Contacts: Metal-metal is always ohmic contact, i.e. ohm’s law valid Schottkycontact/barrier: • When a metal and a semiconductor get into contact a Schottky Barrier forms

© Dr. Somenath Chatterjee, SMIT 2014 Electronic Devices and components Page 19

EC303 Electronic Devices & Components Instructor: Dr. Somenath Chatterjee, E&C Dept.,

Sikkim Manipal Institute of Technology, July-Dec., 2014 Set II

Problem: Consider a semiconductor bar with w =0.1 mm, t= 10 µrn, and L = 5 mm. For B = 10 kg in the

direction shown (1 kG = 10-5Wb/cm2) and a current of 1 mA, we have VAB= - 2 rnV, VCD= -100

mV.Find the type, concentration, and mobility of the majority carrier.

Metal work function: The minimum energy required to move one electron from metal fermi level to the Vacuum level

From Einstein photoelectric experiment we can find the metal work function value.


Contacts: Metal-metal is always ohmic contact, i.e. ohm’s law valid

Schottkycontact/barrier:

• When a metal and a semiconductor get into contact a Schottky Barrier forms

• The Schottky-barrier height is measured in eV

• In the Schottky model, the Schottky-barrier height is equal to the difference of metal work

function and semiconductor electron affinity. For intrinsic,

• For extrinsic p or n,

±+=

i

dags n

orNNkTqE

ln2

χφ

• Vacuum level, E0 -corresponds to energy of free electrons.

• The difference between vacuum level and Fermi-level is called work function, Φ of materials.

• Work function, ΦM is an invariant property of metal. It is the minimum energy required to free

up electrons from metal.

• The semiconductor workfunction, Φs, depends on the doping.

Metal (M) - Semiconductor(S) contacts:

• The band diagram of a metal and a semiconductor before and after being brought into contact

• Soon after the contact formation, electrons will begin to flow from S to M near junction.

FBFCs )( EE −+=Φ χ

Figure (a) and (c) An instant after contact formation

Figure (b) and (d) under equilibrium conditions


• Creates surface depletion layer, and hence a built-in electric field (similar to p+-n junction).

• Under equilibrium, net flow of carriers will be zero, and Fermi-level will be constant.

• A barrier ΦB forms for electron flow from M to S.

• ΦB = ΦM – χ ... ideal MS (n-type) contact. ΦB is called “barrier height”.

• Electrons in semiconductor will encounter an energy barrier equal to ΦM – ΦS while flowing from

S to M.

•

Work function n-type p-type

• sm ϕϕ > • Rectifying • Ohmic

• sm ϕϕ < • Ohmic • Rectifying

Find the barrier height at equilibrium for W-Si contact. Given: ΦM = 4.55eV for W; χ(Si) =

4.01eV; Si doping = 1016 cm−3, Si is n-type.


EC303 Electronic Devices & Components Instructor: Dr. Somenath Chatterjee, E&C Dept.,

Sikkim Manipal Institute of Technology, July-Dec., 2014 SET III

Heterostructure is defined as a semiconductor structure in which the chemical composition changes with position.

• The simplest heterostructure consists of a single heterojunction, which is an interface within a semiconductor crystal across which the chemical composition changes. Examples junctions between GaAs and AlxGa1−xAs solid solutions.

• Applications: Essential elements of the highest performance optical sources and detectors and are being employed increasingly in high-speed and high-frequency digital and analog devices

• Precise control over the states and motions of charge carriers in semiconductors

Quantum well, Quantum wire and Quantum dots: When the size or dimension of a material is continuously reduced from a large or macroscopic size, to

a very small size, the properties remain the same at first, and then small changes begin to occur, until finally when the size drops below100 nm, dramatic changes in properties can occur.

If one dimension is reduced to the nano-scale range while the other dimensions remain large, a structure known as quantum well.

If two dimensions are in nano-scale range and one remains large, the resulting structure is a quantum wire.

The all three dimensions reach the in nanometer range, is called a quantum dot.

Particle in an Infinite Potential Well: Consider first the particle trapped in an infinitely deep

one- dimensional potential well with a specific dimension

Observations: • Energy is quantized, Even the lowest energy level has

a positive value and not zero • The probability of finding the particle is restricted to

the respective energy levels only and not in-between • Classical E-p curve is continuous. In quantum mechanics,

p = hk with k = nπ/l where n = ±1, ±2, ±3 etc.

Bulk Well Wire Dot

Particle in an infinite deep one dimensional potential well. 1. First four allowed energy levels, 2. wave functions, 3. probability of wave functions associated with energy levels


• En = h2k2/8π2m= n2 h2/8ml2 • In fact the negative values are not counted since the probability of finding the electrons in n=1 and

n=-1 is the same and also E is the same at these values • When l is large, energies at En and En+1 move closer to each other => classical systems, energy is

continuous.

Density of states in three dimensional bulk structures of semiconductors: • In this derivation, consider the conduction band electrons to be essentially free. Constraints of the particular

lattice can be included in the effective mass of the electron at the end of the derivation. For a free electron,

the three-dimensional Schrodinger wave equation becomes

• where𝜓 is the wave function of the electron and E is its energy. The form of the solution

• A common approach is to use periodic boundary conditions,in which we quantize the electron energies in a cube of material of side L. Thiscan be accomplished by requiring that

• And is similarly for the y- and z-directions. Thus wave function can be writtenas

• Where A is normalizing factor, where the 2πn/Lfactor in each direction guarantees the condition described by Eq. (3), substituting the value in Schrodinger’s equation,

• Let us determine the number of allowed states per unit volume as a function of energy [the

density of states, N(E)]in various cases such as one, two, or three dimensions.

• We first count states in k-space, then we can use the band structure, E(k), to convert to N(E).

• Consider a sphere in k-space. Associated with this sphere is a volume

where k is the “radius" and k2 = kx

2+ ky2+ kz

2 , again in general,

-------- (1)

-------- (2)

-------- (3)

-------- (4)

-------- (5)

-------- (6)


• These k values arise from the periodic boundary conditions imposed on the carrier’s wavefunction. • We now define a state by the smallest nonzero volume it possesses in k-space.

• Thus, within our imaginedspherical volume of k-space, the total number of states present is

• We are dealing with both electrons and holes, we must consider spin degeneracy, since two carriers, possessing opposite spin, can occupy the same state. As a consequence, eqn (8) can be multiplied by 2 to obtain

• • This represents the total number of available states for carriers, accounting for spin. • We now define a density of states per unit volume, ρ = N2/(LxLyLz

), with units of number per unit volume. This results in

• Finally, considering an energy density, ρenergy = dρ/dE, which unit is number per unit energy per unit

volume, we obtain

which simplifies to

-------- (7)

-------- (8)

-------- (9)

-------- (10)

-------- (11)

-------- (12)

Figure: Volume (shaded) associated with a given state in k-space. A three-dimensional side view.


• This is our desired density-of-states expression for a bulk three dimensional solid. Note that the function

possesses characteristicsquare root energy dependence.

• Since we now explicitly refer to the electron in the conduction band, we replace meff to me to denote

electron effective mass there. We also modify the expression to have a nonzero origin to accountfor the

conduction band starting energy. We thus find

•

• The concentration of electrons in the conduction band from probability function (discussed in the first

chapter)

)()( ∫=α

cEc dEENEfn

Inserting this into our integral then yields

i.e.

Where We can subsequently simplify this as follows,

• However, to stay instructive, let us just consider the situation where (E – EF) >> kT . In this case, the exponential term in the denominator of Equation (15) dominates, so that

So,

-------- (13)

-------- (14)

-------- (15)

-------- (16)

-------- (17)


• Which we can simplify to,

• According to our equation, the values of n in gamma function is 3/2, from equation (19),

• It is well known that the values of 22

3 π=

Γ

• Final expresses the concentration of carriers in the bulk semiconductor conduction band.

Where Nc is the effective density of states and can be written as

23

2232

3

2223 22

2).(2.

21

2).(

=×

=×=

hKTmKTmKTAN ee

cππ

ππ

Which describes the electron concentration in conduction band.

Similarly, the hole concentration in the valance band can be calculated.

Quantum Well (two dimensional) Density of States of semiconductors:

• Two degrees of freedom exist within the (x, y) plane for quantum well

• consider a circular area with radius 22

yx kkk +=

• The one direction of confinement that occurs along the z direction is excluded.

• The associated circular area in k-space is then

--------- (18)

--- (19)

---------------- (20)

Area (shaded) associated with a given state in k-space for a two-dimensional system.


• And encompasses many states having different energies. A given

state within this circle occupies an area of

• With kx = 2π/Lx and ky = 2π/Ly , as illustrated in besides Figure. Recall that Lx = Ly = Na, where N is the number of unit cells along a given direction and a represents an interatomic spacing. Thus

• The total number of states encompassed by this circular area is therefore N1 = Ak/Astate, resulting in

• If we account for spin degeneracy, this value is further multiplied by 2,

• Giving the total number of available states for carriers, including spin. At this point, we can define an area density

• with units of number per unit area, since k =Sqrt(4π2. 2meff E/h2). Our desired energy density is then

ρenergy = dρ/dE and yields

• with units of number per unit energy per unit area. Notice that it is a constant. • Notice also that this density of states in (x, y) accompanies states associated with each value of kz (or

nz). As a consequence, each kz (or nz) value is accompanied by a “subband" and one generally expresses this through

• where nz is the index associated with the confinement energy along the z direction and (E − Enz ) is the

Heaviside unit step function, defined by

Nanowire Density of States:

• The derivation of the nanowire density of states proceeds in an identical manner. The only change is the different dimensionality. Whereas we discussed volumes and areas for bulk systems and quantum wells, we refer to lengths here.


• For a nanowire, consider a symmetric line about the origin in k-space having length 2k:

• The associated width occupied by a given state is

• where k represents any one of three directions in k-space: kx, ky, or kz. For convenience, choose the z

direction. This will represent the single degree of freedom for carriers in the wire. We then have possible kz values of

• Where Lz = Na, N represents the number of unit cells along the z direction, and a is an interatomic

spacing. The smallest nonzero length occurs when nz = 1. As a consequence, the number of states found within Lk is

• If spin degeneracy is considered,

• and describes the total number of available states for carriers. We now define a density

• That describes the number of states per unit length, including spin. This leads to an expression for the

DOS defined as ρenergy = dρ/dE:

Giving,

• The above equation is our desired expression, with units of number per unit energy per unit length.

More generally, since this distribution is associated with confined energies along the other two directions, y and z, we write

• where Enx,ny are the confinement energies associated with the x and y directions and Θ(E − Enx,ny ) is

the Heaviside unit step function. Notice the characteristic inverse square root dependence of the nanowire one dimensional density of states.


Quantum Dot Density of States

• In a quantum dot, the density of states is just a series of delta functions, given that all three dimensions exhibit carrier confinement

• Enx,ny,nz are the confined energies of the carrier, characterized by the indices nx, ny, nz. The factor of 2

accounts for spin degeneracy. To generalize the expression, we write

• which accounts for all of the confined states in the system.

Contacts: Metal-metal is always ohmic contact, i.e. ohm’s law valid Schottkycontact/barrier: • When a metal and a semiconductor get into contact a Schottky Barrier forms

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