In this article we will discuss about the mechanism of current conduction in semiconductors.
When no external field is applied to a semiconductor each carrier (electron or hole) moves in a random way owing to its thermal energy. The frequent change in direction of the path of the carrier occurs due to scattering by the vibrations of the lattice atoms and by the Coulomb field of the ionized donor and acceptor atoms. Application of an external field superimposes on the random motion of the carriers a drift velocity.
In the steady state, the rate of momentum gained from the field equals the rate of loss of momentum due to scattering. A steady drift velocity is thus achieved giving rise to a steady flow of current. In general, the drift velocity is given by v = E, where is the mobility of charge carriers and E is the applied electric field. In an intrinsic semiconductor the flow of current is due to movement of both electrons and holes and is in opposite directions.
Since the electric current flowing in any conductor is given by the amount of charge flow in one second across any plane of the conductor, therefore, the total number of electrons or holes which cross any plane of cross-section A in one second will be nvA, where n is the number of free electrons or holes per unit volume of the semiconductor (in intrinsic semiconductor number of electrons and holes are equal).
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Now, the current due to electrons in an intrinsic semiconductor is Ie = nveAe and the current due to holes is Ih = nvhAe, where v and vh are drift velocities of electrons and holes respectively, and e is the electron or hole charge.
∴ The total current in an intrinsic semiconductor is-
Where, e and h are mobilities of electrons and holes respectively.
Where, V is the applied voltage across the two ends of the semiconductor of length l. Since the resistance offered by material is given by R = V/I, therefore, we have-
Also resistance R and resistivity of the material are related by the relation-
And conductivity σ is given by σ = ne (e + h)
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Thus, the conductivity of semiconductor depends on (i) the number of current carries present per unit volume and (ii) the mobility of the current carriers.
Now the current density J is-
J = I/A = ne (μe + μh) E … (i)
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J = σE
We shall now consider in detail the mechanism of current flow when a voltage is applied to:
1. n-type semiconductor and
2. p-type semiconductor.
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1. n-Type Semiconductor:
An n-type semiconductor has – (i) electrons as majority carriers, (ii) an equal number of immobile positive ions and (iii) holes as minority carriers. Assume hole contribution to current is negligible.
Consider an n-type semiconductor placed between a pair of electrodes across which a voltage is applied. Due to the field produced by the voltage there will be steady drift of the free electrons towards the +ve electrode. The electrons reaching the +ve terminal disappear at the electrode and the immobile positive ions in the vicinity of the negative electrode remain un-neutralized due to the drift of the free electrons.
These ions immediately attract electrons from the negative electrode. Thus a continuous flow of electrons from one terminal of the voltage source to the other terminal, from the -ve electrode to the semiconductor and from semiconductor to the +ve electrode is determined by the applied voltage and conductivity of the semiconductor which account for the current flow.
In n-type semiconductor the electron densities and hole densities are different so the equation (i) cannot be used as such. Suppose the electron and whole densities respectively be n and p.
∴ The current density J = (neμe + peμh) E
and σ = neμe + peμh
Since hole density is negligibly small, therefore, we have;
σ = neμe
2. p-Type Semiconductor:
A p-type semiconductor has (i) holes as majority carriers, (ii) an equal number of immobile negative ions and (iii) electrons as minority carriers. In this case, the electron current is considered negligible.
When a p-type semiconductor is placed between a pair of electrodes, across which a voltage is applied, there will be a constant drift of the holes towards the -ve electrodes. On reaching the negative electrode they combine with the electrons coming out of the metal of the negative electrode and disappear. At the same time, an equal number of holes are generated near the positive electrode.
This is because as the holes drift away from the positive electrode they leave behind the un-neutralized immobile negative charges. These charges and the positive electrode give rise to an electric field which causes the ionizing electrons to leave the acceptor atoms and come to the +ve electrode where they are lost. By losing an electron in the above process the acceptor atom attempts to steal an electron from an adjacent bond.
A hole is thus formed. The generation of holes takes place at the rate equal to the rate at which they disappear at the negative electrode. This rate is again determined by the magnitude of the applied voltage and the conductivity of the semiconductor. The current inside the p-type semiconductor is due to the motion of the holes whereas in the external circuit is due to the motion of electrons.
Again in p-type semiconductor, the electron and the hole densities are not equal and so the equation (i) is not valid in p-type semiconductors. Let n and p be electron and hole densities respectively, therefore, we have current density J as-
J = (neμe + peμh) E
and σ = neμe + peμh
Since electron density is negligibly small, therefore, we have;
σ = peμh
Analysis of Drift and Diffusion Currents:
A semiconductor may be doped with impurity atoms so that the current is due to either electrons or holes in contrast to the metal where current carriers are electrons only. The transportation of the charges in a crystal under the influence of an electric field gives drift current and as a result of non-uniform concentration gradient gives diffusion current.
Both drift and diffusion currents are discussed below:
i. Drift Current:
Let us consider a conductor of length L containing N electrons. If a constant electric field E is applied to the conductor, the electrons would be accelerated with acceleration;
a = qE/m
where, q being the charge of the carriers. If v is average speed of carriers, then
v = μE
where μ is mobility of charge carriers.
Now a charge carrier q takes time t to travel a distance L in the conductor, the total number of electrons passing through any cross-section A of the wire in unit time will be N/t. Thus, the total charge per second passing any area is the current I and is given by-
I = Nq/t = Nqv/L
because L/t is the average or drift velocity of charge carriers. The current in above equation is known as drift current because charge carriers have acquired drift velocity on application of electric field.
If J is the current density, then
J = I/A = Nqv/LaA
Also N/LA represents charge carrier concentration, therefore,
J = nqv
Since this derivation is independent of the form of conducting material, so it is equally applicable to conductors and semiconductors.
ii. Diffusion Current:
The additional mechanism responsible for transportation of charges in semiconductors is diffusion which is not encountered in metals. It is possible to have non-uniform concentration of carriers in a semiconductor. Let us consider that the concentration of holes ‘p’ varies with distance x in the semiconductor and there exists a concentration gradient dp/dx. Let us consider an imaginary surface through the semiconductor (See Fig. 7.9).
Because of thermal energy, the holes will continue to move back and forth across this surface. Now we should expect that in a given time interval, more holes will cross the surface from higher concentration side to lower concentration side. This transportation of holes across the surface constitutes a current in the positive x-direction. The diffusion hole current density Jp is proportional to the concentration gradient and is given by (using Fick’s law of diffusion).
Where, Dp is the diffusion constant for holes whose units are m2 per second. In Fig. 7.9, p is decreasing along x-direction, therefore dp/dx will be negative and Jp will be positive in the positive x-direction.
We can also write the similar equation for diffusion electron current density-
It is possible that both potential gradient and concentration gradient exist simultaneously within the semiconductor. In such a case, the total current will be sum of drift current and diffusion current.
Einstein Relation:
It has been observed by Einstein that the diffusion and mobility are not independent and there exists a relationship between them i.e.;
Above relation is used to calculate the diffusion constant at different temperatures, if mobilities are known.
Continuity Equation:
It is well known that if we disturb the equilibrium concentrations of carriers in a semiconductor material, the concentration of holes or electrons will vary with time. We will now derive the differential equation which governs this relationship.
Let us consider the infinitesimal element of volume of area A and thickness dx within which the hole concentration is p. Here we will consider the hole current Ip only in one direction i.e. along x-axis. If the current entering the volume at x is Ip at any time t and leaving at x + dx is Ip + dIp at the sometime t.
Since the magnitude of charge carrier is q, then dIp/q equals the decrease in the number of holes per second within the volume element Adx. Since the current density Jp = Ip/A, we have 1/qA dIp/dx = 1/q dJp/dx = decrease in hole concentration per second due to current Ip.
Also, we know that due to thermal generation, there is an increase per second of g = p0/τp holes per unit volume, and a decrease per second of p/τp holes per unit volume due to recombination. Since charge can neither be created nor destroyed, the increase in holes per unit volume per second, dp/dt, must be equal to-
Since p and Jp are functions of both t and x, so it is appropriate to write partial differential form i.e.
The above equation is known as the law of conservation of charge or the continuity equation for charge. The continuity equation for electrons can be written as-
τp and τn are life time of holes and electrons respectively. In above equations, p0 and n0 are the equilibrium concentrations of holes and electrons respectively.