Disposi&vos semiconductores Bandas de energa y portadores de carga en

semiconductores

Bandas de energa y portadores de carga...

Captulo 3 Bandas de energa y portadores de carga en semiconductores

El comportamiento de disposi&vos de estado slido -> teora atmica, mecnica cun&ca. Vamos a inves&gar: - La estructura atmica de los tomos - Interaccin de tomos y electrones bajo excitacin externa, como la absorcin y emisin de luz. Estos estudios permi&rn la comprensin del efecto de la red cristalina en los electrones uyendo a travs del cristal (corriente elctrica).

tomos y electrones

Bandas de energa (3.1) Los tomos individuales &enen niveles de energa discretos, pero que pasa en un slido?

Dos tomos aislados

tomos y electrones

Cundo los tomos son aproximados las funciones de onda se sobreponen en amplitud y fase.

tomos y electrones

Las dos combinaciones lineales de orbitales atmicos son denominados: : unin simtrica * : unin an& simtrica

En analoga a las funciones de onda del pozo de potencial, la unin an& simtrica &ene un valor ms grande de energa (un nodo).

tomos y electrones

La energa potencial es menor para H2 que para los dos tomos H separados

En un anlisis similar para He donde el orbital 1s est lleno con

La energa del sistema He-He es ms grande que la de los tomos individuales, luego He existe slo en la naturaleza.

tomos y electrones

Haciendo el mismo anlisis para tres tomos de hidrgeno A, B y C, tendremos tres estados orbitales moleculares diferentes, a, b y c:

- energ&co

+ energ&co

a =1s A( )+1s B( )+1s C( )b =1s A( )1s C( )c =1s A( )1s B( )+1s C( )

tomos y electrones

Sistema con N tomos de Li

tomos y electrones

Efec&vamente, tenemos la formacin de una banda con&nua de energa para N grande Hay una faja con&nua de niveles de energa arriba de los niveles llenos.

tomos y electrones

66 Chapter 3

structures. As the spacing between the two atoms becomes smaller, howev-er, electron wave functions begin to overlap. The exclusion principle dictates that no two electrons in a given interacting system may have the same quan-tum state; thus there must be at most one electron per level after there is a splitting of the discrete energy levels of the isolated atoms into new levels be-longing to the pair rather than to individual atoms.

In a solid, many atoms are brought together, so that the split energy levels form essentially continuous bands of energies. As an example, Fig. 3-3 illustrates the imaginary formation of a silicon crystal from isolated silicon atoms. Each isolated silicon atom has an electronic structure ls22s22p63s23p2 in the ground state. Each atom has available two Is states, two 2s states, six 2p states, two 35 states, six 3p states, and higher states (see Tables 2-1 and 2-2). If we consider N atoms, there will be 2N, 2N, 6N, 2N, and 6N states of type Is, 2s, 2p, 3s, and 3p, respectively. As the interatomic spacing decreases, these energy levels split into bands, beginning with the outer (n = 3) shell. As the "35" and "3/?" bands grow, they merge into a single band composed of a mixture of energy levels. This band of "35-3/?" levels contains 8N available states. As the

Relative spacing of atoms s

i(+) * 1-2

+ 1 0

- 1 -2)

+ 1 0

-1

0

1

0

0

0

j

I 2

VI

n

i

i

o _

0

I3

u 1

Outer shell

Middle shell

Inner shell

Figure 3-3 Energy levels in Si as a function of interatomic spacing. The core levels (n = 1,2) in Si are completely filled with electrons. At the actual atomic spacing of the crystal, the 2 N electrons in the 3s subshell and the 2 N electrons in the 3p subshell undergo sp2 hybridization, and all end up in the lower 4N stales (va-lence band), while the higher-lying 4 N states (conduction band) are empty, separated by a band gap.

Repi&endo el anlisis para Si (semiconductor)

tomos y electrones

El band gap o gap de energa (gap=espacio) es una regin donde no hay estados posibles para que los electrones se ubiquen. Metales, semiconductores y aislantes (3.1.3) Para que una corriente elctrica pueda uir a travs de un slido, los electrones deben ser capaces de pasar de un tomo a otro dentro del slido -> directamente conectado a la conduc&vidad elctrica de los slidos. En el slido de Li, los electrones pueden moverse, mientras en el Si no.

tomos y electrones

68 Chapter 3

Figure 3-4 Typical band

structures at 0 K. Empty

Empty 2,

::::::::!iii:ii. iii* ::::::::;::::::::: !::::: Riled;::::::

Partially filled

. . . . . . . t ' l i ieq.. . . . . .

Insulator Semiconductor Metal

than in insulators. For example, the semiconductor Si has a band gap of about 1.1 eV compared with 5 eV for diamond. The relatively small band gaps of semiconductors (Appendix III) allow for excitation of electrons from the lower (valence) band to the upper (conduction) band by reasonable amounts of thermal or optical energy. For example, at room temperature a semicon-, ductor with a 1-eV band gap will have a significant number of electrons ex-cited thermally across the energy gap into the conduction band, whereas an insulator with Eg = 10 eV will have a negligible number of such excitations. Thus an important difference between semiconductors and insulators is that the number of electrons available for conduction can be increased greatly in semiconductors by thermal or optical energy.

In metals the bands either overlap or are only partially filled. Thus electrons and empty energy states are intermixed within the bands so that electrons can move freely under the influence of an electric field. As ex-pected from the metallic band structures of Fig. 3-4, metals have a high elec-trical conductivity.

3.1.4 Direct and Indirect Semiconductors

The "thought experiment" of Section 3.1.2, in which isolated atoms were brought together to form a solid, is useful in pointing out the existence of energy bands and some of their properties. Other techniques are general-ly used, however, when quantitative calculations are made of band struc-tures. In a typical calculation, a single electron is assumed to travel through a perfectly periodic lattice. The wave function of the electron is assumed to be in the form of a plane wave1 moving, for example, in the x- direction

'Discussions of plane waves are available in most sophomore physics texts or in introductory electromag-netics texts.

Estructuras de banda ]picas a 0 K

Superposicin Aislante Semiconductor Metal En los metales, las bandas de conduccin &enen electrones libres para el movimiento.

tomos y electrones - La principal diferencia entre los aislantes y los semiconductores es en el valor del band gap. - Como la banda de valencia est llena y la de conduccin vaca, no es posible tener un ujo de electrones bajo un campo elctrico externo aplicado. - En los metales tenemos la situacin inversa.

A temperaturas dis&ntas de 0 K existe la probabilidad que excitaciones trmicas hagan que algunos electrones salgan de la banda de valencia hasta la de conduccin. Esto es exponencialmente ms probable en semiconductores que en aislantes debido a la diferencia en Eg. P. Ej. Si-> Eg = 1.1 eV; Diamante-> Eg = 5 eV.

tomos y electrones Semiconductores directos e indirectos (3.1.4) La funcin de onda de un electrn movindose en la direccin x a travs de una red cristalina peridica es dada por: La funcin modula la funcin de onda siguiendo la periodicidad de la red. Los valores posibles para la energa pueden ser gracados vs. k

k x( ) =U kx, x( )e jkxx

U kx, x( )

Energy Bands and Charge Carriers in Semiconductors 69

E

hv = >

Figure 3-5 Direct and indi-rect electron transitions in semiconductors: (a) direct transi-tion with accom-panying photon emission; (b) indi-rect transition via a defect level.

(a) Direct (b) Indirect

with propagation constant k, also called a wave vector. The space-dependent wave function for the electron is

* kM =U(kx,x)e>^ (3-1)

where the function U(kx,x) modulates the wave function according to the pe-riodicity of the lattice.

In such a calculation, allowed values of energy can be plotted vs. the propagation constant k. Since the periodicity of most lattices is different in various directions, the (E, k) diagram must be plotted for the various crystal directions, and the full relationship between E and k is a complex surface which should be visualized in three dimensions.

The band structure of GaAs has a minimum in the conduction band and a maximum in the valence band for the same k value (k = 0). On the other hand, Si has its valence band maximum at a different value of k than its con-duction band minimum. Thus an electron making a smallest-energy transi-tion from the conduction band to the valence band in GaAs can do so without a change in k value; on the other hand, a transition from the minimum point in the Si conduction band to the maximum point of the valence band requires some change in k.Thus there are two classes of semiconductor energy bands; direct and indirect (Fig. 3-5). We can show that an indirect transition, involv-ing a change in k, requires a change of momentum for the electron.

Assuming that U is constant in Eq. (3-1) for an essentially free electron, show that the x-component of the electron momentum in the crystal is given by

tomos y electrones En los semiconductores de gap directo el mnimo de la banda de conduccin es alineado con el mximo de la banda de valencia. Una transicin directa con la emisin de un fotn es posible. En los semiconductores indirectos el mnimo de la BC no coincide con el mnimo de la BV en el espacio-k. As, una transicin es posible slo acompaada de un cambio en momento (p=k) haciendo que la transicin sea menos probable. Semiconductores directos -> emisin de luz. Cambio de bandas de energa en funcin de la composicin de aleacin (3.1.5)

Cambio de bandas de energa en funcin de la composicin de aleacin (3.1.5) AlxGa1-xAs

tomos y electrones

72 Chapter 3

+~ k

2.16 eV

+ k

2

a c PQ

0.4 0.6 0.8 1.0 Aluminum fraction, x

(c) Figure 3-6 Variation of direct and indirect conduction bands in AIGaAs as a function of composition: (a) the (,k) diagram for GaAs, showing three minima in the conduction band; (b) AIAs band diagram; (c) positions of the three conduction band minima in AlxGa]_xAs as x varies over the range of compositions from GaAs (x = 0) to AIAs (x = 1). The smallest band gap, Eg (shown in color), follows the direct T band to x = 0.38, and then follows the indirect X band.

GaAs un semiconductor de gap directo, mientras AlAs es de gap indirecto.

tomos y electrones

72 Chapter 3

+~ k

2.16 eV

+ k

2

a c PQ

0.4 0.6 0.8 1.0 Aluminum fraction, x

(c) Figure 3-6 Variation of direct and indirect conduction bands in AIGaAs as a function of composition: (a) the (,k) diagram for GaAs, showing three minima in the conduction band; (b) AIAs band diagram; (c) positions of the three conduction band minima in AlxGa]_xAs as x varies over the range of compositions from GaAs (x = 0) to AIAs (x = 1). The smallest band gap, Eg (shown in color), follows the direct T band to x = 0.38, and then follows the indirect X band.

tomos y electrones

Electrones y huecos (3.2.1) T > 0 K

BC

Eg BV A temperaturas > 0 K, algunos electrones son excitados a travs del band gap y llegan a la BC que &ene muchos estados vacos.

Energy Bands and Charge Carriers in Semiconductors 73

in this way. Since the semiconductor has a filled valence band and an empty conduction band at 0 K, we must consider the increase in conduction band electrons by thermal excitations across the band gap as the temperature is raised. In addition, after electrons are excited to the conduction band, the empty states left in the valence band can contribute to the conduction process. The introduction of impurities has an important effect on the energy band structure and on the availability of charge carriers. Thus there is considerable flexibility in controlling the electrical properties of semiconductors.

3.2.1 Electrons and Holes

As the temperature of a semiconductor is raised from 0 K, some electrons in the valence band receive enough thermal energy to be excited across the band gap to the conduction band. The result is a material with some elec-trons in an otherwise empty conduction band and some unoccupied states in an otherwise filled valence band (Fig. 3-7).2 For convenience, an empty state in the valence band is referred to as a hole. If the conduction band electron and the hole are created by the excitation of a valence band electron to the conduction band, they are called an electron-hole pair (abbreviated EHP).

After excitation to the conduction band, an electron is surrounded by a large number of unoccupied energy states. For example, the equilibrium number of electron-hole pairs in pure Si at room temperature is only about 1010 EHP/cm3, compared to the Si atom density of 5 X 1022 atoms/cm3. Thus the few electrons in the conduction band are free to move about via the many available empty states.

The corresponding problem of charge transport in the valence band is somewhat more complicated. However, it is possible to show that the effects of current in a valence band containing holes can be accounted for by sim-ply keeping track of the holes themselves.

In a filled band, all available energy states are occupied. For every elec-tron moving with a given velocity, there is an equal and opposite electron mo-tion elsewhere in the band. If we apply an electric field, the net current is zero

.^--.:...9 .* ^ 9 ^ . M.

,__ , _ , , . , , I, . i.

Figure 3-7 Electron-hole pairs in a semiconductor.

2ln Fig. 3-7 and in subsequent discussions, we refer to the bottom of the conduction band as Ec and the lop of the valence band as fv.

tomos y electrones

Los electrones estaban sujetos a un potencial de un tomo. Tras ser echados de la banda de valencia huecos quedan en su lugar. Cundo un electrn es movido de la BV para la BC un hueco es siempre creado en conjunto. Es formado un par electrn-hueco (EHP). Los pocos electrones que se ubican en la BC estn libres para moverse. El transporte de cargas en la banda de valencia es un poco ms complejo:

banda llena

No hay movimiento neto de electrones en una banda llena ->

tomos y electrones

corriente cero. Supongamos que ahora hay un hueco: Cundo hay uno o ms huecos, existe un movimiento neto de electrones en la BV. Podemos visualizar como si lo huecos se mueven en sen&do opuesto a los electrones (misma direccin de ). Ambos los electrones y huecos son responsables por el ujo de corriente elctrica en un semiconductor.

tomos y electrones

Otra forma de ver la banda de valencia llena: Los estados j y j &enen momentos kj y kj, o sea &enen velocidades de igual magnitud pero sen&dos opuestos. -> corriente neta igual a cero.

Energy Bands and Charge Carriers in Semiconductors 75

Figure 3-8 A valence band with all states filled, including states /' and / ', marked for discussion. The /th electron with wave vector k,- is matched by an electron at /' with the opposite wave vector k(-. There is no net current in the band unless an electron is removed. For example, if the /th electron is removed, the motion of the electron at /' is no longer compensated.

charge carriers. We draw valence and conduction bands on an electron ener-gy scale E, as in Fig. 3-8. However, we should remember that in the valence band, hole energy increases oppositely to electron energy, because the two car-riers have opposite charge. Thus hole energy increases downward in Fig. 3-8 and holes, seeking the lowest energy state available, are generally found at the top of the valence band. In contrast, conduction band electrons are found at the bottom of the conduction band.

It would be instructive to compare the (E, k) band diagrams with the "simplified" band diagrams that are used for routine device analysis (Fig. 3-9). As discussed in Examples 3-1 and 3-2, an (E, k) diagram is a plot of the total electron energy (potential plus kinetic) as a function of the crystal-direction-dependent electron wave vector (which is proportional to the momentum and therefore the velocity) at some point in space. Hence, the bottom of the conduction band corresponds to zero electron velocity or kinetic energy, and simply gives us the potential energy at that point in space. For holes, the top of the valence band corresponds to zero kinetic energy. For simplified band diagrams, we plot the edges of the conduction and valence bands (i.e., the potential energy) as a function of position in the device. Energies higher in the band correspond to additional kinetic energy of the electron. Also, the fact that the band edge corresponds to the electron potential energy tells us that the variation of the band edge in space is related to the electric field at dif-ferent points in the semiconductor. We will show this relationship explicitly in Section 4.4.2.

In Fig. 3-9, an electron at location A sees an electric field given by the slope of the band edge (potential energy), and gains kinetic energy (at the ex-pense of potential energy) by moving to point B. Correspondingly, in the (, k) diagram, the electron starts at k = 0, but moves to a nonzero wave vector kB.

tomos y electrones

Huecos &enen carga posi&va. La energa de los huecos en la banda de valencia aumenta de manera inversa a los electrones.

E+ electrones

Espacio E-K

E+ huecos

tomos y electrones

Luego ubicamos ms electrones en el fondo de la banda de conduccin, y ms huecos en el tope de la BV.

BC electrones

BV huecos Los huecos en el tope de la BV &enen energa cin&ca cero. Electrones en el fondo de la BC tambin poseen EC = 0.

tomos y electrones

Vamos ocupar tambin los diagramas de banda simplicados:

-> en funcin de la posicin en el disposi&vo

Un campo elctrico aplicado inclina las bandas de energa en funcin de la posicin:

tomos y electrones 76 Chapter 3

Figure 3-9 Superimposition

ofrhe(E,k) band structure

on the E-versus-position

simplified band diagram for a

semiconductor in an electric field.

Electron energies increase going

up, while hole en-ergies increase

going down. Simi-larly, electron and hole wave vectors point in opposite

directions and these charge car-riers move oppo-

site to each other, as shown.

i k Electron energy

The electron then loses kinetic energy to heat by scattering mechanisms (dis-cussed in Section 3.4.3) and returns to the bottom of the band at B.The slopes of the (E,x) band edges at different points in space reflect the local electric-fields at those points. In practice, the electron may lose its kinetic energy in stages by a series of scattering events, as shown by the colored dashed lines.

3.2.2 Effective Mass

The electrons in a crystal are not completely free, but instead interact with the periodic potential of the lattice. As a result, their "wave-particle" motion cannot be expected to be the same as for electrons in free space. Thus, in ap-plying the usual equations of electrodynamics to charge carriers in a solid, we must use altered values of particle mass. In doing so. we account for most of the influences of the lattice, so that the electrons and holes can be treated as "almost free" carriers in most computations. The calculation of effective mass must take into account the shape of the energy bands in three-dimensional k-space, taking appropriate averages over the various energy bands.

EXAMPLE 3-2 Find the (E, k) relationship for a free electron and relate it to the electron mass.

tomos y electrones

El electrn se ubica en A (k=0) cundo el campo es aplicado, y se mueve hasta el punto B, pero ahora &ene energa cin&ca (momento kB). Eventualmente se choquea con otro tomo y pierde su energa cin&ca como calor, en un evento de dispersion, volviendo a k=0, en la posicin B. En la prc&ca los eventos de dispersin son ml&plos y la energa cin&ca no es perdida en un slo choque. Masa efecIva (3.2.2) Los electrones en un cristal no son completamente libres pero estn sujetados al potencial peridico de la red. Entonces el movimiento onda-par]cula no es lo mismo que para par]culas en el espacio libre.

tomos y electrones

En los clculos de movimiento una masa efec&va que toma en cuenta el efecto de la red en los electrones debe ser usada. Los clculos de la masa efec&va deben considerar la relacin E-k. El clculo de la masa efec&va debe llevar en cuenta la forma de las bandas de energa en el espacio-k. La energa del electrn es parablica con el vector de onda k. Haciendo la segunda derivada (curvatura) de E con relacin a k:

p =mv = !k p2 =m2v2mv2 = p2

m

E = 12mv2 =12p2m =

!2

2m k2

-> La masa del electrn es relacionada con la curvatura de la relacin E-k.

Masa efec&va Por ejemplo em GaAs, la masa efec&va es menos en el fondo de la banda que en las bandas L o X. La curvatura de E-k es posi&va en los minimos de la BC y nega&va en el mximo de la BV. Electrones en el tope de la BV &enen masa efec&va nega&va.

tomos y electrones

d 2Edk2 =

!2

m

m* = !2

d 2Edk2

tomos y electrones

Para una banda cuyo centro es en k=0, la relacin E-k prxima al fondo es parablica. Aplicando esta ecuacin en la expresin de la masa efec&va, tendremos que ella es constante en una banda parablica. Este hecho puede ser ocupado cerca de los mnimos y mximos en las bandas de varios semiconductores. Varias BV en varios semiconductores son duplas

E = !2

2m k2 +Ec

tomos y electrones

78 Chapter 3

For a band centered at k = 0 (such as the T band in GaAs), the (E, k) relationship near the minimum is usually parabolic:

2m* (3 -4 )

Comparing this relation to Eq. (3-3) indicates that the effective mass m* is constant in a parabolic band. On the other hand, many conduction bands have complex (E, k) relationships that depend on the direction of elec-tron transport with respect to the principal crystal directions. In this case, the effective mass is a tensor quantity. However, we can use appropriate averages over such bands in most calculations.

Figure 3-10a shows the band structures for Si and GaAs viewed along two major directions. While the shape is parabolic near the band edges (as indicat-ed in Figure 3-5 and Example 3-2), there are significant non-parabolicities at higher energies.The energies are plotted along the high symmetry [111] and [100] directions in the crystal.The k = 0 point is denoted as r.When we go along

5

4

3

-N 2 >

3 B 0

-1

-2

-3

"Si \\/f\

1 * 1 ?

~ G a A s \ j /

Upper / \ \ " valley 1 / \ \

T TT\ Lower

Eg 1 valley 1 ! 'yfk

Conduction band

L [111] T [100] Wave vector

X L [111] T [100] X

(a) (b)

Figure 3-10 Realistic band structures in semiconductors: (a) Conduction and valence bands in Si and GaAs along [111] and [100] ; (b) ellipsoidal constant energy surface for Si, near the 6 conduction band minima along the X directions. (From Chelikowsky and Cohen, Phys. Rev. B14, 556, 1976).

Varias BV en varios semiconductores son duplas As, tenemos una banda de huecos livianos y una de huecos pesados.