Introduction to Electronic Devices

Introduction to Electronic Devices, Fall 2006, Dr. Dietmar Knipp

Introduction to Electronic Devices (Course Number 300331) Fall 2006 Information: http://www.faculty.iubremen.de/dknipp/

Fundamentals of Semiconductors Dr. Dietmar Knipp Assistant Professor of Electrical Engineering

Source: Apple

Ref.: Apple

Ref.: IBM

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

1

101

Critical dimension (m)

Ref.: Palo Alto Research Center Fundamentals of Semicondutors

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Introduction to Electronic Devices 2 Fundamentals of Semiconductors 2.1 Semiconductors General Information 2.1.1 General Material Properties 2.1.2 Structural Properties of Materials 2.1.2.1 Classification of semiconducting materials 2.1.2.2 The unit cell 2.1.2.3 Diamond crystal structure 2.1.2.4 Crystal Planes and Miller Indices

2.1.3 Basics of Crystal Growth

2.2 Basics of Solid State Physics 2.2.1 The Hydrogen Atom 2.2.2 Energy bands 2.2.3 Band structure in Semiconductors 2.2.4 Energy-Momentum Diagram 2.2.5 Electron energy in a Solid

Fundamentals of Semicondutors

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2.2.6 Material and Carrier Properties 2.2.6.1Carrier Concentration in Semiconductors 2.2.6.2 Density of States 2.2.6.3 Fermi-Dirac Statistic 2.2.6.4 Fermi Energy in Solids

2.2.7 Intrinsic carrier concentration

2.2.8 Donors and Acceptors 2.2.9 Electrons and Holes in Semiconductor 2.1.10 Compensated Semiconductors 2.1.11 Minority and Majority Carriers 2.2.12 Degenerated and Non-degenerated Semiconductors 2.2.13 Bulk Potential

References


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2.1 Semiconductors General Information The purpose of this part of the lecture is to introduce the solid state physics concepts, which are needed to understand semiconductor materials and semiconductor devices. This part of the lecture is kept as comprehensive as possible.

2.1.1 General Material Properties Solid-state materials can be grouped in terms of their conducttivity or resistiviy. Accordingly three classes of materials can be difined: Insulators, Semiconductors and conductors. The conductivity of semiconductors is generally sensitive to temperature, illumination, radiation, magnetic fields and impurity atoms.


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2.1.1 General Material Properties

Range of electrical conductivities σ. Corresponding resistivity:

ρ = 1σ Classification of materials in terms of their conductivity or resititivity. Ref.: M.S. Sze, Semiconductor Devices Fundamentals of Semicondutors

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2.1.1 General Material Properties Periodic table of semiconductor materials

All materials listed in this periodic table are of interest for electronic applications. However, silicon (Si) and gallium arsenide (GaAs) are the most most important materials. Germanium (Ge) is only of interest for niche applications. Silicon has substituted germanium mainly due to the properties of silicon oxide. Ref.: M.S. Sze, Semiconductor Devices Fundamentals of Semicondutors

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2.1.1 General Material Properties Periodic table of semiconductor materials GaAs is a compound semiconductor, meaning it is an alloy of gallium and arsenic. GaAs is non-toxic in its solid state phase. GaAs is a III/V semiconductor, because it is composed of material out of column III and column V of the periodic table. GaAs can be seen as a alloy of gallium and arsenic. Other important materials out of the group of III/V semiconductors are Indium Phosphide (InP), and Gallium Nitride (GaN). The electrical and the optical properties of III/V compound materials are different from the properties of silicon. The materials are of main interest for high speed electronics, photonics, optical communication and high-end solar cells.


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2.1.2 Structural Properties of Materials 2.1.2.1 Classification of semiconducting materials In order to “build” electronic devices we have to understand the electronic transport of charges in the material. However, the electronic properties of electronic material highly depend on the strucutral properties of the material. Based on the strucutral propeties of the material different classes of materials can be distinguished: Amorphous materials, polycrystalline materials and (mono)crystalline materials. The structural order of materials highly depends on the fabrication method and temperatures. In general, the higher the structural order of the material the better the charges can move in the semiconducting material.


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2.1.2.1 Classification of semiconducting materials

Amorphous materials

No long-range order

Poly crystalline materials

Completely ordered in segments

(Mono)Crystalline materials

Entirely ordered solid

Ref.: R.F. Pierret, Semiconductor Fundamentals Fundamentals of Semicondutors

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2.1.2.2 The unit cell The periodic arrangement of atoms is called lattice! A unit cell of a material represents the entire lattice. By repeating the unit cell throughout the crystal, one can generate the entire lattice. A unit cell can be characterized by a vector R, where a, b and c are vectors and m, n and p are integers, so that each point of a lattice can be found. Primitive unit cell.

R=ma+nb+pc The vectors a, b, and c are called the lattice constants.

Ref.: M.S. Sze, Semiconductor Devices Fundamentals of Semicondutors

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2.1.2.2 The unit cell Different unit cells based on cubic unit cells

Simple cubic unit cell

Body centered cubic unit cell

Face centered cubic unit cell

(bcc)

(fcc) Ref.: M.S. Sze, Semiconductor Devices


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2.1.2.3 Diamond crystal structure

Silicon and germanium have a diamond crystal structure. The silicon structure belongs to the class of face center cubic unit cells. A silicon unit cell consists of eight silicon atoms.

Diamond lattice.

The structure can be seen as two interpenetrating face centered crystal sublattices with one sublattice displaced from the other by one quarter of the distance along the body diagonal of the cube.


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2.1.2.3 Diamond crystal structure

Most of the III/V semiconductors grow in a zincblende lattice, which is identical to a diamond lattice except that one of face center cubic cell sublattices has gallium atom and the other arsenic atoms.

Zincblende lattice.


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2.1.2.4 Crystal Planes and Miller Indices

Miller Indices of some important planes in a cubic crystal.

Crystal properties along different planes are different and the electrical, thermal and mechanical properties can be dependent on the crystal orientation. Indices (Miller indices) were introduced to define various planes in a crystal.


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2.1.2.4 Crystal Planes and Miller Indices Example: Determine the crystal plane The plane has interceptions at a, 3a and 2a along the three coordinates. Taking the reciprocals of the intercepts, we get 1, 1/3 and ½. The three smallest integers have the ratio 6, 2, and 3. Thus, the plane is referred to be the (623) plane.


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2.1.2.4 Crystal Planes and Miller Indices Conventions how to define Miller indices: (hkl): For a plane that intercepts the x-axis on the negative side of the origin such as (100). [hkl]: For a crystal direction, such as [100] for the x-axis. By definition, the [100]-direction is perpendicular to the (100)-plane, and the [111]-direction is perpendicular to the (111)-plane.

Ref: M. Shur, Introdcution to Electronic Devices Fundamentals of Semicondutors

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2.1.2.4 Crystal Planes and Miller Indices Conventions how to define Miller indices: {hkl}: For planes of equivalent symmetry such as {100} for (100), (010), (001), (100), (010) and (001) in cubic symmetry.

Ref.: M. Shur, Introdcution to Electronic Devices Fundamentals of Semicondutors

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2.1.3 Basics of Crystal Growth 95% of the material used in semiconductor industry is crystalline silicon. Before growing the silicon ingots, the material (SiO2, sand) is purified. The most common growth method is the Czochralski method. The crucible contains poly crystalline material, which is heated by radio frequency induction up to 1412°C. The system is typically filled with an inert gas like argon to prevent contamination of the single crystalline ingot. A silicon rod is used as the seed for the growth of the silicon crystal. Simplified schematic drawing of the Czochralski puller. Ref.: M.S. Sze, Semiconductor Devices Fundamentals of Semicondutors

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2.1.3 Basics of Crystal Growth

Photo of an ingot. The ingot has a diameter of 200mm. After pulling the single crystalline ingot the material is sawed into wafers of 300-500µm thickness. A more detailed description of the growth of crystalline materials is given in chapter 11 of M.S. Sze‘s book „Semiconductor devices, Physics and Technology“.


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2.2 Basics of Solid State Physics To understand the properties of semiconductors it is essential to understand the properties of their constituent atoms. Based on Bohr‘s model the atom consists of a core, which contains basically the complete mass of the atom. The shell is nearly without a mass. Despite the fact that nearly all the mass is concentrated in the core the diameter of the core is small with 10-15m in comparison to the diameter of the shell 10-10m=0.1nm=1Å (Ångström). The core consists of neutrons and protons. The core is positively charged. The shell (electron shell) is negatively charged due to electrons on is orbital. Overall the atom is not charged or neutral. The electrons behave like satellites. The electrons circulate around the core on defined orbitals. The electrons are stabilized on their orbitals due an equilibrium of centrifugal and Coulomb forces. We will discuss the consequences of the model based on a hydrogen atom, which is the simplest atom.


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2.2.1 The Hydrogen Atom Due to the equilibrium between the centrifugal forces and the electrostatic forces a direct relation exists between the velocity of the electron and the radius to the core. The velocity of each electron is related to radius of the orbital. As an electron can have different energies, the electron can have different radius to the core of the atom. However, the model has the following problems: Centrifugal Based on classical electrodynamics it v force can be expected that a charged q1 particle on a orbital leads to the electron formation of a magnetic dipole, which Electrostatic radiates energy. Due to the loss of force energy the particle would be more r + attracted by the core, which leads to a core q2 spiral like projection. Finally, the particle would fall into the core of the atom. Schematic diagram of a hydrogen atom Fundamentals of Semicondutors

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2.2.1 The Hydrogen Atom To solve this inconsistency Nils Bohr proposed the following postulate: The energy levels of an atom and therefore the radius of the orbitals are quantized. The allowed energy levels for a hydrogen atom are given by

EB En = − 2 n

n = 1,2,3,.....

Hydrogen energy levels

where EB is the Bohr energy and n is the principle quantum number. The Bohr energy is given by

q2 EB = 8πε 0 aB

Bohr engery

where aB is the Bohr radius. q is the charge of the electron, which is the elementary charge and ε0 is the permittivity. Electron energies between these energy levels En are not allowed. Fundamentals of Semicondutors

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2.2.1 The Hydrogen Atom As the electron energies are quantized the radius of the energy levels are quantized as well. The energy levels for each element are unique. The formation or the splitting of these energy levels allows the formation of energy bands. The energies between the defined energy levels are called the forbidden energy bands. The unit of the energy is usually given in electronvolt (eV). The quantity eV (electron volt) is an energy unit corresponding to the energy gained by an electron when its potential is increased by 1V (1eV=1.6*10-19AVs=1.6*10-19J). The Bohr radius is given by

aB =

ε 0h2

π ⋅ me q

2

Bohr radius

where h is the Planck constant and me is the mass of the electron.


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2.2.1 The Hydrogen Atom Bohr's atom model can be combined with Einstein's photon theory (2. Bohr‘s Postulate). The energy difference between two energy levels n and m is given by

En − Em = h ⋅ f n,m

n>m

Photon energy

where En corresponds to the higher energy level. The transition from a higher to a lower energy level leads to an energy loss. The energy can be released in the form of a photon, where f is the frequency of the emitted light. The frequency and the corresponding wavelength of the light is given by

f n ,m

q 4 me 1 1 = 2 2⋅ 2− 2 8ε 0 h m n

λn,m =

c f n ,m


Frequency of the emitted light.

Wavelength of the emitted light.

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2.2.2 Energy Bands Moving from a single atom to a solid. For an isolated atom, the electrons have discrete energy levels. As a number of p isolated atoms are brought together to form a solid, the orbitals of the outer electrons overlap and interact with each other. This interaction includes attraction and repulsion forces between the atoms. The forces between the atoms cause a shift of the energy levels. Instead of forming a single levels, as it is the case for a single atom, p energy levels are formed. These energy levels are closely spaced. When p is large the different levels essentially form a continuous band. The levels and therefore the bands can extend over several eV depending on the interatomic or molecular spacing.

Schematic illustration of the splitting of the degenerated states into a continuous band of allowed states. Ref.: M.S. Sze, Semiconductor Devices Fundamentals of Semicondutors

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2.2.3 Band structure in Semiconductors Energy Band in semiconductors

Schematic representation of an isolated silicon atom

We will now move from the general description of the band structure in a solid to the more specific situation for silicon. An isolated silicon atom has 14 electrons. Of the 14 electrons 10 occupy deeper energy levels. Therefore, the orbital radius is smaller than the intermolecular separation forces in the crystal. The 10 electrons are bound very strongly to the atoms.

Ref.: M.S. Sze, Semiconductor Devices


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2.2.3 Band structure in Semiconductors Energy Band in semiconductors The 4 remaining valence band electrons are bound weakly and can be involved in chemical reactions. Therefore, we can concentrate on the outer shell (n=3 level). The n=3 level consists of a 3s (n=3 and l=0) and a 3p (n=3 and l=1) subshells. The subshell 3s has two allowed quantum states per atom and both states are filled with an electron (at 0 Kelvin). The subshell 3p has 6 allowed states and 2 of the states are filled with the remaining electrons.


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2.2.3 Band structure in Semiconductors Energy Band in semiconductors

Schematic diagram of the formation of the energy bands in silicon as a function of the lattice spacing Fundamentals of Semicondutors

Schematic diagram of the formation of the energy bands in silicon as the interatomic distance decreases and the 3s and 3p subshells overlap. At a temperature of absolute zero, the electrons occupy the lowest energy states, so that all states in the lower band (valence band) will be full and all states in the upper band (conduction band) are empty. Ref.: M.S. Sze, Semiconductor Devices 28


2.2.3 Band structure in Semiconductors Energy Band in semiconductors The bottom of the conduction band is called Ec and the top of the valence band is called Ev. The energy difference between the bottom of the conduction band and the top of the valence band is called bandgap energy Eg. The bandgap energy Eg=(Ec- Ev) between the bottom of the conduction band and the top of the valence band is the width of the forbidden energy gap. Eg is the energy required to break a bond in the semiconductor to free an electron to the conduction band and leave a hole in the valence band. A deficiency of an electron in the valence band is considered to be a hole. The deficiency in the valence band maybe be filled by a neighboring electron, which results in an shift of the deficiency location. A hole is positively charged. Both the electron and the hole contribute to the current flow.


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2.2.4 Energy Momentum Diagram Energy-band diagram for Silicon and Gallium Arsenide If an electron is excited to the conduction band it can move freely in the crystal, since the electron can be treated like a particle in free space. The propagation of the free electron can be described by the wave function, which is the solution of the Schrödinger equation. The wave function for a free electron is given by

ψ = A1 exp(ikx ) + B1 exp(− ikx )

Wave function

where k is the wave vector, which is given by

p k= h 2π

Wave vector

P is the momentum of the electron. Due to this expression the electron energy can be given as a function of the wave factor. We speak about the k-space representation. The energy bands can now be determined as a function of the k-vector. Fundamentals of Semicondutors

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2.2.4 Energy Momentum Diagram Electron energy in free space

me v 2 En = 2 Energy of a free electron

p = me v Momentum of a free electron

En: Energy of a free electron me: mass of a free electron v: velocity of the electron Fundamentals of Semicondutors

Energy momentum diagram for a free electron

me v 2 p2 = En = 2 2me 31


2.2.4 Energy Momentum Diagram Electron energy in free space

h λ= me v hk p= 2π

DeBroglie equation Dualism of waves and matter for electromagentic waves. k: wave vector

We can rewrite the equation so that the wave vector is expressed in terms of the momentum of the electron.

p k= h 2π


Wave vector

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2.2.4 Energy Momentum Diagram Energy-band diagram for for Silicon and Gallium Arsenide Silicon

GaAs


Indirect semiconductor Fundamentals of Semicondutors

Direct semiconductor 33


2.2.4 Energy Momentum Diagram Electron energy in a Solid For a solid the electron energy near the conduction band minimum can be approximated by a parabolic function similar to an electron in free space. However, the electron energy of an electron in a solid is quite different from the energy of an electron in free space. The energy of an electron can be given by:

h2k 2 En (k ) = EC + 2 8π ⋅ mn

Energy of a electron in the conduction band

where mn is the effective mass of the electron. The effective mass can be calculated by:

mn =

1 ∂ En ∂ p 2


2

Effective mass of an electron

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2.2.4 Energy Momentum Diagram Electron energy in a Solid Narrowing the parabola, corresponds to a larger second derivative, the smaller the effective mass. Energy-momentum relation-ship of a special semi-conductor with an electron effective mass of mn=0.25m0 in the conduction band and a hole effective mass of mp=m0. The actual energymomentum relationship (also called energy-band diagram) for silicon and gallium arsenide are much more complex. Ref.: M.S. Sze, Semiconductor Devices Fundamentals of Semicondutors

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2.2.4 Energy Momentum Diagram Electron energy in a Solid The actual energy-momentum relationship (also called energy-band diagram) for silicon and gallium arsenide are quite different from the energy momentum diagram of a free electron. Nevertheless, the general features like the bandgap between the bottom of the conduction band and the top of the valence band can be observed. Second, the minimum and the maximum of the conduction and valence band are parabolic. For silicon the maximum of the valence band occurs for p=0, but minimum of the conduction band is shifted to p=pc. Therefore, in silicon in addition to the energy Eg, which is necessary to excite an electron an momentum pc is necessary. For GaAs the maximum in the valence band and the minimum in the conduction band occur at the same momentum (p=0). Gallium arsenide is called a direct semiconductor, because it does not require a change in momentum for an electron transition from the valence band to the conduction band. Silicon is called an indirect semiconductor, because a change of the momentum is required in a transition. Fundamentals of Semicondutors

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2.2.5 Electron energy in a Solid With the gained knowledge we can schematically explain the enormous differences in conductivity of insulators, semiconductors and conductors in terms of energy bands. Metals or conductors are characterized by a very low resistivity. Depending on the material two different schematic energy band diagrams exist.

The conduction band is either partially filled (e.g. for Cu) or the valance band and the conduction band overlap (e.g. Zn, Pb). Electrons are free to move with only a small applied electric fields.

Energy Band diagram in a conductor Fundamentals of Semicondutors

Ref.: M.S. Sze, Semiconductor Devices 37


2.2.5 Electron energy in a Solid For an insulator the valence electrons are strongly bonded to the neighboring atoms. This bonds are difficult to break and consequently there are no free electrons, which can participate in an current flow. Insulators are characterized by a large bandgap. All energy levels in the valance band are occupied, whereas all energy levels in the conduction band are empty. Thermal energy or an applied electrical field is not sufficient to raise the uppermost electron in the valence band to the conduction band. One of the best insulators is silicon oxide.



Energy Band diagram in an insulator

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2.2.5 Electron energy in a Solid Materials with an bandgap of 0.6eV to 4.0eV are considered to be semiconductors (room temperature). Most of the materials have bandgaps between 1.0eV and 2.0eV (room temperature). Silicon has a bandgap of 1.12eV, Gallium arsenide has a bandgap of 1.42eV. Therefore, the conductivity of a (intrinsic) semiconductors is low at room temperature. The thermal activation energy is not high enough to excite an electron from the valence band to the conduction band.

At room temperature the thermal activation energy is a fraction of the bandgap, Ethermal=kT=0.0256eV=25.6meV, so that a small number of electrons get thermally excited, which contribute to a moderate current flow for low/moderate electric field levels.

Energy Band diagram in a semiconductor. Ref.: M.S. Sze, Semiconductor Devices


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2.2.6 Material and Carrier Properties Intrinsic and extrinsic Semiconductors The material is considered to be an intrinsic semiconductor if the materials contains a relatively small amount of impurities. The material is considered to be an extrinsic semiconductor if the materials contains a relatively large amount of impurities.

Semiconductors in Thermal Equilibrium In the following it is assumed that the semiconductor is an intrinsic semiconductor. Influences of impurities on the semiconductor properties are neglected. Further, it is assumed that the semiconductor is in thermal equilibrium, which means that the semiconductor is not exposed to additional excitements like light, pressure or electric field. The semiconductor material is kept constant temperature throughout the entire sample (no temperature gradient exists in the semiconductor material). Fundamentals of Semicondutors

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2.2.6.1Carrier Concentration in Semiconductors In the following the carrier concentration in the conduction and the valence band will be calculated. The carrier concentration is given by:

n=

ECtop

∫ N e (E )⋅ Fe (E )dE

Electron concentration

ECbot

p=

EVtop

∫ N h (E )⋅ Fh (E )dE

Hole concentration

EVbot

where n and p are the electron and hole concentration [1/cm3] (Number of electrons and holes per unit volume. Ne(E) and Nh(E) are Density of States (Allowed energy states per energy range and per unit volume). Fe(E) and Fh(E) are the Fermi-Dirac distributions for electrons and holes. The Fermi-Dirac distribution is a probability function, which indicates whether a state is occupied by an electron or a hole.


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2.2.6.1Carrier Concentration in Semiconductors In the first step the product of the Density of States Ne(E), Nh(E) and the FermiDirac Distribution Fe(E), Fh(E) is calculated. The product states whether the states in the conduction and the valence band are occupied by free electrons and holes. The product corresponds to a carrier density for a given energy. In order to determine the overall carrier concentration the integral over all energies (conduction and the valence band) has to be determined.


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2.2.6.1 Carrier Concentration in Semiconductors Schematic Band Diagram, Density of States, Fermi-Dirac Distribution and Carrier Concentration of an intrinsic semiconductor in thermal equilibrium

Schematic Band Diagram

Density of States

Fermi-Dirac Distribution

Electron and hole Density


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2.2.6.2 Density of States The density of states can be calculated by the Schrödinger equation. However, the derivation of the density of state function will not be discussed here. Further information is given by M.S Sze, Semiconductor Devices, Appendix H.

N C (E ) =

4π h3

2me3 (E − Ec )

NV (E ) =

4π h3

2mh3 (EV − E )

Density of states for electrons

Density of states for holes

The Density of States is determined by a single material parameter, which is the effective mass of the electron or the hole. Therefore, the density of states for electrons and holes are very often different.


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2.2.6.3 Fermi-Dirac Statistic The Fermi-Dirac statics describes the probability that an electronic state for a given energy E is occupied by an electron. The Fermi-Dirac Statistic is symmetric around the Fermi energy EF. The Fermi energy can be defined as the energy at which the Fermi-Dirac distribution is equal to ½. In general, the Fermi-Dirac statistic is strongly temperature dependent. With decreasing temperature the

k: Boltzmann constant, T: temperature in Kelvin, EF: Fermi energy

transition gets “sharper”. It means that in practical terms an electronic state is very likely to be occupied by an electron if the energy of the electron is a few kT higher than the Fermi energy. Consequently it is very unlikely that an electronic state is occupied by an electron if the energy is a few kT below than the Fermi energy. Ref.: M.S. Sze, Semiconductor Devices


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2.2.6.3 Fermi-Dirac Statistic

Fermi Dirac Distribution F(E)

So far the Fermi-Dirac distribution was only introduced for electrons. The Fermi-Dirac distribution for holes is given by:

1 Fe (E ) = 1 + exp(E − EF kT )

1.0 Fh(h) Fe(E)

Fermi energy for electrons

0.5

Fh (E ) = 1 − Fe (E ) = 0 -0.5 -0.4 -0.3 -0.2 -0.1 0

0.1 0.2 0.3

Energy E-EF [eV]


1 = 1 + exp(EF − E kT ) Fermi energy for holes 46


2.2.6.3 Fermi-Dirac Statistic Thermal equilibrium A semiconduting material is in thermal equilibrium, if the temperature at each position of the crystal is the same, the overall current through the material is 0, and the solid state is not illuminated. Furthermore, we assume that no chemical reaction is taking part. As a consequence the Fermi energy throughout the material is constant.

EF = EF ( x, y, z ) = const.


Thermal equilibrium

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2.2.6.4 Fermi Energy in Solids

E

E

E Ec

EF EC

EC

EF

EF EV

EV

Conductor

Semiconductor

EV

Insulator

Fermi levels for conductors (metal), semicondcutors and insulators.


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2.2.6.4 Fermi Energy in Solids How can we apply now the concept of the Fermi level do different materials like conductors, insulators and semiconductors? In the case of a conductor the Fermi level is in the conduction band. Therefore, the conduction band is always occupied with electrons. The situation is quite different for insulators and semiconductors. In the case of a semiconductor it is assumed that the material is an intrinsic semiconductor. As a consequence the Fermi level is (approximately) in the middle of the bandgap. However, the bandgap of an insulator is much larger than the bandgap of a semiconductor. The bandgap for a semiconductor is in the range of 0.6eV to 4eV, whereas the bandgap of an insulator is larger than 5.0eV. For example silicon oxide, which is the insulator in microelectronics, has a bandgap of 9.0eV. As a consequence it is very difficult to overcome such a high energy barrier.


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2.2.6.5 Boltzmann distribution To calculate the carrier concentration for electrons and holes the Fermi-Integral has to be solved.

n = N C ⋅ ℑ1 2 (− (EC − EF ) kT )


p = NV ⋅ ℑ1 2 (− (EF − EV ) kT )

Hole concentration

However, the Fermi integral cannot be solved analytically. Therefore, an approximation is used to determine the carrier densities. The approximation is called the Boltzmann distribution.

n ≈ N C ⋅ exp(− (EC − EF ) kT )

for

EC − EF ≥ 2kT

Electron concentration, Boltzmann distribution

p ≈ NV ⋅ exp(− (EF − EV ) kT )

for

EF − EV ≥ 2kT

Hole concentration, Boltzmann distribution Fundamentals of Semicondutors

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2.2.6.5 Boltzmann distribution Instead of using the energy dependent Density of States a new parameter is introduced, which is the effective Density of States. The effective Density of States is again defined for electron and holes. The effective Density of States is independent of the energy. Therefore, the effective Density of States is a pure material parameter.

 m kT  N C = 2  2π e 2  h  

3

 m kT  NV = 2  2π h 2  h   Fundamentals of Semicondutors

Effective Density of States in the conduction band

3

Effective Density of States in the valence band

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2.2.7 Intrinsic carrier concentration We already distinguished between intrinsic and extrinsic semiconductors. The material is considered to be an intrinsic semiconductor, if the material contains a relatively small amount of impurities. Under such conditions the number of electrons per volume in the conduction band is equal to the number of holes per volume in the valence band. Therefore, an intrinsic carriers concentration ni can be defined.

n = p = ni Intrinsic carrier concentration

Electron, hole and intrinsic carrier concentration. Fundamentals of Semicondutors

Ref.: M.S. Sze, Semiconductor Devices 52


2.2.7 Intrinsic carrier concentration Based on the intrinsic carrier concentration an intrinisc energy can be determined. For an intrinsic semiconductor in thermal equlibrium the intrinisc energy is equal to the Fermi energy.

EF (n = p = ni ) = Ei The electron and hole concentration is given by

p ≅ NV ⋅ exp(− (E F − EV ) kT )

n ≅ N C ⋅ exp(− (EC − E F ) kT )

NV ⋅ exp(− (Ei − EV ) kT ) = N C ⋅ exp(− (EC − Ei ) kT ) So that we can derive the following expression for the intrinsic energy.

EV + EC kT  NV   Ei = ln + 2 2  NC 


Intrinsic energy

53


2.2.7 Intrinsic carrier concentration The intrinsic energy is again a pure material parameter. The intrinsic energy is not affected by light exposure or pressure. The intrinsic energy is constant for a semiconductor even if the material is not in thermal equilibrium anymore (e.g. a voltage is applied to the sample). At room temperature the second term is much smaller than the first term. Therefore, the intrinsic energy is very close to the middle of the bandgap (EC-EV)/2=Eg/2. For silicon the intrinsic energy deviates from the middle of the bandgap by Ei-(EC+EV)/2≈-kT/2=-13meV. The intrinsic energy is shifted towards the valence band. For Gallium Arsenide the situation is opposite and the intrinsic energy is slightly shifted towards the conduction band: Ei(EC+EV)/2≈3kT/2=39meV


54


2.2.7 Intrinsic carrier concentration Based on n=p=ni the intrinsic carrier concentration can be expressed in terms of the effective density of states for the electrons and holes.

 EC − Ei  N C = ni ⋅ exp   kT 

 E − EV  NV = ni ⋅ exp i   kT  So that the intrinsic concentration results to the following expression:

 Eg   ni = N C NV ⋅ exp −  2kT 

Intrinsic carrier concentration

Intrinsic carrier concentration for silicon and GaAs. Ref.: M.S. Sze, Semiconductor Devices


55


2.2.7 Intrinsic carrier concentration In the next step the expression for the carrier concentration (electrons) can be modified by describing the effective density of states as a function of the intrinsic carrier concentration. As a result a expression for the carrier concentration can be derived which does not require knowledge of the effective density of states for the material.

n ≅ N C ⋅ exp(− (EC − E F ) kT )

for

EC − EF ≥ 2kT

 E − Ei  N C = ni ⋅ exp C   kT 

 E − Ei  n = ni ⋅ exp F   kT   Ei − EF  p = ni ⋅ exp   kT  Fundamentals of Semicondutors


Hole concentration

56


2.2.8 Donor and Acceptors When a semiconductor is doped, the semiconductor becomes extrinsic and impurity levels are introduced. In the following the influence of acceptors and donors on the material properties will be discussed. We will focus here on the doping of silicon.

Schematic silicon lattice for n-type doping with donor atoms (arsenic or phosphorus).

If we introduce donors like arsenic and phosphorus in a silicon single crystal a silicon atom is replaced by an donor atom with five valence electrons. The arsenic or phosphorus atoms form covalent bonds with its neighboring silicon atoms. The 5th electron has a low binding energy to become a conducting electron. The arsenic or phosphorus atom is called a donor and the silicon becomes n-type because of the addition of the negative charge carrier.


57


2.2.8 Donor and Acceptors

If we introduce acceptors like boron in the silicon lattice a silicon atom is replaced by a boron atom with three valence electrons. Additional electrons are „accepted“ to form four covalent bonds. The boron atom is considered as an acceptor and the silicon becomes p-type because of the addition of the positive charge carrier. Schematic silicon lattice for p-type doping with donor atoms (boron).


58


2.2.8 Donor and Acceptors Periodic table of semiconductor materials


Elements out of column III and column V of the perodic table are of particualr interest to intentionally dope silicon. Elements out of column III form acceptor states, whereas elements from column V tend to form donor states.


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2.2.8 Donor and Acceptors The introduction of donors like arsenic in the silicon lattice leads to the formation of energy levels very close to the bottom of the conduction band. At room temperature the thermal energy kT is high enough to thermally excite the excess electron to the conduction band. As a consequence positively charged localized states are left in the material and free and mobile electrons are created in the conduction band. A donor state is neutral when it is occupied by an electron and becomes positively charged if the state donates its electron to the conduction band. Under such conditions the energy level of the donors is very close to the conduction band.

Energy

EC ED

Donor levels

EV

Schematic energy band representation of a semiconductor with donor ions.

Distance Fundamentals of Semicondutors

60


2.2.8 Donor and Acceptors With increasing donor concentration the Fermi level will shift closer to the bottom of the conduction band. Therefore, the energy difference between the Fermi level and the conduction band (EC-EF) gets smaller with increasing donor concentration.

Schematic Band Diagram

Density of States

Fermi-Dirac Distribution

Electron and hole Density M.S. Sze, Semiconductor Devices


61


2.2.8 Donor and Acceptors An analog behavior is observed for increasing acceptor concentration. The higher the acceptor concentration the closer the Fermi level will move to the valence band. At room temperature the thermal activation is already high enough to active an hole from the valence band. As a consequence the acceptor ions get negative and holes are created in the valence band. An acceptor is negatively charged when it is occupied it is occupied by and electron and becomes neutral after accepting an electron from the valence band.

Energy

EC

EA

Acceptor levels

EV

Schematic energy band representation of a semiconductor with acceptor ions.

Distance Fundamentals of Semicondutors

62


2.2.9 Electrons and Holes in Semiconductor The product of the electron and hole concentration is equal to the square of the intrinsic carrier concentration if the semiconductor is in thermal equilibrium. In this case it does not matter, whether the semiconductor is an intrinsic semiconductor or an extrinsic semiconductor. In the second case the semiconductor is doped by acceptors or donors. If the semiconductor is intrinsic the following relationship applies

p = n = ni

and

p ⋅ n = ni2

Intrinsic semiconductor in thermal equilibrium

Doping of a semiconductor leads to the following relationship

p ≠ n, p ≠ ni , n ≠ ni Fundamentals of Semicondutors

and

p⋅n =

ni2

Extrinsic semiconductor in thermal equilibrium 63


2.2.9 Electrons and Holes in Semiconductor If a semiconductor samples is uniformly doped (no internal electric field) and no electric field is applied (external electric field) the semiconductor is neutral. In this case charge neutrality applies. To preserve charge neutrality, the total negative charges (electrons and ionized acceptors) must equal the total positive charges (holes and ionized donors).

n + NA = p + N D

Charge neutrality

If we assume that the material is only doped by donors so that NA=0 the equation is simplified to n=p+ND. Therefore, the semiconductor is an n-type semiconductor. The hole concentration can now be calculated by

pn =

ni2

nn

Hole concentration for an n-type semiconductor

where the index n indicates that we deal with a n-type semiconductor.


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2.2.9 Electrons and Holes in Semiconductor The following expression for the electron concentration can be derived:

nn =

(

1 N D + N D2 + 4ni2 2

)

Electron concentration for an n-type semiconductor

In most of the cases we can assume that the Donor concentration is higher than the intrinsic carrier concentration so that the expression is reduced to

nn ≈ N D

Complete ionization for an n-type semiconductor

If the electron concentration is approximately equal to the Donor concentration complete ionization can be assumed. Complete ionization is observed for (shallow) donors and acceptors, which means that the introduced impurities form defect levels very close to the bands.


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2.2.9 Electrons and Holes in Semiconductor Consequently we get the following term for the hole concentration :

pn = ni2 N D So that the Fermi level can be calculated by using the Boltzmann distribution

N  E F ≈ EC − kT ⋅ ln C   ND 

Fermi level for an n-type semiconductor

The analog behavior can be observed for a p-type doped semiconductor. If we assume that donor concentration is ND=0 we get the following expression for the holes: p=n+NA. The electron concentration can be described by

n p = ni2 p p Fundamentals of Semicondutors

Electron concentration for an p-type semiconductor 66


2.2.9 Electrons and Holes in Semiconductor Subsequently the following expression is obtained for the hole concentration :

(

1 p p = N A + N A2 + 4ni2 2

)

Hole centration for an ptype semiconductor

If we again assume that the defect levels are very close to the band (valence band) most of the acceptors will be ionized so that

pp ≈ N A

Complete ionization for an p-type semiconductor

So that the Fermi level can be calculated by using the Boltzmann distribution

 NV   E F ≈ EV + kT ⋅ ln  NA  Fundamentals of Semicondutors

Fermi level for an p-type semiconductor 67


2.2.9 Electrons and Holes in Semiconductor Various impurities in silicon and gallium arsenide

Si

Measured ionization engeries for various impurities in silicon and GaAs.

GaAs



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2.2.9 Electrons and Holes in Semiconductor Influence of the Doping Concentration on the Fermi Level The energetic position of the Fermi level depends on the concentration of the dopants and the temperature. With increasing temperature the Fermi distribution is getting broader so that the Fermi level is closer to the intrinsic energy level. With increasing doped concentration the Fermi level shifts closer to the bands (conduction and valence band). This behavior is similar for all semiconductor materials.

Influence of the temperature and the doping concentration on the Fermi level in silicon.



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2.2.10 Compensated Semiconductor So far either n-type or p-type semiconductors were considered in the discussion. However, every often in microelectronics the material is doped by donors and acceptors. For example a p-type wafer is doped with arsenic (n-type region) so that a pn-junction is formed. In this case the semiconductor is compensated. In order to preserve charge neutrality both dopant concentrations have to be considered.

n + NA = p + N D

Charge neutrality

However, in most of the cases the concentration of one dopant species is much higher than the concentration of the other species so that the semiconductor properties are determined by the higher dopant concentration.


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2.2.10 Compensated Semiconductor

ND > N A Assumption: (n-type semiconductor)

[

1 nn = ⋅ N D + N D2 + 4ni2 2 N D >> ni ⇒ nn ≈ N D

]

Assumption: (p-type semiconductor)

[

1 p p = ⋅ N A + N A2 + 4ni2 2 N A >> ni ⇒ p p ≈ N A Majority carriers (p-type semiconductor)


[

1 ⋅ N D + N D2 + 4ni2 2 N D >> ni ⇒ pn ≈ ni2 N D

]

Minority carriers (n-type semiconductor)

Majority carriers (n-type semiconductor)

N A > ND

pn =

ni2

]

np =

[

ni2

1 ⋅ N A + N A2 + 4ni2 2 N A >> ni ⇒ n p ≈ ni2 N A

]

Minority carriers (p-type semiconductor) 71


2.2.11 Minority and Majority Carriers As complete ionization can assumed for typical dopants like arsenic or boron the concentration of free carriers is more or less controlled by the dopant concentration. If for example silicon is doped by arsenic the concentration of electrons in the conduction band is much higher than the concentration of holes in the valence band. In this case the electrons in the conduction band are majority carriers and the holes in the valence band are minority carriers. As the name implies, the electrons represent the majority of carriers and the holes represent the minority of carriers. The analog behavior is observed for boron doped material. Here the concentration of holes in the valance is much higher than the concentration of electrons in the conduction band. Consequently the holes are the majority carriers, whereas the electrons are the minority carriers. Electrons are majority and holes are minority carriers in n-type materials! Holes are majority and electrons are minority carriers in p-type materials! For bipolar electronic devices like diodes (e.g. solar cells, LED) or bipolar transistors the electronic transport is controlled by the minority carriers, because the electronic transport is limited by the number or the lifetime of minority carriers. Fundamentals of Semicondutors

72


2.2.12 Degenerated and Non-degenerated Semiconductors For most of the electronic devices the electron and hole concentration is much lower than the effective density of states in the conduction or the valence band. The Fermi level is at least 3kT above the valence band or 3kT below the conduction band. In such a case we speak about a nondegenerated semiconductor. For very high levels of doping the concentration of dopants gets higher than the effective density of states in the valence or the conduction band. In such a case the semiconductor is degenerated and the Fermi levels shifts into the conduction or the valence band. Under such conditions the equations which were derived here does not apply any more. However, the fabrication of degenerated semiconducting materials can be necessary. For example the fabrication of laser diodes require population inversion, which can only achieve if the semiconductor is degenerated.


73


2.2.13 Bulk potential In following we will introduce the bulk potential. The bulk potential is an important parameter if it comes to the explanation of bipolar devices like diodes or bipolar transistors. The bulk potential is directly related to the Fermi level in a material. Therefore, the position of the Fermi level can be expressed by the bulk potential or vice versa. The electron and the hole concentration of an intrinsic semiconductor can be expressed in terms of the intrinsic carrier concentration.

 E − Ei  n = ni ⋅ exp F  kT  

 Ei − EF  p = ni ⋅ exp   kT 

Instead of using the energy difference between the intrinsic energy level and the Fermi level the term can be substituted by the bulk potential.

1 ϕb = − ⋅ (Ei − EF ) q Fundamentals of Semicondutors

Bulk potential

74


2.2.13 Bulk potential The bulk potential is a measure of the energy difference between the intrinsic energy level and the Fermi level. „Bulk“ implies that this parameter is related to the bulk/volume properties of a semiconductor. The complementary term would be the surfec potential, which corresponds to the potential at the surface of a semiconductor. The term surface potential will be introduced in chapter 6, Furthermore, the Boltzmann equation can be simplified by using the temperature voltage

Vth = kT q

Temperature voltage

so that electron and hole concentration results to

n = ni ⋅ exp(ϕb Vth )


p = ni ⋅ exp(− ϕb Vth )

Hole concentration

Therefore, the bulk potential is directly related with the carrier concentration. Fundamentals of Semicondutors

75


2.2.13 Bulk potential In order to directly relate the bulk potential with the material properties we have to rewrite the equation. For an n-type semiconductor the bulk potential results to

(

)

 1 2 2  ϕbn = Vth ⋅ ln  N D + N D + 4 ⋅ ni    2 ⋅ ni In most of the cases the Donor concentration is large than the intrinsic carrier concentration so that:

 ND   > 0 ϕbn ≈ Vth ⋅ ln  ni 

Bulk potential for an n-type semiconductor

Accordingly we can derive an expression for an p-type semiconductor.

 NA   < 0  ni 

ϕbp = −Vth ⋅ ln Fundamentals of Semicondutors

Bulk potential for an p-type semiconductor

76


References Michael Shur, Introduction to Electronic Devices, John Wiley & Sons; (January 1996). (Price: US$100) Simon M. Sze, Semiconductor Devices, Physics and Technology, John Wiley & Sons; 2nd Edition (2001). (Price: US$115) R.F. Pierret, G.W. Neudeck, Modular Series on Solid State Devices, Volumes in the Series: Semicondcutor Fundamentals, The pn junction diode, The bipolar junction transistor, Field effect devices, (Price: US$25 per book)


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