PN Junction in equilibrium
  • PN junctions are important for the following reasons:
    (i) PN junction is an important semiconductor device in itself and used in a wide variety of applications such as rectifiers, Photodetectors, light emitting diodes and lasers etc

    (ii) PN junctions are an integral part of other important semiconductor devices such as BJTs, JFETS and MOSFETs

    (iii) PN junctions are used as test structures for measuring important semiconductor properties such as doping, defect density, lifetime etc

  • The discussion associated with the PN junctions will proceed in the following order:

    (i) PN junction in equilibrium
    (ii) dc IV characteristics in forward bias
    (iii) characteristics in reverse bias
    (iv) dynamic characteristics
    (v) Circuit models
    (vi) Design perspective


    Device Structure :

    The Figure below shows a simplified structure of a PN junction:




  • The structure can be fabricated by diffusing P-type impurity in the n-epilayer grown over an substrate.

  • While the doping in the n-epilayer can be uniform, the doping in the P-region is often either Gaussian or error-function in nature. The doping profiles and the junction are schematically illustrated below:






    1-D Abstraction

  • Even though the doping in both N and P-regions may in general be nonuniform, for simplicity, we shall assume them to be uniform in the initial analysis because the basic device physics remains almost the same

  • A simplified, one-dimensional abstracted view of a PN junction described by the region within the dotted lines of device schematic is shown below:



  • We shall assume that the thicknesses of P and N-regions are large enough so that one can ignore the presence of Ohmic contacts and the heavily doped N-region and consider only the P and N regions for analysis. Such a diode with wide N and P-regions is called a wide-base diode.

  • The PN junction that we shall study will therefore be a 1-D structure with uniformly doped P and N regions with thicknesses sufficiently large to ignore effects of contacts and other layers. It shall be represented simply as





    PN junction in Equilibrium

  • As mentioned earlier, the characteristics of a semiconductor device is completely specified in equilibrium if the variation of potential as a function of position is specified. As a first step to obtaining this potential profile, we shall sketch the energy-band diagram of the device. The energy band diagram would provide us with

    (i) a qualitative variation of potential in the device

    (ii) boundary conditions for solution of Poisson's equation

  • As usual, the energy band diagram of the PN junction will be obtained by combining the energy band diagrams of N and P-type semiconductors separately


    Energy Band Diagram In Equilibrium



    Energy Band diagram of N- and P-regions before equilibrium


  • When the N and P-regions are brought into contact, the electrons would flow from regions of higher Fermi-energy to regions of lower Fermi energy and holes would flow in the opposite direction.

  • Because of loss of electrons, the N-region would acquire a net positive charge due to the uncovered positively charged donor atoms and P-region would acquire a negative charge due to uncovered negatively charged acceptor atoms.

  • At equilibrium there is no net flow of either electrons or holes so that the PN junction has a single constant Fermi level.

  • The transfer of charges will affect only the regions close to the junction so that regions which are far still have the same energy band diagram(i.e. same relative positions of conduction and valence band wrt Fermi energy)

  • As we approach the junction from the N-side, the conduction band must bend upwards away from the Fermi energy to indicate the fact that the region is progressively getting depleted of electrons ( remember .


  • Similarly, as we approach the junction from the P-side, the conduction band must bend downwards towards the Fermi energy to indicate the fact that the region is getting depleted of holes

    Using these principles, the final energy band diagram can be sketched as





  • As a result of transfer of charges from N and P-regions, the region next to the junction is charged and is known as the space charge region.

  • The charge on the N-side is positive and on the P-side negative.

  • As a result, the space charge region will have an electric field directed from the N to the P-region with a maximum value at the junction and zero at the edges of the space charge region.

  • As a result of the electric field, there will be a net voltage across the space charge region known as the built-in voltage.

  • The magnitude of the built-in voltage can be quickly estimated from the energy band diagram. We do this by performing an analog of Kirchoffs voltage law:

    We start from a point in the N-region(away from the space charge region) at the energy and then move to a point in the P-region(away from the space charge region)again at energy via any path other than the Fermi-energy and add up the energy gained or loss at each step of the path, then the net sum should be zero!


    The built-in oltage can be expessed as:


    For Non-degenerate semiconductors:


  • An important result that can be deduced from Eq.(2) is that built-in voltage will be higher for semiconductors with larger bandgap.

    Using the relationship , the expression for built-in voltage for a
    PN junction having non-degenerate semiconductors can be written as


    Example 1.1 Determine the built in voltage for a uniformly doped Silicon PN junction with at room temperature. Will the built-in voltage increase or decrease with increase in temperature?

    Substitution of the doping values in Eq. (4) gives



    The built-in voltage decreases with increase in temperature due to exponential increase of intrinsic carrier concentration with temperature. The pre-factor kT/q in Eq.(4) has a much lesser influence.



  • There is another method by which the magnitude of built-in voltage can be obtained. In this case we start with the fact that in equilibrium, the net electron current is zero:


    Use of Einstein's relation : allows the above expression to be re-written as:



    Integrating the above expression across the space charge region gives:


    where are the potentials in the bulk of N- and P- regions respectively.
    is the electron density in the N-region and is the elecron density in the P-region.


    Example 1.2 Can the built-in voltage of the PN junction be measured by simply connecting a voltmeter across its two terminals?

    The answer is NO and this can be explained in several ways:

  • Although there is a net voltage across P and N-regions, the built-in voltage does not appear across the external terminals. If it did, then upon connection of a resistor across it, a current would begin to flow. This contradicts the fact that no current can flow in equilibrium.

    So how does the voltage across the external terminals become zero?

  • The built-in voltage is cancelled by voltage drop across the contacts made to N and P-regions.





    The net voltage between anode and cathode terminals can be written as



    The first term on the RHS represents the contact potential or barrier height for the Anode/P metal-semiconductor junction. Keeping in mind that contact potential between any two materials is simply the difference of their work-functions, we obtain



    where and are the work functions of P-type, N-type, cathode and anode metals respectively. For simplicity we asume that both anode and cathode metals are same ( say aluminium ) so that Using four equations given above, it is easy to see that
    Poisson's equation

  • The energy band diagram gives only a qualitative variation of potential across the space charge region. The detailed nature of this potential can be obtained through the solution of Poisson equation:





    Analytical Solution of Poisson's equation

  • Because of the exponential terms in the expression for charge density, the analytical solution of the Poisson's equation becomes difficult.

  • This difficulty is overcome through the assumption that the electron and hole density within the space charge region is negligible as compared to the ionized donor or acceptor atom density. This approximation, known as the depletion approximation, allows the Poisson equation to be simplified to:



    Henceforth, we shall also assume that all donor and acceptor atoms are ionized.

  • The table below shows the charge density as a function of potential within the space charge region for a PN junction with same doping in N and P regions for simplicity.


  • The data in the table shows that over a large range of potential, the depletion approximation is valid. Only for regions close to the space charge edge, does the approximation become weak.







    Simplified Charge density

    With the depletion approximation , the charge density can be expressed as


    The space charge region is often called the depletion region




    Simplified Poisson Equation


    The Poisson's equation for P and N-regions of the depletion region can be written as


    The boundary conditions can be written as:

    The boundary conditions can be written as:


    Outside the space region the charge density is zero so that


    This implies that electric field outside the depletion region is constant. However, to be consistent, this electric field must be zero, otherwise it would imply a non-zero current, some applied bias etc.





  • The electric field at x=0 must be continuous, otherwise it would imply an infinite charge density.

  • Similarly, the potential at x=0 also must be continuous


  • The Poisson equation with these boundary conditions can be easily solved to obtain the following results.

    Solution:

    Electric field: It is max. at the junction







    Potential:




  • The variation of potential across the depletion region is parabolic.

    Using the boundary condition that potential must be continuous at the junction:

  • Deletion widths: Using the relation , we can obtain





  • The depletion widths vary inversely with the doping.



    Example 1.3 Determine the total depletion width and the magnitude of maximum electric field for a symmetrical Si PN junction at equilibrium for doping densities of
    Using Eq.(23) and (27), we can obtain the following set of values



    The depletion width increases with decrease in doping but the magnitude of maximum electric field decreases even though the space charge region gets wider. This is because while the width of the space charge region increases as , the charge density with in the space charge region decreases as as the doping is reduced. This results in a net decrease in charge and therefore the electric field at the junction.



    Example 1.4 Determine the built-in voltage for a Silicon PN junction with uniformly doped P region with and an N-region which consists of two uniformly doped regions but of different doping values as illustrated below.



    The difficulty in this problem is that while it is clear that in Eq.(4), it is not clear whether the N-type doping should be
    or 5 x. The answer depends on where the depletion edge in N-region lies. Let us assume that it lies in the lightly doped region so that we take =.This gives a of 0.7 volts.We have to check whether our assumption is correct or not. Use of Eq.(27) shows that depletion width is 4257 thereby validating our aasumption. If assumption had been wrong, we would have to redo our calculations with =5 x .

    As the PN junction is reverse biased, the depletion width increases so that eventually the depletion edge would lie in the higher doped N-region. In that case also a new value of built-in voltage would have to calculated and used in the expressions for depletion width, electric field etc.




    Example 1.5 Suppose in the example above, the thickness of the lightly doped region is 2500 only. Calculate the depletion width at equilibrium.

    Using the previous example, we know that the depletion edge will lie in the higher doped N-region so that


    To find the depletion widths , we can adopt the methodology used for uniformly doped PN junctions except that solution of Poisson's equation is carried out in three regions, with region I being P-type , region II being N-type with doping and region III with N-type doping of The boundary conditions are similar except that two new boundary conditions describing continuity of potential and electric field will have to be used at the boundary of regions II and III.

    An alternative to working out the solution by beginning from Poisson's equation is to use some of the results already obtained with uniformly doped PN junctions. For example, we know that the electric field will vary linearly and can be sketched as






    Using the concept of charge neutrality, meaning that net charge on the P-side must be balanced by net charge on the N-side, we can write



    The slopes of electric field in each region can be written straight from Poisson equation.
    For example, in region II, so that



    and similarly using Poisson equation on the P-side in region I



    In these equation refers only to the magnitude of the maximum electric field. The area under the curve is simply the total voltage across the junction so that



    Solution of the above equations will give values for and therefore the total depletion width.
    Comparison With Exact Numerical calculations

    The Figure below shows a comparison of an actual charge profile computed using a 1-D device simulator and charge profile under depletion approximation for a doping of .




    The Figure above shows that the transition region is about 600 , almost same as the depletion width(735 ) predicted by the depletion approximation!

  • The depletion approximation therefore appears to be a poor assumption. However, a careful look shows that the depletion assumption overestimates the charge in region I but underestimates the charge in region II. Since, the electric field and potential are determined by the integral of charge density, the error in electric field and potential profile is not large!
    Example 1.6 Instead of approximating the charge density profile by an abrupt transition region, a better approximation would be to have a linear approximation to the transition region as illustrated below for a PN junction with same value of doping in both N and P regions.



    Obtain expressions for electric field and potential

    Integration of Poisson's equation in regions 1 and 2 and matching the electric field at the boundary gives




    The maximum electric field is given by the expression:



    Integration of electric field with the condition that the net voltage across the space charge region is , gives

    Example 1.7 So far we have discussed PN junctions in which both P and N-regions are made out of the same semiconductor. Let us consider next an heterojunction and sketch its band diagram at equilibrium and find its barrier height.


    Figure below shows the band diagram of the two semiconductors, when they are far apart.





    Using the principles described earlier, the band diagram after equilibrium can be sketched as




    There exists a discontinuity in conduction band and valence band at the junction. Their magnitudes can be expressed as



    where is the difference in the bandgaps of the two semiconductors

    The barrier height can be determined by performing an analog of Kirchoffs law. We start from a point at Fermi energy in the P-type GaAs far from the junction and arrive again at the Fermi energy but on the side of N-AlGaAs, again far from the junction and add up all the energy increments along the way:



    The first term is the usual term that is present in the expression for built-in voltages of homojunctions also. The second term is the additional term that results from the presence of conduction-band discontinuity.