capacidad calor´ calor´ıfica electr´onica onica a T = T = 1 K, 4.2 K, 77 K, 300 K y 933 K (temperatura de fusi´ on). on). 6. 8. Determina Determina en un modelo de electrones libres libres el recorrido recorrido libre medio medio de los electrones electrones de conducci´ on on en una muestra de C u cuyo tiempo de relajaci´ on o n es τ = 10−9 s. El ˚. par´ametro ametro de la celda (centrada en caras) es a = 3.61 A 6. 9. Se considera una muestra de C de C u que se ha impurificado de manera controlada con un 0.2% de atomos a´tomos de Ag. Ag . A una temperatura lo suficientemente baja para que la contribuci´ on de las vibraciones sea despreciable, la resistividad de la muestra es on 2 × 10−10 Ω · m. Suponiendo que los electrones se comportan como libres, calcula la secci´ on on eficaz de difusi´on on de los electrones por las impurezas de Ag. Ag . La energ´ıa ıa de Fermi del C u es de E F F = 7 eV 6. 10. Se aplica un campo magn´ etico etico B sobre una muestra met´ alica alica cuyo tiempo de re−10 lajaci´ on o n es τ = 3 × 10 s. Determina el valor del campo para que el electr´ on on describa una orbita o´rbita completa, en la aproximaci´ on de los electrones libres. on 6. 11. Las propiedades de transporte de una muestra se ven afectadas por su tama˜ no cuando ´este este es lo suficientemente peque˜ no para ser comparable al recorrido libre no medio de los electrones. Considera una muestra met´ alica de longitud L longitud L y y de secci´ on on transversal d2 (con d < < L). L). La densidad de electrones de conducci´ on o n es n y el camino libre medio, debido a las impurezas, λ. Utilizando el mode de Drude, realiza un c´alculo alculo aproximado de la dependencia de las siguientes magnitudes con la dimensi´on on transversal de la muestra: a) El camino libre medio electr´ onico onico l(d) . b) La resistivida re sistividad d el´ectrica ectri ca ρ(d). c) La resistenc re sistencia ia el´ectrica ectri ca R(d).
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22
Semiconductores
7. Semiconductores 7. 1. La estructura del Si es del tipo diamante, con par´ ametro ametro de red a red a = = 5.43 ˚ A. a) Calcula la densidad de atomos a´tomos de Si por c por cm m3. b) Se dopa un cristal de Si con atomos a´tomos de P P con una concentraci´ on o n de N d = 1016 cm3 . Suponiendo que todos los atomos a´tomos de P substituyen substi tuyen a ´atomos atomo s de Si, Si , ¿ qu´e propo pro porci rci´ on o´n de atomos a´tomos de S de Sii han sido substituidos? ¿ Cu´al al es la distancia
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Estado S´ olido. Curso 2010-2011.
23
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24
Semiconductores
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