Chapter 3 Band Structures in Solids Outline

3.1 Bonding forces and energy bands in solids 3.2 Charge carriers and occupation statistics 3.3 Carrier concentration 3.4 Drift of carriers in electric and magnetic fields 3.5 Behavior of the equilibrium Fermi level

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3.1 Bonding forces and energy bands in  solids •

•

Bonds in Solids • The electrons in the outermost shell of the constituent  atoms, also known as the valence electrons, of a solid  contribute most to the binding force in the solid. There are different spatial distributions (wave functions) of electrons for  different energy band structures. hence solids can also be classified  according to the distributions of the electrons :

1)   Ionic bonds • Distribution has a high degree of localization, giving rise to charge centers  consisting of positive and negative ions. • NaCl (common salt) is an example of such a solid. • Ionic solids are typically insulators. ELEC4510‐Semiconductor Materials and Devices (Fall‐2011)

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1. Ionic bonds  Electron transfers from one  constituent atom to a dissimilar  neighbor atom giving rise to positive  and negative ions.  Electrostatic attraction among  oppositely charged ions is largely  responsible for the stability of the  solid.  The constituent atoms of an ionic  solid usually come from groups at the  extreme ends of the Periodic Table  (e.g. Groups I and VII).  Ionic crystals are brittle, do not  conduct electricity except on melting,  and often have cubic structures. ELEC4510‐Semiconductor Materials and Devices (Fall‐2011)

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2.   Metallic bonds  Distribution is highly non‐localized, giving rise to a more or less uniform distribution of electrons (sea of electrons).  The force holding the lattice together arise from an interaction between the positive ion cores and the surrounding free electrons. This is one type of metallic bonding.  Na (alkali metal) is an example of such a solid.  Metallic solids are typically good electrical conductors.  Valence electrons are distributed over the entire crystal. Hence they are shared among all the ions in the solid.  About three quarters of all elements are metals.  Metallic bonds are non‐directional. Hence metals are ductile since distortions of the solid arrangement do not “break” the bonds.  “Sea” of free electrons to conduct electricity and heat. ELEC4510‐Semiconductor Materials and Devices (Fall‐2011)

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3. Covalent bonds  Distribution has a high degree of directionality, giving to localization (and sharing) of electrons between adjacent atoms. There are two electrons per bond with opposite spins, in accordance with the Pauli Exclusion Principle  Si is an example of such a solid  Covalent solids are typically insulators or semiconductors  Electrons shared between neighboring constituent atoms in directional bonds.  Group IV elements (C, Si, Ge, ...)  Solids at room temperature  Each atom shares its four valence electrons with its four nearest neighbors  Brittle, semiconducting

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Electronic States in Solids In atoms, QM leads to the quantization of the energy levels of bound electrons. Put in another way: there are “gaps” between allowed energy levels. In solids, the electronic states are described by QM, but will be different because: • The potential energy are very different for a solid and that for an atom.

• • • • • •

Electrons in a solid are under the influence of the electrostatic force of a large number of constituent atoms (density: ~1022 atoms/cm3) in the solid. The electrons are no longer localized around any particular atom. The electrons are “distributed” over the entire crystal. The quantization of the energy levels in a solid has the following  structure: Bands of very closely spaced allowed energy levels separated by Gaps of forbidden energy

Note: thus one can see that the structure of the energy levels in a  solid is quite similar to that in an atom, except that in a solid,  allowed energy levels are clustered in bands separated by gaps  rather than distinct levels separated by gaps as is the case in an  atom. ELEC4510‐Semiconductor Materials and Devices (Fall‐2011)

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Formation of Energy Bands in Solids • A regular array of constituent atoms (say N) with large separation; Large separation ‐> valence electrons of the atoms not influenced by the potentials of the other atoms. Electronic energy structure of this “solid” is identical to that of the constituent atoms, only the degeneracy of each level is increased by N‐fold. • Separation is gradually decreased to approach the equilibrium value in the solid, the valence electrons will increasingly be influenced by the potentials of the neighboring atoms ‐> energy levels of electrons altered. • N‐fold degeneracy is removed and the energy levels broaden into a band of allowed energy levels. Silicon: valence electrons in the atoms can be thought to be 4‐fold degenerate (sp3 hybridization), or 4N‐fold degenerate when there are N silicon atoms. • 4N levels broaden and eventually split into two bands of allowed states. Energy levels in Si as a function of inter‐atomic  • T=0, lower band is completely filled with electrons. This Fig. 3‐3 spacing. The core levels (n = 1, 2) in Si are completely filled with  band is called the valence band. electrons. AT the actual atomic spacing of the crystal, the 2N  • Upper band is completely empty. This band is called the electrons in the 3s sub‐shell and the 2N electrons in the 3p sub‐shell  conduction band. undergo sp3 hybridization, and all end up in the lower 4N states  • No electrons are allowed to have energy level in the (valence band), while the higher lying 4N states (conduction band)  range between the lowest energy of the conduction are empty, separated by a bandgap. band (Ec) and the highest energy of the valence band (Ev). The bands are said to be separated by an energy band gap . ELEC4510‐Semiconductor Materials and Devices (Fall‐2011) 7

Classification of Solids by Resistivity Classical Physics offered no explanation for the tremendous range of electrical resistivity for solids at room temperature : from less than in good conductors to almost in the best insulators. With the energy band structure derived from QM, the classification can finally be explained  Insulators :  Completely filled valence band,  Completely empty conduction band,  large energy band gap.  Semiconductors :  Insulators at 0 Kelvin.  Small energy band gap.  Small number of electrons in the conduction band at temperature > 0 Kelvin.  Conductors : Valence band is only partially filled.

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Direct and Indirect Bands

In QM, energy of electrons depend on quantities . related to the “motion” of the electron described by crystal momentum, wave vector Band theory of solids:  energy of an electron : where n is a label for the energy band to  which the electron belongs is called the crystal momentum and is  related to the motion of the electron through  the crystal.

Energy plotted as a function of momentum (E vs. )

Crystal momentum (p) of an electron is  given by :  p  k

 Direct gap semiconductor : the minimum energy of  the conduction band and the maximum energy of  the valence band occur at the same value of .  GaAs is an example of such a semiconductor.  Indirect gap semiconductor : the minimum energy  of the conduction band and the maximum energy  of the valence band occur at different values of  .   Si is an example of such a semiconductor.

Fig. 3‐5 Direct and indirect electron transitions in  semiconductors : (a) direct transition with accompanying  photon emission; (b) indirect transition via a defect level.

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Electronic Transitions 1. Direct band : • Direct transition from the minimum energy of the conduction band to the  maximum energy of the valence band is possible. • This transition involves no change in the crystal momentum  of the  electron. • The energy involved in such a transition is balanced by the emission or  absorption of a photon.  .

2. Indirect band : • Direct transition from the minimum energy of the conduction band to  the maximum energy of the valence band is highly unlikely. • Since different  values are involved, the crystal momentum of the  electron is changed as a result of the transition. • The energy involved in such a transition is balanced either by heat  (typically lattice vibration) or a combination of heat and photon. Semiconductor

Band gap

Si

Indirect

1.1eV

GaAs

Direct

1.43eV

Indirect

1.73eV

Direct

>> 2.16eV

AlAs

Indirect 2.16eV ELEC4510‐Semiconductor Materials and Devices (Fall‐2011)

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3.2 Charge carriers and occupation statistics Electron‐Hole Pair • T>0, electrons can be promoted  thermally from the valence band to  the conduction band. • Empty states, unoccupied by  electrons, are left behind in the  valence band. These empty states are  Fig. 3‐7 called “holes”. •

• •

•

Electron‐hole pairs in a semiconductor.

For intrinsic (undoped) silicon, there are about 1010 cm 3 electron‐hole pairs (EHP)  at room temperature. This is really a very small number compared to the number of  valence electrons available in the system : ~ 1022cm  3. Holes contribute to electrical conduction. They can be modeled as positively charge  carriers with positive mass. The existence of hole conduction makes semiconductors significantly different from  ordinary metals, in which conduction takes place via the flow of negatively charged  electrons only. In semiconductors, both the flow of electrons and the flow of holes contribute to  electrical conduction. The fact that we can select and control electron and hole  flows makes it possible to realize many of the devices so vital to the making of  integrated circuits. ELEC4510‐Semiconductor Materials and Devices (Fall‐2011) 11

Effective Mass deviation‐I To model the behavior of electrons in a solid, we must begin with the concept of the effective mass of an electron. 1. Unlike electrons in a vacuum tube, electrons in a solid are not totally “free” but interact with each other and with the ions in the solid. 2. However, it can be shown that the electrons in a solid can be modeled as “free” particles, but with a modified mass different from the free electron mass (m0) , and respond to electromagnetic field similar to free electrons 3. This mass is called the “effective” mass (m* ) of an electron in a solid. 4. The crystal momentum (p) of an electron is given by :

p  k Therefore, the total energy (E) of a “free” electron with effective mass (m*) is  given by: p2 2 2 E  k 2m * 2 m *

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Effective Mass deviation‐II Taking the second derivative of E w.r.t. k yields :

Therefore

m* 

2 d2E  dk 2 m*

2 d2E dk 2

5. The derivation shows that the effective mass (m*) is inversely proportional to the  curvature of E in the k‐space. • The larger the curvature, the smaller the effective mass.

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Effective Mass of semiconductors 6.

Effective mass of an electron is different for different solids:

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Intrinsic Semiconductor‐I • Semiconductors with no impurities or lattice defects. • Electron‐Hole pairs (EHPs) are the only mobile charged carriers. • EHPs are generated by promoting electrons from the valence band to the conduction band, thus an energy of Eg is required per EHP. • Similarly EHPs are annihilated by the recombination of electrons and holes, thus an energy of Eg is released per EHP recombined.

Fig. 3‐11 Electron‐hole pairs in the covalent  bonding model of the Si crystal.

• If the equilibrium concentrations of electrons and holes are denoted respectively by n0 and p0 and since electrons and holes are generated in pairs, we must have :

n0  p0  ni

where ni is called the intrinsic carrier concentration of the semiconductor. ELEC4510‐Semiconductor Materials and Devices (Fall‐2011)

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Intrinsic Semiconductor‐II • In fact, this equilibrium concentration is maintained through a dynamic process : EHPs are constantly being generated and EHPs are constantly being annihilated by recombination. • At equilibrium, the generation rate (gi) and the recombination rate (ri) are equal :

gi  ri •

The recombination rate obeys the familiar Law of Mass Action : ri   r n 0 p 0   r n i2 where

•

is a proportionality constant.

ni is a strong function of temperature.

• For silicon at room temperature, ni ~ 1010/cm3. ELEC4510‐Semiconductor Materials and Devices (Fall‐2011)

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Extrinsic Semiconductor •

Semiconductors are purposely doped with impurities.

•

Electrons and holes no longer appear in pairs : predominance of either electrons or  holes can be achieved.

•

Extrinsic semiconductor can be classified into:  n‐type : more electrons (majority) than holes (minority).  p‐type : more holes (majority) than electrons (minority).

•

Impurities introduced into the lattice actually create additional energy levels,  usually within the forbidden band.  Group V impurities, such as P, As, and Sb, introduce energy levels (donor levels)  near the bottom of the conduction band in Ge and Si. Each donor impurity  provides one extra electron to the lattice.  Group III impurity, such as B, introduces energy levels (acceptor levels) near the  top of the valence band in Ge and Si. Each acceptor impurity provides one  extra hole to the lattice.

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N or P type n‐type • Si doped with a Group V element, say P : donor levels are filled with electrons at 0 Kelvin. • electrons are thermally excited to the conduction band at higher temperature, e.g. room temperature. • When this occurs, n0 >> p0 , the material becomes n‐type. p‐type

Fig. 3‐12c Donor and acceptor atoms in  the covalent bonding of a Si crystal.

• Si doped with a Group III element, say B : acceptor levels are filled with holes at 0 Kelvin. • electrons are thermally excited to the acceptor levels from the valence band at higher temperature, e.g. room temperature. • When this occurs, p0 >> n0 , the material becomes p‐type. Say if 1015/cm3 of P is introduced into Si, the concentration of the conduction electrons increases by 5 orders of magnitude. Consequently, the resistivity reduces from 2×105 Ω‐cm to 5 Ω‐cm. ELEC4510‐Semiconductor Materials and Devices (Fall‐2011)

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Donator in conduction band Fig. 3‐12a

Donation of electrons from a donor level to the conduction band.

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Acceptor in valence band Fig. 3‐12b Acceptance of valence band electrons by an acceptor level, and the  resulting creation of holes.

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