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By combing doping and semiconductor material, many different semiconductor junctions can be formed. If more than one type of semiconductor forms the junction region, the semiconductor is known as a heterojunction and in many semiconductor devices thesedifferent junctions are vital or lead to ehanced performance of the device.

When two semiconductors with different energy gaps are combined, a heterojunction is formed. The most important difference between these semiconductors in heterostructure devices is the energy gap and refractive index. Also, the threshold current density was improved at room temperature and it was achieved by cladding the active layer material, such as GaAs, with wider energy gap material, for instance Al _x Ga _1-x As has very good lattice constant matching with GaAs. In general, heterostructure lasers will give you both good optical and current confinement, such as P-n-N or P-p-N double heterostructure laser. The conductivity type of the smaller energy gap crystal is denoted by a lower case n or p and that of the larger energy gap crystal is denoted by an upper case N or P.

The p-n Junction

Junctions are crucial to many semiconductor applications. The oldest method of making a p-n junction is by diffusion. The dopant diffuses in under heating so that the surface acceptor concentration exceeds the donor concentration. A junction appears when N _d =N _a

Another doping technique is ion implantation. The starting material, n-type only, is bombarded with the required species of ions, say acceptors. This produces sharper junctions, but causes damage to the crystal lattice structure increasing the number of dislocations and interstitial atoms.

Epitaxial deposition techniques are now widely established. The starting material is a single crystal in all cases, so it is possible to grow further crystal layers which are in register with the starting crystal. The most precise although most expensive way of achieving this is Molecular Beam Epitaxy (MBE). Ions of the semiconductor together with dopants are fired at the crystal surface. Under the right conditions (ultrahigh vacuum, correct ion fluxes, correct substrate temperature) the crystal grow epitaxially with the required dopant included. This technique can produce very sharp junctions and there is no counter-doping, i.e. no donors in the p-type region. Discussion on fabrication from crystal growth to etching, other expitaxial growth methods, up to metalisation and etching is outlined here . We are supposed to be discussing p-n junctions!

We now consider a p-n junction in the absence of voltage bias, so that it is in thermodynamic equilibrium. This means that the chemical potential must be constant across the junction. Since is near the valence band edge in a p-type region and near the conduction band in an n-type region, the bands must bend through the junction as shown below:

In the region where the bands are bending, is near the middle of the gap and therefore n<n ₀ , p<p ₀ , where n ₀ and p ₀ are the concentrations deep inside the n and p type regions. This is therefore called the depletion region. If we assume the junction is sharp, with N _a and N _d changing abruptly at the junction, and assume N _d =N _a , p ₀ =N _a , thus we see that within the depletion region p<N _A on the left and n<N _d on the right. Since N _A and N _D are the densities of ionised acceptor and donor ions, this means that there is a net negative charge on the p-type side of the junction, and a net positive charge on the n-type side. These separated charges generate an electric field, which is the physical cause of the band bending. The overall picture is summarised below.

We now calculate charge, electric field and potential. First we find the band offset. , i.e. the difference in the height of above the valence band on the two sides. In the n-type region, the value of relative to the local valence band is given by

(1)

assuming N _A =0 on the n-side. On the p-type side, we have

(2)

assuming N _D =0 on this side. These give

(3)

Using the definition of the intrinsic density n _i (24, semiconductor basics) , gives

(4)

This is the difference in the electrostatic potential between the two sides since is the energy difference between electrons at the bottom of the conduction band on the two sides, as may be seen from the diagram.

The variation of and the electric field

(5)

across the junction can be calculated as long as the variations with x of N _D and N _A are known. If we assume an abrupt junction then the charge distribution has the form:

(6)

where w _p and w _n are the semi-widths of the depletion region on the p-side and n-side; values for them will be found later.

The electric field satisfies Gauss' law:

(7)

since with constant, and in the present 1-d case , this gives

(8)

The boundary conditions are E=0 for x<-w _p and x>w _n since the junction is in equilibrium. The solution of (8) is therefore,

.

(9)

Furthermore, E must be continuous at x=0, which gives

(10)

As seen from the figure, this is simply the condition of electrical neutrality of the whole depletion region. The variation of E with x given is shown in (d)

Equations (5) and (8) together give

(11)

must be continuous at x=0; this gives a second relation between w _p and w _n :

(12)

Recall that is already known from (4) . The variation of with x given by (11) and is shown in (c). As stated, (4) , (11) can be solved for the values of w _p and w _n ; they are,

(13)

(14)

The dependence on T and on the doping deserves comment. Assuming for simplicity N _A =N _D (equal doping on both sides), so that the factors and are both one, substitution of (12) for gives:

(15)

The logarithmic dependence is very weak compared with the denominator, so:

w increases as T increases

w decreases as doping N _A +N _D increases

The application of p-n junctions depends on having an applied voltage so that a current flows through the junction. With a voltage V applied a difference eV appears between the values of on the p and n side. We distinguish between forward and reverse bias, as in the diagram below:

Forward Bias

Reverse Bias

The calculation of etc. go through much as before, and give:

(16)

(17)

There is a charge separation in the depletion region. Thus the depletion region behaves like a capacitor, and the capacitance is given by

(18)

Where A is the area of the junction (in the y-z plane). This has the useful property that the capacitance can be varied by applied voltage. In practice, reverse bias is needed so that the current flow is small. A p-n junction device used as a voltage-variable capacitor is known as a varactor diode.

The important property of a p-n junction is the current-voltage characteristics. A derivation requires discussion of diffusion and recombination of carriers. The result is

(19)

where the predominant temperature dependence of I ₀ is given by

(19)

The first semiconductor lasers where made from heavily doped p-n junctions. Under conditions of forward bias the electrons and holes would recombine at the barrier junction producing some laser emission at high currents. These devices were inefficient and had high threshold currents as the majority carriers tended to drift away from the junction interface. It was soon discovered that more efficient lasers could be produced by the implementation of a heterostructure design.

PIN diode Structures

Overview

A p-i-n diode is a p-n junction with a doping profile tailored so that an intrinsic layer, the 'i region,' is sandwiched between a p layer and an n layer. In practice, however, an idealized i region is approximated by either a high-resistivity p layer (referred to as p layer) or a high-resistivity n layer ( n layer). The nature of the low doping in the i region, causes most of the potential will drop across this region. To model this diode structure,we must first define the space charge distribution and then work out the electric field in each region. Applying continuity of displacement vector at each of the boundaries, the electric field can be deduced. The electrostatic potential profile is then obtained by integrating the electric field. Let us start by defining the space charge distribution of PIN diode.

(1)

From Gauss's law

(2)

Integrating equation (2) and we obtain

(3)

Boundary condition gives

(4)

Electric field is the negative gradient of electrostatic potential, therefore from equation (3) and using the fact that electrostatic potential continuous at each boundary. We obtain,

(5)

where

(6)

P- p -N and P - n -N depletion length x _p Calculation

We are now interested in modelling the P- p -N or even P- n - N. The space charge distribution is:

(7)

where N _A , N _a , and N _D are the dopant concentration in P, p , and N region respectively. Then Gauss's law gives

(8)

The corresponding electric field is

(9)

Boundary conditions at W

and hence

(10)

The electrostatic potential is

(11)

The built in voltage is defined as

(12)

PIN Modelling

Figure 1. This is the PIN Homojunction with W=0 m m, Al _x Ga _(1-x) As, x=0.3 on each side, N _A =N _D =1x10 ¹⁷ cm ^-3 , and temperature T=300 K. With the same material on both sides the electric field is continuous at the junction. W _p =0.10 m m, W _n =0.10 m m, V _bi =1.627 V

Heavily doped in p and n region

Isotype n-N Heterojunction Modelling

Anisotype p-N Heterojunction Modelling

Anisotype P-n Heterojunction Modelling

PIN under forward bias

PiN under reverse bias

Approaching a Quantum Well

n-P Heterojunction

If the narrow band gap semiconductor is doped n-type and the larger gap semiconductor is doped P-type. When they are brought together, a space charge will exist due to the diffusion and redistribution of free carriers at thermal equilibrium.

The space charge distribution r (x) is

(1)

The electric field is proportional to integal of the charge distribution, Therefore the field will be

(2)

where the boundary condition at x = 0 gives

(3)

The electrostatic potential is

(4)

where

(5)

Therefore with equations 3, 4 and 5 the depletion length of x _p is as

(6)

The energy band edges function E _c (x) and E _v (x) can be obtained from –qV(x)

(7)

(8)

nP HeteroJunction Modelling

Using the software, we show the potential, electric, field, and energy band profiles for a n-GaAs, P AlGaAs heterojunction under equilibrium conditions of zero bias and then under reversed and forward bias respectively.

Figure 1: nP heterojunction Modelling at thermal equilibrium, i.e. V _a = 0. The two figures on the bottom, on the left is the junction under reverse bias, on the right is forward bias.

p-N Heterojunction

Consider a narrow band gap material of p-type semiconductor and a larger band gap material of N-type semiconductor are brought into contact. The Fermi level will be lined up to form an equilibrium state, in doing so the energy band will be bent, and a built-in electric field will be created between the junction. The modelling is similar to that of abrupt homojunction. I will not derive it again but just amend the previous derivation

Space charge distribution of p-N heterojunction:

(1)

where N _a and N _D are the net ionized acceptor and donor concentration at p and N side, respectively. With Gauss's law, the electric field will be

(2)

where ε _p and ε _N are the permittivity of p and N side, respectively. If we compare equation 2.26 with 2.9, we will find that the electric field is not continuous at x=0, but the normal displacement vector D =ε E is continuous at x =0; this will give you a same expression as

(3)

The electric field is a negative gradient of electrostatic potential, which means that the slope of the potential profile is given by the negative of the electric field profile. If we choose the reference potential to be zero for x < x _p , we have

(4)

Again the contact potential is evaluated using the bulk values of the Fermi levels F _p and F _N measured from the valence or conduction band edges E _vp and E _CN, respectively, before contact, figure 5:

(5)

(6)

In figure 1,where Φ and χ are the work function that is the energy difference between the vacuum level and the Fermi level, and the electron affinity, which is the energy required to take an electron from the conduction band edge to the vacuum level.

The depletion length in the p side will be calculated once V ₀ is known. Using equation 2.28 and 2.7, we have

(7)

and therefore

(8)

If the junction is under bias simply replaces V ₀ to V ₀ - V _a

(9)

The energy band edge will be

(10)

(11)

(12)

HeteroJunction Modelling

If you look at the electric field at x=0, you will find that the electric field is not continuous, simply because the permittivity of GaAs and Al _x Ga _1-x As, are different between the two materials. However, the displacement vector is continuous at x=0. The space charge distribution is sketched in the second plot. In this control panel, the energy gap of AlGaAs was designed desirable in direct gap region, if the Al mole fraction was to excess 0.45 the system will indicate and warn you that you are using the indirect gap material that is not good for laser design.

This program also add-on an increment and decrement control on each concentration input. The radio buttons indicate that this is a p-GaAs and N-Al _x Ga _1-x As Heterojunction. As usual the eye button is to call the program to plot potential profile, electric field, space charge distribution and energy band edges profiles.

p-N Heterojunction Under Foward and Reverse Bias

Figure 2: Reverse bias (-7V): X _p =0.044mm, X _n =0.221mm, V _tot =8.5832V, electric field at x=0 –611kVcm ^-1
Forward bias(1.3V): X _p =0.008mm,X _n =0.0405mm, V _tot =0.289V, electric field at x=0 –112kVcm ^-1

p-P and n-N Heterojunctions

nP and pN that we have covered in the previous section are called anisotype heterojunctions. Where the doping in the two semiconductor materials, as in nN and pP, they are called isotype heterojunctions. The energy band diagrams for isotype heterojunctions utilise the same expressions as were derived for the anisotype cases . Again, the energy band diagram is based on ΔE _c and ΔE _v , together with the requirement that the Fermi level is constant at thermal equilibrium. To model the situation exactly requires a bit more theory. In this isotype heterojunction model, we simply modify the equations derived previously and reconsider the built in voltage for the isotype heterojunctions. In general, we will derive the voltage drop at x = 0 also shown a way to model these isotype heterojunctions.

When p-type and P-type semiconductors are contacted, a space charge region is created. The charge distribution is

(1)

Note that far away from the junction

The electric field satisfies

(2)

Since electric field is zero at minus infinity and x = x _p . Equation (2) can be integrated from minus infinity to 0 _- to obtain

(3)

where

(4)

The boundary condition gives

(5)

and

(6)

or

(7)

The electric field profile will be

(8)

This electric field profile is very similar to nP heterojunction in figure 8. Using the Boltzmann distribution

(9)

where

Using the relationship between electrostatic potential and electric field, we got the relationship between valence energy band edge and electric field, that is ,

(10)

where

(11)

and

(12)

Using the above equations, we obtain a differential equation for E _v (x) from equation (8)

(13)

Using relation (12)

(14)

Integrating from x= minus infinity to 0 _-

(15)

with   equation (14) becomes

(16)

Figure 1. Bandedge profile of heterojunction before contact.

If we look at the energy band diagram before contact as shown in figure 1. The built voltage will be

(17)

The difference has been taken as F _p -F _P , so that forward bias is for the positive potential connected to the P-side. In this case, V ₀ will be

positive if

negative if

(18)

or even

For positive or zero built in voltage, it will generally be near zero, the energy band diagram is then given by expressions similar to the nP case, simply replacing all the donor concentration, which is N _d , to acceptor concentration that is N _a . However, if built in voltage were too negative, the energy band diagram is given by expressions similar to that of pN heterojunction case with N _D replaced by N _A .

From equation (17), we and obtain an expression for P side, which is

at P side.

(19)

we obtain at x= 0

(20)

(21)

Therefore

(22)

Thus from equations (16) and (17)

(23)

, and , and hence

(24)

The solution using Matlab is given by

(25)

where LambertW( x ) is the solution to w *exp( w ) = x and LambertW( x ) is linear when x <<1. B is usually very large, so that we can ignore the lambertW and hence

(26)

Semiconductor n-N Heterojunction

For nN heterojunction the built in Voltage will be

(27)

ΔE _c is generally larger than the difference (E _cn - F _n ) – (E _cN – F _N ), the built in voltage will be positive. The expressions for the energy band diagrams are similar to the p-N heterojunction case with Na replaced by N _d.

The energy diagram before contact is shown at figure 2

 

Figure 2. Bandedge profile before contact in p-P heterojunction.

Following the similar steps as for the p-P heterojunction, we can find an equation at x =0 for electric fields at both sides. The displacement vector must continuous at zero, therefore we obtain:

(27)

the only unknown V _on is obtained by solving equation (27)

n-N Heterojunction and p-P Heterojunction Modelling