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

Introduction to Electronic Devices (Course Number 300331) Fall 2006 Electronic Transport Information: http://www.faculty.iu- Dr. Dietmar Knipp bremen.de/dknipp/ Assistant Professor of Electrical Engineering

Source: Apple

Ref.: Apple

Ref.: IBM Critical 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 1 101 dimension (m) Ref.: Palo Alto Research Center

Electronic Transport 1 Introduction to Electronic Devices, Fall 2006, Dr. Dietmar Knipp Introduction to Electronic Devices 3 Electronic Transport 3.1 Thermal movement of carriers 3.2 Carrier Drift 3.3 Carrier Diffusion 3.4 Einstein Relation 3.5 Current Density Equation 3.6 Thermal equilibrium processes 3.7 Non-thermal equilibrium processes 3.8 Generation Processes 3.9 Recombination Processes 3.9.1 Direct (Band-to-band) recombination 3.9.2 Donor-Acceptor recombination 3.9.3 Trapping 3.9.4 Auger recombination 3.10 Quasi Neutral Semiconductor 3.11 Quasi Fermi Levels 3.12 Continuity Equation 3.12 Poisson Equation 3.14 Summary of Semiconductor Equations References

Electronic Transport 2 Introduction to Electronic Devices, Fall 2006, Dr. Dietmar Knipp

3 Electronic Transport Electrons in the conduction band and holes in the valence band are able to move upon thermal activation, a gradient or an applied electric field. In the following the concepts of electronic transport in crystalline materials will be described.

3.1 Thermal movement of carriers Electrons in the conduction or holes in the valence band can essentially be treated as free carriers or free particles. Even in the absence of an electric field the carriers follow a thermally activated random motion. In thermal equilibrium the average thermal energy of a particle (electron or hole) can be obtained from the theorem for equipartition of 3 E thermal = kT Average thermal energy of an electron / hole average 2 The thermal energy of the particle is equal to the kinetic energy of the electron, so that the velocity of the particle can be calculated. The mass of the electron is equal to the effective mass of the electron.

Electronic Transport 3 Introduction to Electronic Devices, Fall 2006, Dr. Dietmar Knipp

3.1 Thermal movement of carriers

Furthermore, the velocity of the electron corresponds to the thermal velocity of the electron, so that the thermal velocity can be determined by:

1 2 E = m v Kinetic energy of an electron / hole kin 2 eff th

At room temperature the average thermal velocity of an electron is about 105m/s in silicon and GaAs. 3kT vth = Thermal velocity of an electron meff Thermal motion of free carriers can be seen as random collision (scattering) of the free carriers with the crystal lattice. A random motion of an electron or hole leads to zero net displacement of the free carrier over a sufficient long distance / period of time. The average distance between two collisions within the crystal lattice is called mean free path. Associated to the mean free path we can introduce a mean free time τ. A typical mean free path is in the range of 100nm and the mean free time is in the range of 1ps.

Electronic Transport 4 Introduction to Electronic Devices, Fall 2006, Dr. Dietmar Knipp

3.2 Carrier Drift When a small electric field is applied to the semiconductor material each free carrier will experience an electro static force Force = −qF So that the carrier is accelerated along the field (in opposite direction of the field).

F=0 F

Schematic path of an electron in a semiconductor (a) random thermal motion, (b) combined motion due to random thermal motion and an applied electric field. Ref.: M.S. Sze, Semiconductor Devices

Electronic Transport 5 Introduction to Electronic Devices, Fall 2006, Dr. Dietmar Knipp

3.2 Carrier Drift An additional velocity component will be superimposed upon the thermal motion of the electron. The additional velocity is caused by an applied electric field F. The additional component is called drift velocity. The drift of the electrons can be described by a steady state motion since the gained momentum is lost due to collisions of the electrons and the lattice.

P = mnvn P = −q ⋅ F ⋅τ C

Based on momentum conservation the drift velocity can be calculated. The drift velocity is proportional to the applied electric field F.

qτ C vn = − F Electron drift velocity mn

Electronic Transport 6 Introduction to Electronic Devices, Fall 2006, Dr. Dietmar Knipp

3.2 Carrier Drift

The electric field is multiplied with a prefactor. The prefactor depends on the mean free time, the effective mass of the free carrier, and the charge of the carrier. The prefactor is called

the hole or electron mobility µn,p in units of cm2/Vs.

Drift velocity as a function of the electric field in silicon and GaAs. Ref.: M.S. Sze, Semiconductor Devices

Electronic Transport 7 Introduction to Electronic Devices, Fall 2006, Dr. Dietmar Knipp

3.2 Carrier Drift

q ⋅τ q ⋅τ C V µ ≡ µ p ≡ Electron and hole mobility n m mn p The mobility is an important electronic transport parameter. The mobility is directly related to the material properties. Rewriting of the expression for the drift velocity leads to

v vp µ = n µ = Electron and hole mobility n F p F

The mobility is directly related to the mean free time between two collisions, which is determined by various scattering mechanisms. The most important scattering mechanisms are lattice scattering and impurity scattering. Lattice scattering is caused by thermal vibrations of the lattice atoms at any temperature above 0K. Due to the vibrations energy can be transferred from the carriers and the lattice.

Electronic Transport 8 Introduction to Electronic Devices, Fall 2006, Dr. Dietmar Knipp

3.2 Carrier Drift

Since lattice vibrations increase with increasing temperature the influence of lattice scattering is getting the determinant effect at high temperatures. As a consequence the mobility is reduced. Theoretical analysis show that the mobility is decreased in proportion to T-3/2. Impurity scattering is observed when a charged carrier (in our case a free carrier) interacts with dopants / impurities (acceptors or donors). The charged carriers will be deflected due to Coulomb interaction between the two charges. The probability of impurity scattering depends on the total concentration of impurities (sum of the positively and negatively charged ions). Impurity scattering is becoming less dominant for higher temperatures. Theoretical 3/2 calculations reveal that the impurity scattering scales with T /NT, where NT is the total impurity concentration. The probability of a collision can be expressed in terms of the mean free time τ 1 τ 1 1 = + Probability of a collision C C, lattice τ C, impurity

Electronic Transport 9 Introduction to Electronic Devices, Fall 2006, Dr. Dietmar Knipp

3.2 Carrier Drift The probability of a collision is

proportional to 1/τC. The mobility can be described by

µ1 1 1 = µ + L µI

where µL is the mobility as a function of the lattice scattering and µI is the mobility as a function of the impurity concentration.

Measured temperature dependent electron mobility in silicon for various donor concentrations. The insert shows the limiting electron transport mechanisms for high and low temperatures. Ref.: M.S. Sze, Semiconductor Devices

Electronic Transport 10 Introduction to Electronic Devices, Fall 2006, Dr. Dietmar Knipp

3.2 Carrier Drift

Measured electron and hole mobilities in silicon and gallium arsenide at 300K for various impurity concentrations.

Ref.: M.S. Sze, Semiconductor Devices

Electronic Transport 11 Introduction to Electronic Devices, Fall 2006, Dr. Dietmar Knipp

3.2 Carrier Drift In the next step the influence of electric fields on the energy bands will be discussed. Applying an electric field to a semiconductor leads to a “tilt” of the energy bands. The tilt of the band is called band bending. The contacts are considered to be ohmic (ideal contacts) The behavior of contacts will be discussed as part of the sessions on diodes.

F

Conduction process in an n-type semiconductor under biasing conditions. Ref.: M.S. Sze, Semiconductor Devices

Electronic Transport 12 Introduction to Electronic Devices, Fall 2006, Dr. Dietmar Knipp

3.2 Carrier Drift The potential energy of the charges in the bands (electrons in the conduction band and holes in the valence band) can be described in terms of the energy of the conduction band and the valence band. The potential energies are related to electric potential ϕ as follows:

EC = −qϕ + const.

EV = −qϕ − Eg + const.

If an electric potential is applied to the semiconductor the bands get “tilted”. The charges experience a force due the electric field. The force applied to the charges is given by the negative gradient of the potential energy leading to

− qF = −grad()EC = −q ⋅ grad(ϕ )

Therefore, the electric field applied to an electrons is equal to the negative gradient of the electric potential.

Electronic Transport 13 Introduction to Electronic Devices, Fall 2006, Dr. Dietmar Knipp

3.2 Carrier Drift If we reduce the problem to one dimension the equation simplifies to

E dϕ − qF = − C = −q dx dx

The bottom of the conduction band corresponds to the electric potential of the electron. Since we are interested in the gradient of the energy and not the energy itself, we can use any energy level of the band diagram. We will therefore simply use the intrinsic energy level.

1 dE 1 dE F = C = i q dx q dx

The potential energy of homogenous semiconductor increases linearly with the distance. Therefore, the electric field distribution throughout the semiconductor is constant.

Electronic Transport 14 Introduction to Electronic Devices, Fall 2006, Dr. Dietmar Knipp

3.2 Carrier Drift The transport of carriers under the influence of an applied electric field produces a current called drift current. The drift current is calculated by

Jn = qnvn = qnµnF Electron drift current density

Hole drift current density J p = qpvp = qpµ pF

2 where Jn,p is the drift current density in [A/cm ] and n, p are the carrier concentration per cm3.

Current flow through a uniformly doped semiconductor

Ref.: M.S. Sze, Semiconductor Devices

Electronic Transport 15 Introduction to Electronic Devices, Fall 2006, Dr. Dietmar Knipp

3.2 Carrier Drift The overall current flowing through a semiconductor sample is equal to the sum of electron and the hole current density.

Drift current density J = J p + Jn = qpµ pF + qnµnF

Based on the expression for the current densities we can derive the following terms for the electron and hole conductivity

σ n = qnµn σ p = qpµ p Electron and hole conductivity

So that we get the following expression for the overall conductivity.

σ = σ n +σ p = qpµ p + qnµn =1 ρ Conductivity

The inverse of the conductivity is the resistivity.

Electronic Transport 16 Introduction to Electronic Devices, Fall 2006, Dr. Dietmar Knipp

3.2 Carrier Drift Therefore, we can directly relate the conductivity and the resistivity with the current density.

σ F J = F = Drift current density ρ

In extrinsic semiconductors, only one component is significant, because of the distinct difference in the carrier densities.

σ ≈ σ = qnµ Conductivity of a n-type n n semiconductor Conductivity of a p-type σ ≈ σ p = qpµ p semiconductor

Electronic Transport 17 Introduction to Electronic Devices, Fall 2006, Dr. Dietmar Knipp

3.2 Carrier Drift

Measured room temperature resistivity for n-type and p-type silicon and gallium arsenide as a function of the impurity concentration.

Ref.: M.S. Sze, Semiconductor Devices

Electronic Transport 18 Introduction to Electronic Devices, Fall 2006, Dr. Dietmar Knipp

3.3 Carrier Diffusion So far we assumed that the semiconductor is uniformly doped. As a consequence the concentration of free charges in the semiconductor is uniform. Charges are moved by an electric field applied to the sample. In the case of a non uniform doping profile (e.g. in a diode) the influence of carrier diffusion has to be considered. A diffusion process is the consequence of a carrier concentration gradient or spatial variation of the carrier concentration in the semiconductor. An understanding of diffusion processes is essential to describe the electronic transport in a pn-junction and a bipolar transistors.

dn/dx dp/dx

Illustration of carrier n p - n(x) + p(x) diffusion in an gradually doped semiconductor. jnD jpD

x

Electronic Transport 19 Introduction to Electronic Devices, Fall 2006, Dr. Dietmar Knipp

3.3 Carrier Diffusion

In order to illustrate the diffusion of carriers the semiconductor can be separated in to finite boxes. The concentration of carriers increases from right to the left. Due to thermal energy the carriers move around in order to achieve thermal equilibrium. As a consequence of the thermal activated movement a net (diffusion) charges towards the right is observed.

Illustration of the diffusion current.

x

Ref.: M. Böhm, Semiconductor Electronics

Electronic Transport 20 Introduction to Electronic Devices, Fall 2006, Dr. Dietmar Knipp

3.3 Carrier Diffusion The carriers tend to move from the region of high carrier concentration to the region of low carrier concentration. The movement is caused by thermal energy. As a consequence the flow of a diffusion current is observed. In the following it is assumed that the temperature and activation energy throughout the semiconductor is constant. Only the carrier concentration varies throughout the semiconductor. If that is the case the electron diffusion current can be described by

dn j = qD Electron diffusion current density nD n dx

The diffusion current is proportional to the spatial derivate of the electron concentration. The diffusion current results from the random thermal motion of carriers in a concentration gradient. The electron diffusion current density is 2 measured in A/cm . The constant Dn is the electron diffusion coefficient or electron diffusivity. The dimension of the diffusion constant is given in cm2/s. The diffusion coefficient is defined as the product of the thermal velocity and the mean free path length for electrons.

Electronic Transport 21 Introduction to Electronic Devices, Fall 2006, Dr. Dietmar Knipp

3.3 Carrier Diffusion The mean free path length can be expressed as the product of the thermal velocity and the meanτ free time. τ 2 Dn ≡ vthl = vth ⋅()vth µC = vth ⋅τ C 2 · The square of the thermal velocity can be substituted by 2E=(vth) mn =kT, so that the diffusion coefficient can be relatedµ to the mobility  m   kT  m  kT 2 2 n n   n n Dn = vth C = vth ⋅  =    = µn  q   mn  q  q The hole diffusion current can be expressed in a similar way. The electron current density is assumed to be positive, whereas the hole current density is

negative. The constant Dp is the hole diffusion coefficient or hole diffusivity.

dp j = −qD Hole diffusion current density pD p dx

Electronic Transport 22 Introduction to Electronic Devices, Fall 2006, Dr. Dietmar Knipp

3.4 Einstein Relation The overall current density is given by the sum of the electron and the hole current density.

jD = jnD + jpD Diffusion current density

The relationship between the diffusivity and the mobility is given by the Einstein relationship. The Einstein relation correlated the diffusion with the drift properties of the carriers.

 kT  Dn =   ⋅ µn Electron Diffusion coefficient  q 

 kT  Hole Diffusion coefficient Dp =   ⋅ µ p  q 

Electronic Transport 23 Introduction to Electronic Devices, Fall 2006, Dr. Dietmar Knipp

3.5 Current Density Equation The electronic transport in a semiconductor can be described by the drift and the diffusion of carriers. The overall current flowing through a semiconductor is given by the sum of the drift and the diffusion current densities.

dn J = J + J = qµ nF + qD Electron current n nF nD n n dx density equation

dp J = J + J = qµ nF − qD Hole current p pF pD p p dx density equation

J J J = n + p Current density equation

Electronic Transport 24 Introduction to Electronic Devices, Fall 2006, Dr. Dietmar Knipp

3.5 Current Density Equation The electric field and the diffusion of carriers have an influence on the Fermi level. Therefore, we can express the current density as a function of the Fermi level: 1 dE F = ⋅ i Electric field expressed in terms µ q dx of the intrinsic energy level J p = J pF + J pD dp = q pF − qD µ p p dx kT D = µ Einstein relation p q p  1 dEi  dp = q p p ⋅  − kTµ p  q dx  dx

dp p  dEi dEF   Ei − EF  = ⋅ −  p = ni ⋅exp   kT  dx kT  dx dx  Boltzman distribution Derivative of the carrier concentration

Electronic Transport 25 Introduction to Electronic Devices, Fall 2006, Dr. Dietmar Knipp

3.5 Current Density Equation µ dp p  dE dE  = ⋅ i − F  dx kT  dx dx   1 dEi  dp J p = q p p ⋅  − kTµ p  q dx  dx µ

dEi  dEi dEF  J p = p p − pµ p  −  dx  dx dx 

dE J = pµ F Hole current density equation p p dx

dEF J = nµ Electron current density equation n n dx

Electronic Transport 26 Introduction to Electronic Devices, Fall 2006, Dr. Dietmar Knipp

3.6 Thermal equilibrium processes

A semicondutor sample or an electronic device is in thermal equlibrium if the Fermi level throughout the entire sample is constant. As a consequence the overall current flow through sample is zero.

dE dE J = pµ F = 0 J = nµ F = 0 Thermal equilibrium p p dx n n dx This means that the current flow for each position in the sample is zero. Therefore, the electron and the hole current has to be zero. This can only be achieved if the drift current is equal to the diffusion current. This means that the current caused by the electric field is compensated by the current caused by the carrier gradient. j = 0 jdrift = − jdiff The drift current is equal to the diffusion current n n n for electrons in thermal equilibrium

drift diff The drift current is equal to the diffusion current jp = 0 jp = − jp for holes in thermal equilibrium

Electronic Transport 27 Introduction to Electronic Devices, Fall 2006, Dr. Dietmar Knipp

3.7 Non-thermal equilibrium processes In thermal equilibrium the product of the electron and hole concentration is 2 equal to the square of the intrinsic carrier concentration (n·p=ni ). If additional carriers are introduced in the semiconductor the thermal equilibrium is disturbed and the product of the electron and hole concentration is larger than 2 the square of the intrinsic carrier concentration (n·p>ni ). The semiconductor is now in non-thermal equilibrium. Additional carriers can be introduced by the injection of carriers via contacts (carrier injection) or the generation of carriers via light exposure of the semiconductor. For example forward biasing of a diode leads to the injection of carriers. Additional carriers can be generated by illuminating a sample with light. For example the absorbed light leads to the generation of additional carriers in a photodiode. During device operation an electric field is applied to the semiconductor device device which leads for example to the injection of charge. Subsequently the device is in non-thermal equilibrium. The thermal equilibrium of a semiconductor can be restored by the recombination of the introduced carriers.

Electronic Transport 28 Introduction to Electronic Devices, Fall 2006, Dr. Dietmar Knipp

3.7 Non-thermal equilibrium processes Depending on the nature of the recombination process the released energy can be emitted as a photon or dissipated as heat to the lattice. When a photon is emitted the process is called radiative recombination. Otherwise the process is a non-radiative recombination process. In the following different generation and recombination processes will be discussed.

3.8 Generation processes If sufficient thermal or optical energy is provide an electron can be excited from the valence band to the conduction band. In terms of the energy band diagram the process can be described as the generation of an electron/hole pair. The thermal or optical energy enables the electron to transfer from the valence band to the conduction band leaving a hole in the valence band behind. This process is called the generation of carriers. The generation of carriers is represented by the (carrier) generation rate G (number of electron-hole pairs generated per cm3 and second).

Electronic Transport 29 Introduction to Electronic Devices, Fall 2006, Dr. Dietmar Knipp

3.8 Generation processes The inverse process – the transition from the conduction band to the valence band – is called recombination. The process is represented by the (carrier) recombination rate R. The generation rate is larger than zero, G>0, if the carrier concentration is below the carrier concentration in thermal equilibrium. The generation rate is G>0 for diodes under reverse bias (extraction of carriers) and under illumination conditions. For all optoelectronic devices like solar cells, optical detectors the generation rate is larger than zero G>0. The energy of the photon has be be high enough to excite an electron from the valence band to the conduction band. With other words: The excitation has to be high enough to separate an electron hole pair.

Direct generation and recombination of electron-hole pairs.

Ref.: M.S. Sze, Semiconductor Devices

Electronic Transport 30 Introduction to Electronic Devices, Fall 2006, Dr. Dietmar Knipp

3.9 Recombination processes The recombination rate is larger than zero (R>0) if the carrier concentration is higher than the carrier concentration in thermal equilibrium (Injection of carriers). The recombination rate is larger than zero (R>0) for forward biased diodes. Carriers are injected via the electrodes. It has to be distinguished between radiative and non-radiative recombination. In the case of radiative recombination the energy is released via the emission of a photons. In the case of non-radiative recombination the energy is dissipated via heat to the lattice (leads to lattice scattering / phonon scattering). Radiative and non-radiative processes can be again classified. Direct or band-to-band recombination is a radiative recombination process. The same applies for radiative band-to-impurity recombination processes. Typical non- radiative recombination processes are Auger recombination or recombination via impurities and/or traps.

Electronic Transport 31 Introduction to Electronic Devices, Fall 2006, Dr. Dietmar Knipp

3.9 Recombination processes In the following the different recombination processes will be discussed.

Radiative Recombination Non-radiative Recombination

Band-to-band Donor Acceptor Auger Impurity (trap) recombination recombination recombination recombination

Eth

EC E E EC C C ED Eth

Et

E E E E A th V V EV EV

Ref.: M. Böhm, Semiconductor Electronics

Electronic Transport 32 Introduction to Electronic Devices, Fall 2006, Dr. Dietmar Knipp

3.9.1 Direct (Band-to-band) recombination When it comes to recombination in a direct-bandgap semiconductor the probability is high that electrons and holes directly recombine. In this case the carriers will directly recombination from the bottom of the conduction band to the top of the valence band. The rate of recombination is proportional to the electron and hole concentration. R ∝ n ⋅ p In thermal equilibrium the recombination rate is equal to the generation rate

Gth = Rth and 2 Rth ∝ n0 ⋅ p0 = ni The recombination rate is proportional to the product of the electron concentration and hole concentration in thermal equilibrium. The subscript 0 indicates the thermal equilibrium conditions.

Electronic Transport 33 Introduction to Electronic Devices, Fall 2006, Dr. Dietmar Knipp

3.9.1 Direct (Band-to-band) recombination If we shine light on a semiconductor and the photon energy is larger than the

bandgap electron-hole pairs are created at a generation rate GL and the carrier concentration is above the equilibrium.

G = GL + Gth The recombination rate can be described by

R ∝ np − n0 p0

To maintain charge neutrality the access carrier concentration ∆n=n-n0 for electrons and the access carrier concentration for holes ∆p=p-p0 has to be equal (quasi neutral semiconductor). Indirect semiconductors like silicon are typically characterized by non-radiative recombination processes. Carriers typically recombine via impurities or traps. Shockley, Reed and Hall developed a theory to describe the recombination for such recombination processes. The theory is called Shockley Reed Hall theory (SRH). The SRH will not be derived as part of the lectures.

Electronic Transport 34 Introduction to Electronic Devices, Fall 2006, Dr. Dietmar Knipp

3.9.2 Donor-Acceptor Recombination

Donor-Acceptor recombination is comparable with direct recombination. It is a radiative recombination process and the energy is emitted via photons. The recombination rate is proportional to the product of electrons and holes.

3.9.3 Trapping Trapping processes are non-symmetric recombination processes. Either electron or holes are captured in „traps“. The carriers are captured or “trapped” for a certain period time before they are released again. Accordingly, we can define a carrier lifetime. The lifetime is the average time period between the release and the capturing of a carrier by a trap. Throughout this time period the carrier stays in the conduction or the valence band. For example with increasing doping or impurity concentration the lifetime of carriers is reduced. Trapping is well know for all semiconducting materials. However, trapping plays a major for less structural ordered semiconducting materials. Such effects are observed for amorphous materials like amorphous silicon, interfaces (e.g. silicon / silicon nitride interface). Traps can lead to hysteresses effects. As consequence parameters of electronic devices (e.g. threshold voltage) change over time.

Electronic Transport 35 Introduction to Electronic Devices, Fall 2006, Dr. Dietmar Knipp

3.9.4 Auger Recombination

For highly doped materials an additional recombination mechanism plays an important role: Auger recombination. In this case an electron recombines with a hole without involving a trap level. The released energy is dissipated by the crystal lattice via vibrations (heat). Since two electrons and a hole are involved in the process the recombination rate is proportional to

2 RAuger ∝ n ⋅ p

Electronic Transport 36 Introduction to Electronic Devices, Fall 2006, Dr. Dietmar Knipp

3.10 Quasi Neutral Semiconductor The equations derived so far can be applied for semiconductors under thermal equilibrium. However, electronic devices in operation are usually in non-thermal equilibrium. In the following the concept of quasi neutral semiconductors will be introduced which allows a simple and analytical description of electronic devices in non-thermal equilibrium. In order to describe the electronic devices the term neutral or quasi neutral will be introduced. For example a semiconductor can be considered to be quasi neutral if electrons and holes are generated uniformly throughout a semiconductor due to light exposure. In this case the semiconductor remains neutral or quasi neutral even though the semiconductor is not in thermal equilibrium anymore.

2 Mass-action law in Thermal equilibrium np = ni

2 np ≠ ni Mass-action law in Non-Thermal equilibrium

Electronic Transport 37 Introduction to Electronic Devices, Fall 2006, Dr. Dietmar Knipp

3.10 Quasi Neutral Semiconductor The electron and hole concentration in quasi neutral semiconductors can be described by the electron/hole concentration in thermal equilibrium plus the access carrier concentration for electrons and holes.

n = ∆n + n p = ∆p + p Carrier concentration in a 0 0 quasi neutral semiconductor so that

2 ()()∆n + n0 ⋅ ∆p + p0 ≠ ni

Quasi neutrality assumes now that

2 n0 p0 = ni and ∆n ≈ ∆p Quasi Charge Neutrality

In the following the concept of the Fermi-Dirac Statistic will be applied to quasi neutral semiconductors.

Electronic Transport 38 Introduction to Electronic Devices, Fall 2006, Dr. Dietmar Knipp

3.11 Quasi Fermi Levels Under thermal equilibrium conditions, the electron and hole concentration of a non-degenerated semiconductor can be described as a function of the Fermi level (Boltzmann distribution):

 EF − EC  Electron concentration n = NC ⋅ exp   kT 

 EV − EF  p = NV ⋅ exp   kT  Hole concentration

Under non-thermal equilibrium conditions these equations are no longer valid. The product of electron and hole concentration is larger (generation of carriers) or smaller (recombination of carriers) than the square of the intrinsic carrier concentration. Under such conditions it seems appropriate to introduce quasi (n) (p) Fermi levels for electrons and holes. EF and EF are the electron and hole quasi Fermi levels, so that the expressions for the electron and hole concentration can be rewritten as

Electronic Transport 39 Introduction to Electronic Devices, Fall 2006, Dr. Dietmar Knipp

3.11 Quasi Fermi Levels  E()n − E   F C  n = NC ⋅ exp  Quasi electron concentration  kT   E − E(p)   V F  Quasi hole concentration p = NV ⋅ exp   kT  Under thermal equilibrium conditions the quasi Fermi levels for electrons and holes are identical, so that the Fermi potential is identical for electrons and holes.

()n p ( ) EF = EF = EF EF = −qϕF

(n) (p) Under non-equilibrium conditions EF is not equal to EF and both might be a function of the position and the time. The mass-action law can be rewritten as

ϕ  E()n − E(p)   p −ϕ n  2  F F  2  F F  Mass-Action law for p ⋅n = ni ⋅exp  = ni ⋅exp  quasi neutrality  kT   Vth 

Electronic Transport 40 Introduction to Electronic Devices, Fall 2006, Dr. Dietmar Knipp

3.11 Quasi Fermi Levels

2 (n) (p) Since in thermal equilibrium np=ni , the difference of EF and EF can be used as the measure of the deviation from the thermal equilibrium.

E E

EC EC n n EF ϕF

EF n p EF0 ϕb ϕb ϕF ϕb p p E ϕb F Ei Ei

EV EV ϕ=-E/q ϕ=-E/q Energy band and potential for an n- Energy band and potential for an n-type type semiconductor under thermal semiconductor under non-thermal equilibrium. equilibrium. Additional carriers are generated for example by light exposure.

Electronic Transport 41 Introduction to Electronic Devices, Fall 2006, Dr. Dietmar Knipp

3.11 Quasi Fermi Levels In terms of the Fermi level the transition from thermal equilibrium to non-thermal equilibrium is called the splitting of the Fermi level. The further the product of electrons and holes moves away from the square of the intrinsic carrier

concentration the further the Fermi level moves away from the Fermi level EF0 in thermal equilibrium. It is now important to distinguish two different cases: In the first case the product of the electron and hole concentration is getting larger than the square of the intrinsic carrier concentration. This is usually caused be an increase of the electron and the hole concentration. A typical example for an increasing concentration of carriers would be a forward biased diode. The concentration of electrons and holes increases and therefore the quasi Fermi levels for electrons and holes move closer to the conduction band and the valence band. In this case the term injection of carriers is used. Additional carriers are injected in the semiconductor. In the opposite case the number of electrons and holes is getting smaller, therefore, the product of the electron and hole concentration is getting smaller than the square of the intrinsic carrier concentration. Consequently the quasi Fermi levels are moving further away from the band. In such a case free carriers get extracted from the semiconductor. The term extraction of carriers is used.

Electronic Transport 42 Introduction to Electronic Devices, Fall 2006, Dr. Dietmar Knipp

3.11 Quasi Fermi Levels A typical example for extraction of carriers is a diode under reverse bias conditions. Under forward bias carriers are injected and under reverse bias carriers are extracted from the device. The influence of injection and extraction of carriers can be described by the following relationships: In the case of injection the product of electrons and holes is larger than the square of the intrinsic carrier concentration. The energy difference of the quasi Fermi levels for the electrons and holes is positive and the difference between the quasi Fermi potential for the electrons and holes is negative. The opposite is observed for the extraction of carriers. The product of the electron and hole concentration is smaller than the square of the intrinsic carrier concentration and subsequently the difference of the electron and hole quasi Fermi level is negative, whereas the difference of the electron and the hole quasi Fermi potential is getting positive.

2 ()n (p) ϕ n p Injection n ⋅ p > ni EF − EF > 0 F −ϕF < 0 ϕ 2 ()n (p) n p Extraction n ⋅ p < ni EF − EF < 0 F −ϕF > 0

Electronic Transport 43 Introduction to Electronic Devices, Fall 2006, Dr. Dietmar Knipp

3.11 Quasi Fermi Levels Injection and Extraction for an n-type semiconductor

Injection Extraction E E E C EC E n p n F EF ϕF p ϕF n p EF0 ϕ ϕ p n EF0 b ϕb F ϕ ϕb ϕ p b F p E n n ϕb F ϕ EF E b i Ei

E V EV ϕ=-E/q ϕ=-E/q 2 2 n⋅ p > ni n⋅ p < ni ()n p ( ) ()n (p) EF − EF > 0 EF − EF < 0

Electronic Transport 44 Introduction to Electronic Devices, Fall 2006, Dr. Dietmar Knipp

3.11 Quasi Fermi Levels

As we move from a Fermi level to a quasi Fermi level a new expression for the current density has to be derived. dn j = qnµ F + qD Current density equation for electrons n n n dx

Again we have to express dn/dx and F in terms of the Fermi level.

 E n − E   F i  Boltzman distribution n = ni ⋅exp   kT 

n dn n dEF dEi  = ⋅  −  Derivative of the carrier concentration dx kT  dx dx 

Electronic Transport 45 Introduction to Electronic Devices, Fall 2006, Dr. Dietmar Knipp

3.11 Quasi Fermi Levels

dE n dE n n dE j = nµ i + qD ⋅ F − qD ⋅ i n n dx n kT dx n kT dx

kT D = ⋅ µ Einstein relation for electrons n q n

Total current density j = jn + jp dEn j = µ n F Current density equation for electrons n n dx dE p j = µ p F Current density equation for holes p p dx

Electronic Transport 46 Introduction to Electronic Devices, Fall 2006, Dr. Dietmar Knipp

3.12 Continuity Equation So far individual effects were discussed like the carrier drift as a consequence of an electric field, the diffusion of carriers as a consequence of a carrier concentration gradient, the generation and recombination through recombination centers. In the following the relationship between these effects will be studied. The generation of carriers can be quantitatively described by the generation rate, G, which corresponds to the concentration of electrons or holes generated within a second. The opposite behavior is quantitatively described by the recombination rate, R, which is equal to the concentration of electrons or holes recombining within a second. In an uniform semiconductor dn dp = = G − R dt dt Under thermal equilibrium G=R. However, for an electronic device under operation the assumption of an uniform semiconductor does not apply.

Electronic Transport 47 Introduction to Electronic Devices, Fall 2006, Dr. Dietmar Knipp

3.12 Continuity Equation To derive the continuity equation we consider an incremental volume of a semiconductor. Under such conditions the continuity equation is given by:

∂n 1 ∂J = n + G − R ∂t q ∂x Continuity equation for electrons

∂p 1 ∂J p = − + G − R Continuity equation for holes ∂t q ∂x

Illustration of the continuity - - - - equation based on an incremental - volume, where 3 electrons enter a - - volume, where 3 electrons get - generated and 2 electrons - - - - recombine. Consequently 4 x electrons leave the volume.

Electronic Transport 48 Introduction to Electronic Devices, Fall 2006, Dr. Dietmar Knipp

3.12 Continuity Equation

Under steady state conditions dn/dt=dp/dt=0, so that the contitnuity equation gets simplified: 1 ∂J n + G − R = 0 Electron Continuity equation under steady q ∂x state conditions 1 ∂J − p + G − R = 0 Hole Continuity equation under steady state q ∂x conditions

Now the continuity equations can be combined with the current density equation. The continuity equation can be rewritten in the form: µ ∂F ∂n ∂2n n + µ F + D + G − R = 0 n ∂x n ∂x n ∂x2

Continuity equation for electrons

Electronic Transport 49 Introduction to Electronic Devices, Fall 2006, Dr. Dietmar Knipp

3.12 Continuity Equation

µ ∂F ∂p ∂2 p − p − µ F + D + G − R = 0 p ∂x p ∂x p ∂x2 Continuity equation for holes

The continuity equation considers drift and diffusion contributions.

Electronic Transport 50 Introduction to Electronic Devices, Fall 2006, Dr. Dietmar Knipp

3.13 Poisson Equation

In addition to the continuity equation the Possion equation has to be satisfied, ϕ dF d 2 1 = − = ρ Poisson equation 2 ε ε s dx dx 0 s

where ρS is the space charge density and εS is the semiconductor permittivity. The space charge density is given by

ρs = q()p − n + N D − N A

Electronic Transport 51 Introduction to Electronic Devices, Fall 2006, Dr. Dietmar Knipp

3.13 Poisson Equation

Example of the space charge, electric field and potential distribution

Space charges + Position ρS ϕ - - dF d 2 1 = − = ρ 2 ε ε S dx dx 0 S

Electric Filed The electric field distribution F can be determined by the integration of the space charge density. The potential can be Potential determined by the integration of ϕ the electric field.

Electronic Transport 52 Introduction to Electronic Devices, Fall 2006, Dr. Dietmar Knipp

3.14 Summary of Semiconductor Equations µ dn dE n ∂n 1 dj j = q nF + qD ⋅ = µ n F = ⋅ n + G − R n n n dx n dx ∂t q dx

Current density for electrons Continuity equation for electrons µ dp dE p ∂p 1 dj j = q pF − qD ⋅ = µ p F = − ⋅ p + G − R p p p dx p dx ∂t q dx

Current density for holes Continuity equation for holes

ϕ dF d 2 1 = − = ρ 2 ε ε S dx dx 0 S Poisson equation

Electronic Transport 53 Introduction to Electronic Devices, Fall 2006, Dr. Dietmar Knipp

References Michael Shur, Introduction to Electronic Devices, John Wiley & Sons; (January 1996). (Price: US$100), Audience: under graduate students

Simon M. Sze, Semiconductor Devices, Physics and Technology, John Wiley & Sons; 2nd Edition (2001). (Price: US$115), Audience: under graduate students

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), Audience: under graduate students

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