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4.- PROPIEDADESELÉCTRICASDELOSSÓLIDOS
FÍSICADELESTADOSÓLIDOII
4.Propiedadeseléctricasdelossólidos
• Conductividad eléctrica.• Metales, semiconductores y aislantes.• Semiconductores intrínsecos y extrínsecos.• Dieléctricos.• Ferroelectricidad.• Piezoelectricidad.
PROPIEDADESELÉCTRICAS
ConductividadEléctricaOBJETIVO:Estudiarlaconductividadeléctricaenlosmaterialesyestimarsuutilidad
comomaterialeselectrónicos.
Ordendemagnituddelaconductividad:
• Superconductores:Resistenciacero.
•Metales:conductividadesmuyaltas.
• Semiconductores:“Conducenenunamplio
rangodevalores”.
•Aislantesydieléctricos:Malosconductores.
PROPIEDADESELÉCTRICAS
ConductividadEléctrica
©2003 Brooks/Cole, a division of Thomson Learning, Inc. Thomson Learning™ is a trademark used herein under license.
PROPIEDADESELÉCTRICAS
ConductividadEléctrica
Representacióndelasbandasdeenergíadediferentesmateriales:a. Aislanteb. Semiconductorintrínseco.c. Semiconductorextrínsecotipond. Semiconductorextrínsecotipope. Metalf. Semimetal
PROPIEDADESELÉCTRICAS
ConductividadEléctricaConceptosbásicosparaentenderlaleydeOhmylaconductividadeléctrica.
ü Densidad de corriente – corriente que fluye por unidad de área.
ü Velocidad de arrastre – velocidad a la que los portadores de carga se mueven porun material bajo los efectos de un campo eléctrico o magnético aplicado.
ü Movilidad – facilidad con que los portadores de carga se mueven por un material.
ü Constante dieléctrica – relación entre la permitividad de un material y la delvacío. Describe la habilidad relativa de un material para polarizarse y almacenarcarga.
LEYDEOHM: 𝐽 = 𝜎𝐸
𝐽 = 𝑛𝑞𝑣* 𝑣*=velocidaddearrastre
PROPIEDADESELÉCTRICAS
ConductividadEléctrica
PROPIEDADESELÉCTRICAS
ConductividadEléctricaRecordamosdeltemaanteriorquelaconductividadpuedeescribirseentérminosdelamovilidaddelosportadoresdecarga
𝜎 = 𝑛𝑒𝜇-𝑑𝑜𝑛𝑑𝑒𝑙𝑎𝑚𝑜𝑣𝑖𝑙𝑖𝑑𝑎𝑑𝑣𝑖𝑒𝑛𝑒𝑑𝑎𝑑𝑎𝑝𝑜𝑟𝜇- = 𝑒𝜏𝑚-∗
Y𝜏 eseltiempoentresucesivosprocesosde“scattering”,llamadotambiéntiempoderelajación.
𝜎 = 𝑛𝑒𝜇- = 𝑛𝑒8𝜏𝑚-∗ ⇒ 𝜌 =
1𝜎 =
𝑚-∗
𝑛𝑒8𝜏
𝑅𝑒𝑔𝑙𝑎𝑀𝑎𝑡ℎ𝑖𝑠𝑠𝑒𝑛
1𝜏 =
1𝜏BC
+1𝜏E-F
+1
𝜏GHI
J ⇒ 𝜌 = 𝑚-∗
𝑛𝑒8𝜏 =𝑚-∗
𝑛𝑒8𝜏BC+
𝑚-∗
𝑛𝑒8𝜏E-F+
𝑚-∗
𝑛𝑒8𝜏GHI
PROPIEDADESELÉCTRICAS
ConductividadEléctrica𝜌 =
𝑚-∗
𝑛𝑒8𝜏 =𝑚-∗
𝑛𝑒8𝜏BC+
𝑚-∗
𝑛𝑒8𝜏E-F+
𝑚-∗
𝑛𝑒8𝜏GHI= 𝜌KLMLM. + 𝜌O-F-P. + 𝜌QHIRS.
𝜌 = 𝜌QE-*T + 𝜌U-VGER*T W𝜌QE-*T = 𝐹𝑜𝑛𝑜𝑛𝑒𝑠
𝜌U-VGER*T = 𝐶𝑜𝑛𝑡𝑟𝑖𝑏. 𝑒𝑥𝑡𝑟í𝑛𝑠𝑒𝑐𝑎𝑠
PROPIEDADESELÉCTRICAS
ConductividadEléctrica
Movimientodeunelectrónatravésde:a. Uncristalperfectob. Uncristalsometidoaalta
temperatura.c. Uncristalquecontienedefectosa
nivelatómico.
Losprocesosde“Scattering”deelectronesreducenlamovilidadyportantolaconductividad
PROPIEDADESELÉCTRICAS
ConductividadEléctrica(METALES)
Controldelaconductividadenmetalesü Recorrido libre medio – distancia promedio que recorren los electrones de carga sin
ser dispersados por los átomos.ü Temperatura – al aumentar la temperatura de un metal, los átomos vibran, fonones
de la red.ü Defectos a nivel atómico – las imperfecciones de la red dispersan electrones, lo que
reduce la movilidad y conductividad de un metal.ü Regla de Matthiessen – la resistividad de un material metálico es la suma de una
resistividad que tiene en cuenta los efectos de la temperatura (ρT), y una resistividadindependiente de la temperatura a la que contribuyen los defectos a nivel atómico,incluidas las impurezas (ρd).
ü Efectos de procesado y reforzado
PROPIEDADESELÉCTRICAS
(Figuradeltema3)
El efecto de la temperatura en laresistividad eléctrica de un metal sindefectos, a alta T.
La pendiente de la curva se llamaCOEFICIENTE DE LA RESISTIVIDAD
ConductividadEléctrica(METALES)
PROPIEDADESELÉCTRICAS
ConductividadEléctrica(METALES)
PROPIEDADESELÉCTRICAS
(a) Efecto de endurecimiento del material en frío sobre la conductividad eléctrica delcobre y (b) efecto de la adición de ciertas impurezas seleccionadas en laconductividad del cobre.
CONTRIBUCIONESEXTRÍNSECASALARESISTIVIDAD
ConductividadEléctrica(METALES)
PROPIEDADESELÉCTRICAS
ConductividadEléctrica(METALES)
PROPIEDADESELÉCTRICAS
ConductividadEléctrica(SUPERCONDUCTORES)
La resistividad eléctrica de un superconductor es cero pordebajo de una cierta temperatura crítica. Dichatemperatura no es universal y depende del material.
Variacióndelaresistenciadeunmetalnormal,comparadoconunsuperconductorcuandolatemperaturaseacercaa0K.
PROPIEDADESELÉCTRICAS
ConductividadEléctrica(SUPERCONDUCTORES)
Temperaturasdetransiciónsuperconductoraparaciertosmateriales
La superconductividad es un mecanismocuántico, uno de los pocos que se manifiestan enel mundo “macroscópico”.
La temperatura a la cual el material pasa a sersuperconductor se denomina Temperatura detransición superconductora (Tc).
La mayoría de los metales son superconductorespor debajo de los 10K. El actual record detemperatura Tc está en los 138 K.
PROPIEDADESELÉCTRICAS
ConductividadEléctrica(SEMICONDUCTORES)
Descripción simplificada de la estructurade bandas planas para un semiconductorintrínseco
Conductividad eléctrica frente a la temperaturade un semiconductor intrínseco y de un metal.Notese el corte en el eje y.
PROPIEDADESELÉCTRICAS
ConductividadEléctrica(SEMICONDUCTORES)
ü Semiconductor intrínseco - semiconductor en el que sus propiedades secontrolan por sus elementos constituyentes no por impurezas.
ü Semiconductor extrínseco - semiconductor al que se han añadido dopantes, yéstos controlan el número y tipo de portadores.
ü Dopado – incorporación deliberada de pequeñas cantidades controladas de otroselementos para modificar el nº de portadores en un semiconductor.
ü Termistor – dispositivo de semiconductor que es sensible a los cambios detemperatura.
ü Recombinación radiativa - Recombinación de huecos y electrones que conduce ala emisión de luz.
Conceptosbásicosparaentenderlossemiconductores
PROPIEDADESELÉCTRICAS
ConductividadEléctrica(SEMICONDUCTORES)
PROPIEDADESELÉCTRICAS
ConductividadEléctrica(SEMICONDUCTORES)SEMICONDUCTORINTRÍNSECO
Distribución de electrones y huecosen las bandas de valencia yconducción:
(a) a cero absoluto(b) a una temperatura elevada.
Entodomomentohadecumplirse:
𝒏𝟎 = 𝒑𝟎⇒ 𝒏𝟎𝒑𝟎 = 𝒏𝒊𝟐 = 𝒑𝒊𝟐
n0 eselnúmerodeelectronesenunsemiconductorintrínseco
p0 eselnúmerodehuecosenunsemiconductorintrínseco
PROPIEDADESELÉCTRICAS
ConductividadEléctrica(SEMICONDUCTORES)
Cuando aplicamos un campo eléctrico a un semiconductor, los electrones semueven en la banda de conducción mientras que los huecos se mueven en labanda de valencia en dirección opuesta.
SEMICONDUCTORINTRÍNSECO
PROPIEDADESELÉCTRICAS
ConductividadEléctrica(SEMICONDUCTORES)SEMICONDUCTORINTRÍNSECO
are called III–V semiconductors because they arecompounds of elements in groups III and V (nowgroups 13 and 15) of the periodic table. The lastcompounds listed are called II–VI semiconductorsbecause they are compounds of elements in groupsII and VI (now groups 12 and 16) of the periodictable.The decrease in band gap with size of atom is
simply a consequence of the fact that the outerorbitals of larger atoms overlap more and give riseto wider bands. As a consequence, the gaps arenarrower.
13.2.2 Carrier concentrations in intrinsicsemiconductors
To determine more accurate values for the numberof holes and electrons present in an intrinsic semi-conductor it is appropriate to use Fermi–Diracstatistics and the density of states at the bottom ofthe conduction band (see Section S4.8). To a goodapproximation, it is found that the number ofelectrons in the conduction band per unit volume,
ni, which is equal to the number of intrinsic holes inthe valence band, pi, at an absolute temperature, T ,is:
ni¼ pi¼ 4:826"1021m#em
#h
m2e
! "3=4
T3=2 exp $ Eg
2kT
! "
ð13:2Þ
where m#e is the effective mass of the electron, m#h isthe effective mass of a hole, and Eg is the band gap.In an intrinsic semiconductor
n ¼ p ¼ ni ¼ pi
Equation (13.2) shows that, at a given temperature
n p ¼ constant
The temperature dependence of the equilibriumconstant is given by:
np ¼ 2:33" 1043m#e m
#h
m2e
! "3=2
T3 exp $ Eg
k T
! "
Assuming that the effective mass of electrons andholes is independent of temperature, we obtain
np / T3 exp $ Eg
kT
! "
This equation applies to doped semiconductors aswell as to intrinsic semiconductors, a finding ofconsiderable practical importance. To a goodapproximation, the Fermi energy lies at the centreof the band gap (Section S4.8):
EF ¼1
2Eg þ
3
4kT ln
m#hm#e
! "
( 1
2Eg
Writing the total conductivity, !(total), as
!ðtotalÞ ¼ ne"e þ pe"h
we obtain
!ðtotalÞ ¼ 773:1m#em
#h
m2e
! "3=4
T3=2ð"e þ "hÞ
" exp $ Eg
2kT
! "
ð13:3Þ
Table 13.1 Approximate values for the band gap ofsome semiconductors
Symbol orCompound formula Band gap/eV
Elements:Diamond C 5.47Silicon Si 1.12Germanium Ge 0.66Grey tin Sn 0.08
III–V semiconductors:Gallium nitride GaN 3.36Gallium phosphide GaP 2.26Gallium arsenide GaAs 1.42
II–VI semiconductors:Cadmium sulphide CdS 2.42Cadmium selenide CdSe 1.70Cadmium telluride CdTe 1.56
Note: the band gap is normally given in electron volts in mostcompilations; 1 eV is equal to 1:60219" 10$19 J.
398 ELECTRONIC CONDUCTIVITY IN SOLIDS
by n; the number of holes is given by p; and themobility is given by !.It is possible to determine the number of electrons
excited into the conduction band by thermal energyin an intrinsic semiconductor using Fermi–Diracstatistics (see Section 2.3.7 and Section S4.12). Itis found that:
n / exp ! Eg
2kT
! "
where n is the number of electrons in the conductionband, Eg is the band gap between the top of thevalence band and the bottom of the conductionband, k is the Boltzmann constant, and T is theabsolute temperature. As the number of holes isequal to the number of electrons in an intrinsicsemiconductor,
p / exp ! Eg
2kT
! "
The total conductivity, ", which is proportional tothe number of electrons and holes, can be expressedin the form:
" ¼ "0 exp !Eg
2kT
! "
where "0 is a constant, Eg is the band gap betweenthe top of the valence band and the bottom of theconduction band, k is the Boltzmann constant, and Tis the absolute temperature. Taking logarithms ofeach side of this equation, we obtain
ln " ¼ ln "0 !Eg
2kT
The gradient of a plot of conductivity versus 1/Twill therefore yield a value for the (thermal) bandgap (Figure 13.8).An alternative method of obtaining the magnitude
of the band gap is via the absorption of radiation. Ina semiconductor, almost all of the energy levelsbelow the conduction band are occupied. Thismeans that low-energy radiation directed at a crystalwill not interact with the electrons, and the crystal
will be transparent. As the energy graduallyincreases, eventually the energy will just be suffi-cient to promote an electron from the top of thevalence band to the bottom of the conduction band.The radiation will now be absorbed and the crystalwill become opaque. The (optical) band gap can beequated to the energy at which this change occurs.Thus,
Eg ¼ h#g
where h#g is the energy of the photon required topromote an electron from the valence band andcreate a hole in its place.
Note that the absorption of radiation is morecomplex than this simple model suggests, and amore complete description will be given in Section14.1.3. Moreover, the measured value of the opticalband gap is usually slightly different from thethermal band gap. This is simply a reflection ofthe fact that the bands in a semiconductor are notflat as drawn in Figure 13.7 but have a morecomplex curved shape.
Approximate values for the band gap in somesemiconductors is given in Table 13.1. The bandgap decreases as the atom size increases (i.e. as onemoves down the relevant group in the periodictable). Thus, within the group of elements listed,diamond is best regarded as an insulator, whereasgrey tin is regarded as a metal. The second group
Figure 13.8 The variation of resistivity versus recipro-cal temperature for an intrinsic semiconductor
SEMICONDUCTORS 397
by n; the number of holes is given by p; and themobility is given by !.It is possible to determine the number of electrons
excited into the conduction band by thermal energyin an intrinsic semiconductor using Fermi–Diracstatistics (see Section 2.3.7 and Section S4.12). Itis found that:
n / exp ! Eg
2kT
! "
where n is the number of electrons in the conductionband, Eg is the band gap between the top of thevalence band and the bottom of the conductionband, k is the Boltzmann constant, and T is theabsolute temperature. As the number of holes isequal to the number of electrons in an intrinsicsemiconductor,
p / exp ! Eg
2kT
! "
The total conductivity, ", which is proportional tothe number of electrons and holes, can be expressedin the form:
" ¼ "0 exp !Eg
2kT
! "
where "0 is a constant, Eg is the band gap betweenthe top of the valence band and the bottom of theconduction band, k is the Boltzmann constant, and Tis the absolute temperature. Taking logarithms ofeach side of this equation, we obtain
ln " ¼ ln "0 !Eg
2kT
The gradient of a plot of conductivity versus 1/Twill therefore yield a value for the (thermal) bandgap (Figure 13.8).An alternative method of obtaining the magnitude
of the band gap is via the absorption of radiation. Ina semiconductor, almost all of the energy levelsbelow the conduction band are occupied. Thismeans that low-energy radiation directed at a crystalwill not interact with the electrons, and the crystal
will be transparent. As the energy graduallyincreases, eventually the energy will just be suffi-cient to promote an electron from the top of thevalence band to the bottom of the conduction band.The radiation will now be absorbed and the crystalwill become opaque. The (optical) band gap can beequated to the energy at which this change occurs.Thus,
Eg ¼ h#g
where h#g is the energy of the photon required topromote an electron from the valence band andcreate a hole in its place.
Note that the absorption of radiation is morecomplex than this simple model suggests, and amore complete description will be given in Section14.1.3. Moreover, the measured value of the opticalband gap is usually slightly different from thethermal band gap. This is simply a reflection ofthe fact that the bands in a semiconductor are notflat as drawn in Figure 13.7 but have a morecomplex curved shape.
Approximate values for the band gap in somesemiconductors is given in Table 13.1. The bandgap decreases as the atom size increases (i.e. as onemoves down the relevant group in the periodictable). Thus, within the group of elements listed,diamond is best regarded as an insulator, whereasgrey tin is regarded as a metal. The second group
Figure 13.8 The variation of resistivity versus recipro-cal temperature for an intrinsic semiconductor
SEMICONDUCTORS 397
by n; the number of holes is given by p; and themobility is given by !.It is possible to determine the number of electrons
excited into the conduction band by thermal energyin an intrinsic semiconductor using Fermi–Diracstatistics (see Section 2.3.7 and Section S4.12). Itis found that:
n / exp ! Eg
2kT
! "
where n is the number of electrons in the conductionband, Eg is the band gap between the top of thevalence band and the bottom of the conductionband, k is the Boltzmann constant, and T is theabsolute temperature. As the number of holes isequal to the number of electrons in an intrinsicsemiconductor,
p / exp ! Eg
2kT
! "
The total conductivity, ", which is proportional tothe number of electrons and holes, can be expressedin the form:
" ¼ "0 exp !Eg
2kT
! "
where "0 is a constant, Eg is the band gap betweenthe top of the valence band and the bottom of theconduction band, k is the Boltzmann constant, and Tis the absolute temperature. Taking logarithms ofeach side of this equation, we obtain
ln " ¼ ln "0 !Eg
2kT
The gradient of a plot of conductivity versus 1/Twill therefore yield a value for the (thermal) bandgap (Figure 13.8).An alternative method of obtaining the magnitude
of the band gap is via the absorption of radiation. Ina semiconductor, almost all of the energy levelsbelow the conduction band are occupied. Thismeans that low-energy radiation directed at a crystalwill not interact with the electrons, and the crystal
will be transparent. As the energy graduallyincreases, eventually the energy will just be suffi-cient to promote an electron from the top of thevalence band to the bottom of the conduction band.The radiation will now be absorbed and the crystalwill become opaque. The (optical) band gap can beequated to the energy at which this change occurs.Thus,
Eg ¼ h#g
where h#g is the energy of the photon required topromote an electron from the valence band andcreate a hole in its place.
Note that the absorption of radiation is morecomplex than this simple model suggests, and amore complete description will be given in Section14.1.3. Moreover, the measured value of the opticalband gap is usually slightly different from thethermal band gap. This is simply a reflection ofthe fact that the bands in a semiconductor are notflat as drawn in Figure 13.7 but have a morecomplex curved shape.
Approximate values for the band gap in somesemiconductors is given in Table 13.1. The bandgap decreases as the atom size increases (i.e. as onemoves down the relevant group in the periodictable). Thus, within the group of elements listed,diamond is best regarded as an insulator, whereasgrey tin is regarded as a metal. The second group
Figure 13.8 The variation of resistivity versus recipro-cal temperature for an intrinsic semiconductor
SEMICONDUCTORS 397
by n; the number of holes is given by p; and themobility is given by !.It is possible to determine the number of electrons
excited into the conduction band by thermal energyin an intrinsic semiconductor using Fermi–Diracstatistics (see Section 2.3.7 and Section S4.12). Itis found that:
n / exp ! Eg
2kT
! "
where n is the number of electrons in the conductionband, Eg is the band gap between the top of thevalence band and the bottom of the conductionband, k is the Boltzmann constant, and T is theabsolute temperature. As the number of holes isequal to the number of electrons in an intrinsicsemiconductor,
p / exp ! Eg
2kT
! "
The total conductivity, ", which is proportional tothe number of electrons and holes, can be expressedin the form:
" ¼ "0 exp !Eg
2kT
! "
where "0 is a constant, Eg is the band gap betweenthe top of the valence band and the bottom of theconduction band, k is the Boltzmann constant, and Tis the absolute temperature. Taking logarithms ofeach side of this equation, we obtain
ln " ¼ ln "0 !Eg
2kT
The gradient of a plot of conductivity versus 1/Twill therefore yield a value for the (thermal) bandgap (Figure 13.8).An alternative method of obtaining the magnitude
of the band gap is via the absorption of radiation. Ina semiconductor, almost all of the energy levelsbelow the conduction band are occupied. Thismeans that low-energy radiation directed at a crystalwill not interact with the electrons, and the crystal
will be transparent. As the energy graduallyincreases, eventually the energy will just be suffi-cient to promote an electron from the top of thevalence band to the bottom of the conduction band.The radiation will now be absorbed and the crystalwill become opaque. The (optical) band gap can beequated to the energy at which this change occurs.Thus,
Eg ¼ h#g
where h#g is the energy of the photon required topromote an electron from the valence band andcreate a hole in its place.
Note that the absorption of radiation is morecomplex than this simple model suggests, and amore complete description will be given in Section14.1.3. Moreover, the measured value of the opticalband gap is usually slightly different from thethermal band gap. This is simply a reflection ofthe fact that the bands in a semiconductor are notflat as drawn in Figure 13.7 but have a morecomplex curved shape.
Approximate values for the band gap in somesemiconductors is given in Table 13.1. The bandgap decreases as the atom size increases (i.e. as onemoves down the relevant group in the periodictable). Thus, within the group of elements listed,diamond is best regarded as an insulator, whereasgrey tin is regarded as a metal. The second group
Figure 13.8 The variation of resistivity versus recipro-cal temperature for an intrinsic semiconductor
SEMICONDUCTORS 397
Sepuededemostrarque:
Donde𝑚-∗ eslamasaefectivadeloselectrones,𝑚C
∗ lamasaefectivadeloshuecos,𝑚-8 lamasadelelectrónalcuadradoyklaconstantedeBoltzman.Portanto
tenemosque,engeneral:
Dadoquelaconductividadelproporcionalalaconcentracióndeportadores….
PROPIEDADESELÉCTRICAS
ConductividadEléctrica(SEMICONDUCTORES)
Comportamientodelaconductividadenunsemiconductorintrínseco.
Larepresentacióndelaconductividadfrentealinversodelatemperaturaeslinealylapendienteestárelacionadaconelintervalodeenergíaprohibida(Eg)delsemiconductor.
SEMICONDUCTORINTRÍNSECO
PROPIEDADESELÉCTRICAS
ConductividadEléctrica(SEMICONDUCTORES)SEMICONDUCTORINTRÍNSECO
are called III–V semiconductors because they arecompounds of elements in groups III and V (nowgroups 13 and 15) of the periodic table. The lastcompounds listed are called II–VI semiconductorsbecause they are compounds of elements in groupsII and VI (now groups 12 and 16) of the periodictable.The decrease in band gap with size of atom is
simply a consequence of the fact that the outerorbitals of larger atoms overlap more and give riseto wider bands. As a consequence, the gaps arenarrower.
13.2.2 Carrier concentrations in intrinsicsemiconductors
To determine more accurate values for the numberof holes and electrons present in an intrinsic semi-conductor it is appropriate to use Fermi–Diracstatistics and the density of states at the bottom ofthe conduction band (see Section S4.8). To a goodapproximation, it is found that the number ofelectrons in the conduction band per unit volume,
ni, which is equal to the number of intrinsic holes inthe valence band, pi, at an absolute temperature, T ,is:
ni¼ pi¼ 4:826"1021m#em
#h
m2e
! "3=4
T3=2 exp $ Eg
2kT
! "
ð13:2Þ
where m#e is the effective mass of the electron, m#h isthe effective mass of a hole, and Eg is the band gap.In an intrinsic semiconductor
n ¼ p ¼ ni ¼ pi
Equation (13.2) shows that, at a given temperature
n p ¼ constant
The temperature dependence of the equilibriumconstant is given by:
np ¼ 2:33" 1043m#e m
#h
m2e
! "3=2
T3 exp $ Eg
k T
! "
Assuming that the effective mass of electrons andholes is independent of temperature, we obtain
np / T3 exp $ Eg
kT
! "
This equation applies to doped semiconductors aswell as to intrinsic semiconductors, a finding ofconsiderable practical importance. To a goodapproximation, the Fermi energy lies at the centreof the band gap (Section S4.8):
EF ¼1
2Eg þ
3
4kT ln
m#hm#e
! "
( 1
2Eg
Writing the total conductivity, !(total), as
!ðtotalÞ ¼ ne"e þ pe"h
we obtain
!ðtotalÞ ¼ 773:1m#em
#h
m2e
! "3=4
T3=2ð"e þ "hÞ
" exp $ Eg
2kT
! "
ð13:3Þ
Table 13.1 Approximate values for the band gap ofsome semiconductors
Symbol orCompound formula Band gap/eV
Elements:Diamond C 5.47Silicon Si 1.12Germanium Ge 0.66Grey tin Sn 0.08
III–V semiconductors:Gallium nitride GaN 3.36Gallium phosphide GaP 2.26Gallium arsenide GaAs 1.42
II–VI semiconductors:Cadmium sulphide CdS 2.42Cadmium selenide CdSe 1.70Cadmium telluride CdTe 1.56
Note: the band gap is normally given in electron volts in mostcompilations; 1 eV is equal to 1:60219" 10$19 J.
398 ELECTRONIC CONDUCTIVITY IN SOLIDS
Lasecuacionesanterioresdemuestranque𝑛 · 𝑝 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡𝑒 aunatemperaturadada.
are called III–V semiconductors because they arecompounds of elements in groups III and V (nowgroups 13 and 15) of the periodic table. The lastcompounds listed are called II–VI semiconductorsbecause they are compounds of elements in groupsII and VI (now groups 12 and 16) of the periodictable.The decrease in band gap with size of atom is
simply a consequence of the fact that the outerorbitals of larger atoms overlap more and give riseto wider bands. As a consequence, the gaps arenarrower.
13.2.2 Carrier concentrations in intrinsicsemiconductors
To determine more accurate values for the numberof holes and electrons present in an intrinsic semi-conductor it is appropriate to use Fermi–Diracstatistics and the density of states at the bottom ofthe conduction band (see Section S4.8). To a goodapproximation, it is found that the number ofelectrons in the conduction band per unit volume,
ni, which is equal to the number of intrinsic holes inthe valence band, pi, at an absolute temperature, T ,is:
ni¼ pi¼ 4:826"1021m#em
#h
m2e
! "3=4
T3=2 exp $ Eg
2kT
! "
ð13:2Þ
where m#e is the effective mass of the electron, m#h isthe effective mass of a hole, and Eg is the band gap.In an intrinsic semiconductor
n ¼ p ¼ ni ¼ pi
Equation (13.2) shows that, at a given temperature
n p ¼ constant
The temperature dependence of the equilibriumconstant is given by:
np ¼ 2:33" 1043m#e m
#h
m2e
! "3=2
T3 exp $ Eg
k T
! "
Assuming that the effective mass of electrons andholes is independent of temperature, we obtain
np / T3 exp $ Eg
kT
! "
This equation applies to doped semiconductors aswell as to intrinsic semiconductors, a finding ofconsiderable practical importance. To a goodapproximation, the Fermi energy lies at the centreof the band gap (Section S4.8):
EF ¼1
2Eg þ
3
4kT ln
m#hm#e
! "
( 1
2Eg
Writing the total conductivity, !(total), as
!ðtotalÞ ¼ ne"e þ pe"h
we obtain
!ðtotalÞ ¼ 773:1m#em
#h
m2e
! "3=4
T3=2ð"e þ "hÞ
" exp $ Eg
2kT
! "
ð13:3Þ
Table 13.1 Approximate values for the band gap ofsome semiconductors
Symbol orCompound formula Band gap/eV
Elements:Diamond C 5.47Silicon Si 1.12Germanium Ge 0.66Grey tin Sn 0.08
III–V semiconductors:Gallium nitride GaN 3.36Gallium phosphide GaP 2.26Gallium arsenide GaAs 1.42
II–VI semiconductors:Cadmium sulphide CdS 2.42Cadmium selenide CdSe 1.70Cadmium telluride CdTe 1.56
Note: the band gap is normally given in electron volts in mostcompilations; 1 eV is equal to 1:60219" 10$19 J.
398 ELECTRONIC CONDUCTIVITY IN SOLIDS
Enunsemiconductorintrínsecotenemosque
are called III–V semiconductors because they arecompounds of elements in groups III and V (nowgroups 13 and 15) of the periodic table. The lastcompounds listed are called II–VI semiconductorsbecause they are compounds of elements in groupsII and VI (now groups 12 and 16) of the periodictable.The decrease in band gap with size of atom is
simply a consequence of the fact that the outerorbitals of larger atoms overlap more and give riseto wider bands. As a consequence, the gaps arenarrower.
13.2.2 Carrier concentrations in intrinsicsemiconductors
To determine more accurate values for the numberof holes and electrons present in an intrinsic semi-conductor it is appropriate to use Fermi–Diracstatistics and the density of states at the bottom ofthe conduction band (see Section S4.8). To a goodapproximation, it is found that the number ofelectrons in the conduction band per unit volume,
ni, which is equal to the number of intrinsic holes inthe valence band, pi, at an absolute temperature, T ,is:
ni¼ pi¼ 4:826"1021m#em
#h
m2e
! "3=4
T3=2 exp $ Eg
2kT
! "
ð13:2Þ
where m#e is the effective mass of the electron, m#h isthe effective mass of a hole, and Eg is the band gap.In an intrinsic semiconductor
n ¼ p ¼ ni ¼ pi
Equation (13.2) shows that, at a given temperature
n p ¼ constant
The temperature dependence of the equilibriumconstant is given by:
np ¼ 2:33" 1043m#e m
#h
m2e
! "3=2
T3 exp $ Eg
k T
! "
Assuming that the effective mass of electrons andholes is independent of temperature, we obtain
np / T3 exp $ Eg
kT
! "
This equation applies to doped semiconductors aswell as to intrinsic semiconductors, a finding ofconsiderable practical importance. To a goodapproximation, the Fermi energy lies at the centreof the band gap (Section S4.8):
EF ¼1
2Eg þ
3
4kT ln
m#hm#e
! "
( 1
2Eg
Writing the total conductivity, !(total), as
!ðtotalÞ ¼ ne"e þ pe"h
we obtain
!ðtotalÞ ¼ 773:1m#em
#h
m2e
! "3=4
T3=2ð"e þ "hÞ
" exp $ Eg
2kT
! "
ð13:3Þ
Table 13.1 Approximate values for the band gap ofsome semiconductors
Symbol orCompound formula Band gap/eV
Elements:Diamond C 5.47Silicon Si 1.12Germanium Ge 0.66Grey tin Sn 0.08
III–V semiconductors:Gallium nitride GaN 3.36Gallium phosphide GaP 2.26Gallium arsenide GaAs 1.42
II–VI semiconductors:Cadmium sulphide CdS 2.42Cadmium selenide CdSe 1.70Cadmium telluride CdTe 1.56
Note: the band gap is normally given in electron volts in mostcompilations; 1 eV is equal to 1:60219" 10$19 J.
398 ELECTRONIC CONDUCTIVITY IN SOLIDS
are called III–V semiconductors because they arecompounds of elements in groups III and V (nowgroups 13 and 15) of the periodic table. The lastcompounds listed are called II–VI semiconductorsbecause they are compounds of elements in groupsII and VI (now groups 12 and 16) of the periodictable.The decrease in band gap with size of atom is
simply a consequence of the fact that the outerorbitals of larger atoms overlap more and give riseto wider bands. As a consequence, the gaps arenarrower.
13.2.2 Carrier concentrations in intrinsicsemiconductors
To determine more accurate values for the numberof holes and electrons present in an intrinsic semi-conductor it is appropriate to use Fermi–Diracstatistics and the density of states at the bottom ofthe conduction band (see Section S4.8). To a goodapproximation, it is found that the number ofelectrons in the conduction band per unit volume,
ni, which is equal to the number of intrinsic holes inthe valence band, pi, at an absolute temperature, T ,is:
ni¼ pi¼ 4:826"1021m#em
#h
m2e
! "3=4
T3=2 exp $ Eg
2kT
! "
ð13:2Þ
where m#e is the effective mass of the electron, m#h isthe effective mass of a hole, and Eg is the band gap.In an intrinsic semiconductor
n ¼ p ¼ ni ¼ pi
Equation (13.2) shows that, at a given temperature
n p ¼ constant
The temperature dependence of the equilibriumconstant is given by:
np ¼ 2:33" 1043m#e m
#h
m2e
! "3=2
T3 exp $ Eg
k T
! "
Assuming that the effective mass of electrons andholes is independent of temperature, we obtain
np / T3 exp $ Eg
kT
! "
This equation applies to doped semiconductors aswell as to intrinsic semiconductors, a finding ofconsiderable practical importance. To a goodapproximation, the Fermi energy lies at the centreof the band gap (Section S4.8):
EF ¼1
2Eg þ
3
4kT ln
m#hm#e
! "
( 1
2Eg
Writing the total conductivity, !(total), as
!ðtotalÞ ¼ ne"e þ pe"h
we obtain
!ðtotalÞ ¼ 773:1m#em
#h
m2e
! "3=4
T3=2ð"e þ "hÞ
" exp $ Eg
2kT
! "
ð13:3Þ
Table 13.1 Approximate values for the band gap ofsome semiconductors
Symbol orCompound formula Band gap/eV
Elements:Diamond C 5.47Silicon Si 1.12Germanium Ge 0.66Grey tin Sn 0.08
III–V semiconductors:Gallium nitride GaN 3.36Gallium phosphide GaP 2.26Gallium arsenide GaAs 1.42
II–VI semiconductors:Cadmium sulphide CdS 2.42Cadmium selenide CdSe 1.70Cadmium telluride CdTe 1.56
Note: the band gap is normally given in electron volts in mostcompilations; 1 eV is equal to 1:60219" 10$19 J.
398 ELECTRONIC CONDUCTIVITY IN SOLIDS
EngeneralenunsemiconductorintrínsecoelniveldeFermiestásituadoenelcentrodelintervaloprohibidodeenergía(enprimeraaproximación).
PROPIEDADESELÉCTRICAS
ConductividadEléctrica(SEMICONDUCTORES)Considérese el Germanio a 25 oC. Estímese (a) el número de potadores decarga, (b) la fracción del total de electrones en la banda de valencia que seexcitan a la banda de conducción y (c), el valor de la constante n0
SOLUCIÓN:
Delosdatosindicadosanteriormente,𝜌 =43𝛺·cma25oC;porlotanto𝜎 = e
f= 0.023Ωke𝑐𝑚ke
DelamismatablaextraemosqueEg (germanio)=0.67eV a250CLasmovilidadesparahuecosyelectronesvienendadaspor:
𝜇M = 3900𝑐𝑚8
𝑉 · 𝑠 ; 𝜇I = 1900𝑐𝑚8
𝑉 · 𝑠
2𝑘p𝑇 = 2 8.63𝑥10kt𝑒𝑉𝐾 273 + 25 = 0.0514𝑒𝑉(𝑎25 𝐶L )
PROPIEDADESELÉCTRICAS
ConductividadEléctrica(SEMICONDUCTORES)SOLUCIÓN(Continuación):
Porlotantoyatendiendoalasexpresionesyaindicadas:
𝜎{LB*T = 𝜎-T-P + 𝜎CR- = 𝑛𝑞𝜇M + 𝑝𝑞𝜇I = 𝑛𝑞 𝜇M + 𝜇I
𝑛 = 𝜎
𝑞(𝜇M + 𝜇I)=
0.023(1.6𝑥10ke|)(3900 + 1900) = 2.5𝑥10e}
𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛𝑒𝑠𝑐𝑚}
(b)Elparámetrodereddelgermaniocúbicotipodiamantees5.6575x10-8 cm.Portantoelnúmerototaldeelectronesenlabandadevalenciadelgermanioserá:
𝐸𝑙𝑒𝑐𝑡𝑟𝑜𝑛𝑒𝑠𝑡𝑜𝑡𝑎𝑙𝑒𝑠 = (8 á𝑡𝑜𝑚𝑜𝑠 𝑐𝑒𝑙𝑑𝑎⁄ )(4 𝑒𝑙𝑒𝑐𝑡 á𝑡𝑜𝑚𝑜)⁄
(5.6575𝑥10k�𝑐𝑚)} = 1.77𝑥108}
𝐹𝑟𝑎𝑐𝑐𝑖ó𝑛𝑒𝑥𝑐𝑖𝑡𝑎𝑑𝑎 = 2.5𝑥10e}
1.77𝑥108} = 1.41𝑥10ke�
PROPIEDADESELÉCTRICAS
ConductividadEléctrica(SEMICONDUCTORES)SOLUCIÓN(Continuación):
Porúltimo,elvalorden0 será:
𝑛� = 𝑛
exp −𝐸�2𝑘�𝑇�
= 2.5𝑥10e}
exp −0.670.0514�
= 1.14𝑥10e| 𝑝𝑜𝑟𝑡𝑎𝑑𝑜𝑟𝑒𝑠
𝑐𝑚}
PROPIEDADESELÉCTRICAS