Electronic Devices and Circuits : Basics of Semiconductor Physics

In this article, you will find the Study Notes on Semiconductors-1 which will cover the topics such as Introduction, Semiconductor materials, intrinsic and extrinsic semiconductors, Properties of semiconductor materials, and important terms.

1. Introduction

Electronic Devices: It is the branch of physics that deals with the emission and effects of electrons emission and effects of electrons and the use of electronic devices.
Electronic Circuits: Components are connected together to create an electronic circuit with a particular function. Active components are sometimes called devices.

2. Semiconductor Materials

The term conductor is applied to any material that will support a generous flow of charge when a voltage source of limited magnitude is applied across its terminals.
An insulator is a material that offers a very low level of conductivity under pressure from an applied voltage source.
A semiconductor, therefore, is a material that has a conductivity level somewhere between the extremes of an insulator and conductor.
Band Theory: A bonding of atoms, strengthened by the sharing of electrons, is called covalent bonding. In the crystal, closely spaced energy levels form a band called as energy band. Each orbit has a separate energy band. A band of energy levels associated with valence shells is called as valence band. Electrons from other bands cannot be removed but electrons from valence band can be removed by supplying a little energy. The conduction band is generally empty. The valence band and conduction band are separated by a gap called forbidden energy gap.
Compound Semiconductors: Such as Gallium Arsenide (GaAs), Cadmium Sulphide (CdS), Gallium Arsenide Phosphide (GaAsP), Gallium Nitride (GaN) are constructed by two or more semiconductor materials of different atomic structures are called compound semiconductors.

3. Intrinsic Semiconductors

At 0^oK, no free carriers are available, Si behaves as an insulator.
At room temperature, a few covalent bonds will be broken by the thermal energy, electron‐hole pair generation as free carriers.
Both electrons and holes are free to move, can contribute to current conduction

4. Extrinsic semiconductor

Extrinsic (doped) semiconductor = intrinsic semiconductor + impurities
According to the species of impurities, the extrinsic semiconductor can be either n‐type or p‐type.

n‐type semiconductor

The donor impurities have 5 valence electrons are added into silicon.
P, As Sb, are commonly used as a donor.
The Si atom is replaced by a donor atom.
Donor ions are bounded in the lattice structure and thus donate free electrons without contributing holes.
By adding donor atoms into an intrinsic semiconductor, the number of electrons increases (n > p) → n‐type semiconductor.
Majority carrier: electron.
Minority carrier: hole

p‐type semiconductor

The acceptor impurity has 3 valence electron (Boron).
Th Si atom is replaced by an acceptor atom.
The boron lacks one valence electron. It leaves a vacancy in the bond structure.
This vacancy can accept electron at the expense of creating a new vacancy.
Acceptor creates a hole without contributing free electron.
By adding acceptor into an intrinsic semiconductor, the number of holes increase (p > n) → p‐type semiconductor.
Majority carrier: hole.
Minority carrier: electron

5. Properties of Semiconductor Materials

Various materials are classified based on the width of forbidden energy gap. In metal, there is no forbidden gap and valence and conduction band are overlapped. In an insulator, the forbidden gap is very large up to 7eV while in semiconductors it is up to 1eV. The silicon and germanium are widely used semiconductors. Intrinsic materials are those semiconductor that has been carefully refined to reduce the impurities to a very level-essentially' as pure as can be made available through modern technology.

The conductivity of an intrinsic semiconductor is very less. The properties like conductivity can be changed by adding an impurity to the intrinsic semiconductor. The process of adding impurity is called doping.
A semiconductor doped with trivalent impurity atoms forms p-type material. It is called acceptor impurity with concentration N_A atoms per unit volume.
A semiconductor doped with pentavalent impurity atoms forms n-type material. It is called donor impurity with concentration N_D atoms per unit volume.
In p-type, holes are majority carriers and in n-type electrons are majority carriers.
When a material is subjected to an electric field, electrons move in a particular direction with steady speed called drift speed and current drift current.

Negative Temperature Coefficient: Those parameters decreasing with the temperature have a negative temperature coefficient, e.g., an energy gap (E_g).

where, constant β₀ = 2.2 × 10^–4 (for Ge)

= 3.6 × 10^–4 (for Si)

Mobility (μ), μ ∝ T^–m

Positive Temperature Coefficient: Those parameters increasing with temperature have a positive temperature coefficient.

6. Important terms

Drift velocity V_d = μE
Current density J = nq μE
Conductivity σ = nq μ
Concentration of free electrons per unit volume
Semiconductor conductivity σ = (nμ_n + pμ_p)q
In intrinsic semiconductor, n = p = n_i Hence, conductivity σ_i = n_i (μ_n + μ_p)q Intrinsic concentration
In extrinsic semiconductor, the conductivity is given by, For n-type, σ_n = (n_nμ_n + p_nμ_p)q For p-type, σ_n = (n_pμ_n + p_pμ_p)q But in n-type p_n < < n_n N_D = Concentration of donor impurity N_A = Concentration of acceptor impurity n_p = Number of electrons (concentration) in p-type P_p = Number of holes (concentration) in p-type and n_n ≅ N_D while in p-type n_p < < p_p and p_p ≅ N_AHence, conductivity can be calculated as, σ_n = N_Dμ_nq and σ_p = N_Aμ_pq
Mass-action law np = n_i²

In n-type, n_np_n = n_i² , hence

In p-type, p_pn_p = n_i², hence

7. Hall Effect

When a magnetic field is applied to a current-carrying conductor in a direction perpendicular to that of the flow of current, a potential difference or transverse electric field is created across a conductor. This phenomenon is known as the Hall Effect.

According to this effect the statements are:

If a specimen (metal or semiconductor) carrying a current I is placed in a transverse magnetic field B, an electric field E is induced in the direction perpendicular to both I and B. This phenomenon, known as the Hall effect, is used to determine whether a semiconductor is n- or p-type and to find the carrier concentration. Also, by simultaneously measuring the conductivity σ, the mobility μ can be calculated.

Consider the figure shown below. Here current l is in +x-direction, magnetic field B is in +z direction then induced electric field will be in negative y-direction.

Hence a force will be exerted in the negative y-direction on the current carriers.

The current l may be due to holes moving from left to right or to free electrons travelling from right to left in the semiconductor specimen. Hence, independently of whether the carriers are holes or electrons, they will be forced downward toward side 1 of above figure.

If the semiconductor is n-type material, so that the current is carried by the electrons, these electrons will accumulate on side 1, and this surface becomes negativity charged with respect to side 2. Hence a potential, called the Hall voltage, appears between surface 1 and 2.

Now under the equilibrium condition

Where ρ is the charge density, w is the width of the specimen and d is the distance between surfaces 1 and 2.

It is customary to introduce the Hall coefficient R_H defined by

By hall experiment mobility of charge carriers is given as

⇒ Hall coefficient, R_H Temperature coefficient of resistance of given specimen.

⇒ For metals, σ is larger, V_H is small.

⇒For semiconductors, σ is small, V_H is large.

7.1 Hall Effect in an n-type semiconductor:

If the magnetic field is applied to an n-type semiconductor, both free electrons and holes are pushed down towards the bottom surface of the n-type semiconductor. Since the holes are negligible in n-type semiconductor, so free electrons are mostly accumulated at the bottom surface of the n-type semiconductor.
This produces a negative charge on the bottom surface with an equal amount of positive charge on the upper surface. So in n-type semiconductor, the bottom surface is negatively charged and the upper surface is positively charged.
As a result, the potential difference is developed between the upper and bottom surface of the n-type semiconductor. In the n-type semiconductor, the electric field is primarily produced due to the negatively charged free electrons. So the hall voltage produced in the n-type semiconductor is negative.

7.2 Hall Effect in a p-type semiconductor:

If the magnetic field is applied to a p-type semiconductor, the majority carriers (holes) and the minority carriers (free electrons) are pushed down towards the bottom surface of the p-type semiconductor. In the p-type semiconductor, free electrons are negligible. So holes are mostly accumulated at the bottom surface of the p-type semiconductor.
So in the p-type semiconductor, the bottom surface is positively charged and the upper surface is negatively charged.
As a result, the potential difference is developed between the upper and bottom surface of the p-type semiconductor. In the p-type semiconductor, the electric field is primarily produced due to the positively charged holes. So the hall voltage produced in the p-type semiconductor is positive. This leads to the fact that the produced electric field is having a direction in the positive y-direction.

7.3 Hall voltage

The expression for the Hall voltage is given by:

7.4 Applications of Hall Effect

Measurement of magnetic flux density.
Measurement of displacement.
Measurement of current.
Measurement of power in Electro-Magnetic waves.
Determination of mobility of semiconductor material.
Hall effect is used in many applications as following:
[Note: Minority carrier mobility (μ) and diffusion coefficient(D) can be measured independently with the help of Haynes-Shockley experiment.]

Example-1:

A n type Ge Sample has a donor density N_D = 10²¹ atoms/m³. It is arranged in a Hall experiment having a magnetic field B =0.2 Wb/m² and current density J = 500 A/m². Find the hall voltage generated when the thickness of sample is 2 mm. Also calculate the field intensity induced in magnitude ?

8. Fermi Level in Intrinsic and Extrinsic Semiconductors

Electrons in solids obey Fermi-Dirac statistics. The distribution of electrons over a range of allowed energy levels at thermal equilibrium is:

Where k s Boltzmann’s constant (k = 8.62 × 10^–5 eV/K = 1.38 × 10^–23 J/K).

The function f(E), the Fermi-Dirac distribution function, gives the probability that an available energy state of E will be occupied by an electron at absolute temperature T. The quantity E_F is called the Fermi level.
If E = E_F then = 0.5 or 50%

If E > E_F then f(E) < ½

If E > E_F then f(E) > ½

A closer examination of f(E) indicates that at 0 K the distribution takes the simple rectangular form shown in figure. At temperature higher than 0K, some probability exists for states above the Fermi level to be filled.

8.1 Fermi Level in Intrinsic Semiconductor

In intrinsic semiconductor Fermi level E_F is given by

where, N_C = density of states in conduction band

N_v = density of states in valence band

In pure Semiconductor at T = 0K, Fermi level lies in the middle of bandgap.

8.2 Fermi Level in n-type Semiconductor

Fermi level in n-type semiconductor is given by

Where, N_D = doping concentration.

Fermi level in n-type semiconductor depends on temperature as well as on doping concentration.
At 0K Fermi level coincides with that of lowest energy level of conduction band.
As doping increases Fermi level moves towards conduction band.
Shift in Fermi level in n-type semiconductor with respect to Fermi level of intrinsic semiconductor is

shift = kT

shift ≅ kT

8.3 Fermi Level in p-type Semiconductor

Fermi level in p-type semiconductor is given by

In p-type semiconductor Fermi level depends on both temperature as well as on doping concentration N_A.
As temperature increases Fermi level moves away from E_V i.e. towards middle of band gap.
As 0K Fermi level coincides with highest energy level E_V of valence band.
As doping concentration increases Fermi level moves toward E_V or away of middle of band gap
Shift in Fermi level in p-type semiconductor with respect to Fermi level of intrinsic semiconductor as

shift = kT

shift ≅ kT

Example-2:

If the Fermi energy in silicon is 0.22 eV above the valence band energy, what will be the values of n₀ and p₀ for silicon at T = 300 K respectively?

Solution:

Given that Fermi energy in silicon is 0.22 eV above the valence band energy, i.e.

E_F – E_v = 0.22 eV

So, we obtain the hole concentration as

Now, the energy bandgap for silicon is 1.12 eV, i.e.

E_g = E_c – E_v = 1.12 eV

Therefore, we obtain

Hence, the hole concentration is

9. Introduction

Any motion of free carriers in a semiconductor leads to a current. This motion can be caused by an electric field due to an externally applied voltage since the carriers are charged particles. We will refer to this transport mechanism as carrier drift. In addition, carriers also move from regions where the carrier density is high to regions where the carrier density is low. This carrier transport mechanism is due to the thermal energy and the associated random motion of the carriers. We will refer to this transport mechanism as carrier diffusion. The total current in a semiconductor equals the sum of the drift and the diffusion current.
As one applies an electric field to a semiconductor, the electrostatic force causes the carriers to first accelerate and then reach a constant average velocity, v, due to collisions with impurities and lattice vibrations. The ratio of the velocity of the applied field is called mobility. The velocity saturates at high electric fields reaching the saturation velocity. Additional scattering occurs when carriers flow at the surface of a semiconductor, resulting in lower mobility due to the surface or interface scattering mechanisms.
Diffusion of carriers is obtained by creating a carrier density gradient. Such gradient can be obtained by varying the doping density in a semiconductor or by applying a thermal gradient.
Both carrier transport mechanisms are related since the same particles and scattering mechanisms are involved. This leads to a relationship between the mobility and the diffusion constant called the Einstein relation

10. Recombination

A process whereby electrons and holes are annihilated or destroyed.

Energy Bands in Recombination Processes

Band-to-Band Recombination: Also referred to as direct thermal recombination, is the direct annihilation of a conduction band electron and a valence band hole. The electron falls from an allowed conduction band state into a vacant valence band state. This process is typically radiative with the excess energy released during the process going into the production of a photon (light).

R-G Center Recombination: A defect often causes an energy state in the mid-gap region of the bandgap that can act as traps for carriers. In this process, a conduction band electron gets trapped at this defect and energy state and then a valence band hole is trapped and recombines with the electron or vice versa. This process is typically nonradiative.

Recombination via Shallow Levels: Donor and acceptor sites can also act as intermediaries in the recombination process similar to the R-G Center recombination with the exception that the recombination is quite often radiative.

Recombination Involving Excitons: Conduction band electrons and valence band holes can form a slightly lower energy state by becoming bound to each other in a hydrogen-like arrangement. The electron and hole can then recombine and result in a photon that has energy slightly less than the bandgap.

Auger Recombination: A nonradioactive recombination event where a conduction band electron loses energy to another conduction band electron which gains energy to result in one electron recombining with a hole in the valence band and one high energy electron which rapidly loses energy by phonon emission and relaxes back down to the conduction band.

11. Generation

A process whereby electrons and holes are created.

The recombination processes can be reversed resulting in generation processes.
Band-to-Band generation occurs when an electron from the valence band is excited by light or heat to the conduction band.

Generation of an electron and hole by an R-G centre intermediary can be done in a couple of ways including an electron from the valence band being excited to the trap state and then to the conduction band creating a hole in the valence band and electron in the conduction band.

Also, an electron in the trap state can be excited to the conduction band while the hole is excited to the valence band.

Another common carrier generation is carrier generation via impact ionization where a high energy electron loses energy to produce an electron-hole pair.

12. Carrier Diffusion

Diffusion: particle movement in response to concentration gradient
Elements of diffusion:
- a medium (Si crystal)
- a gradient of particles (electrons and holes) inside the medium
- collisions between particles and medium send particles off in random directions: → overall, particle movement down the gradient.

Key diffusion relationship (Fick’s first law):

Diffusion flux ∝ - concentration gradient
Flux ≡ number of particles crossing unit area per unit time [cm⁻² · s⁻¹]
For electrons:

For holes:

Dn ≡ electron diffusion coefficient [cm²/s]
Dp ≡ hole diffusion coefficient [cm²/s]
D measures the ease of carrier diffusion in response to a concentration gradient:
Diffusion current density = charge × carrier flux

5. Carrier Drift

Net velocity of the charged particles ⇒ electric current.
Drift current density ∝ carrier drift velocity ∝ carrier concentration ∝ carrier charge.
Drift currents:

Total drift current:

Has the shape of Ohm’s Law:

Where:

Then:

Einstein's Relation: In a semiconductor, this relation gives the relationship between diffusion constant, mobility and thermal voltage.

V_T = KT and is 26mV at 27°C

Mobility:
- The electron mobility and hole mobility have a similar doping dependence: For low doping concentrations, the mobility is almost constant and is primarily limited by phonon scattering. At higher doping concentrations, the mobility decreases due to ionized impurity scattering with the ionized doping atoms. The actual mobility also depends on the type of dopant.
- Mobility is linked to the total number of ionized impurities or the sum of the donor and acceptor densities.
- The minority carrier mobility also depends on the total impurity density. The minority-carrier mobility can be approximated by the majority-carrier mobility in a material with the same number of impurities.
- The mobility at a particular doping density is obtained from the following empiric expression:

Resistivity:
- The resistivity is defined as the inverse of the conductivity. The conductivity of a material is defined as the current density divided by the applied electric field. Since the current density equals the product of the charge of the mobile carriers.
- To include the contribution of electrons as well as holes to the conductivity, we add the current density due to holes to that of the electrons.
- or:

The conductivity due to electrons and holes is then obtained from:

The resistivity is:

13.Poisson’s Equation

Poisson's equation correlates the electrostatic potential Φ to a given charge distribution ρ.

Using the relation between the electric displacement vector and the electric field vector,

where ε is the permittivity tensor. This relation is valid for materials with time independent permittivity. As materials used in semiconductor devices normally do not show significant anisotropy of the permittivity, can be considered as a scalar quantity ε in device simulation. The total permittivity is obtained from the relative ε_r and the vacuum permittivity ε₀ as ε = ε_r ε₀.

can be expressed as a gradient field of a scalar potential field.

Consider the permittivity a scalar, which is constant in homogeneous materials, we obtain Poisson's equation.

The space charge density ρ consists of :

ρ = q(p – n + C),

where q is the elementary charge, p and n the hole and electron concentrations, respectively, and the concentration of additional, typically fixed, charges. These fixed charges can originate from charged impurities of donor (N_D) and acceptor (N_A) type and from trapped holes (ρ_p) and electrons (ρ_n),

C = N_D – N_A + ρ_p - ρ_n

Poisson's equation commonly used for semiconductor device simulation:

14. Continuity Equations

The continuity equations can be derived using the following:

By applying the divergence operator: , to the equation and considering that the divergence of the curl of any vector field equals zero.

Separating the total current density into hole and electron current densities,
and
When we consider the charged impurities as time-invariant and introduce a quantity to split up the above equation into separate equations for electrons and holes, we get