Theory of electrical characterization
of (organic) semiconductors

by Peter Stallinga
Universidade do Algarve




Device structures
Bulk samples, ohmic contacts

Schottky barrier
   Depletion width
   AC: Capacitance
      non-uniform doping
   DC: Current
       space-charge limited current (SCLC)
      displacement current
   AC: Conductance, Loss, Loss-tangent and Spectra
   Deep levels
   Interface states
   Summary of AC measurements
   Admittance spectroscopy, Cole-Cole plots

Measuring mobility
   Field-Effect Transistor (FET)
   Hall measurements
   Transient electro luminescence, Time of flight (ToF)

Optical effects

Time resolved measurements
  Emission and capture of carriers
  Capacitance-transient spectroscopy
     majority traps
     minority traps

Experimental setup

Measurement summary



      Peter Stallinga (
page location:
page last changed:
     26 February 2004


The following text is intended to help in understanding the electrical measurements on semiconductors. In the time that I have been working here in Faro I have made notes about the measurements and made them into a file. I thought it would be nice if I share the accumulated knowledge with others.
With the help of the text, you will be able to understand most of the electrical measurements of other people. Also, the text can function as a starting point for new researchers in the area.
The text is not complete. Also, the text is probably full of typos and mistakes in the equations. If you have any comments please e-mail me at
These pages need a recent browser. Without it you will not be able to see the equations correctly. (In older versions there is no possibility to display Greek symbols or sub- and superscript). The pages were written with Netscape Communicator 4.76 which you can get at Netscape. Some of the pictures look bad on screen, some bad on the printer. It is not possible to have high quality images for both output devices at the same time. Use "view image" to see an enlarged picture when the screen quality is insufficient in normal viewing mode.
© The copyright of this document belongs to the University of the Algarve.
P.S., Faro, October 2001


A good question is: how can a polymer be a metal? Our daily life experience tells us that polymers ("plastics") are very good isolators. Indeed, the conductive properties of organic materials were discovered by accident (see Scientific American of July 1995 ); not many people expected such properties. The obvious answer to the question is that the conducting polymers have an abundance of free (or very loosely bound) electrons.
Then the next question is: why are some organic materials (polymers) semiconductors? The answer is well described in section IVB of Chapter 2 of "Advances in Synthetic Metals, Twenty years of Progress in Science and Technology" for the basic conducting polymer poly acetylene. In all systems of conductive organic materials, the conducting chain or "backbone" of the molecule consists of unsaturated carbon atoms (example --CH-CH-CH--; each atom is only threefold coordinated). Each carbon atom contributes a single pz-electron to a bond with the neighboring atoms.  This situation is unstable to small symmetry-breaking distortions (so called Jahn-Teller).  In this case an alternating shortening and lengthening of the bonds, or alternating double and single bonds. The band gap that is opened up by this symmetry lowering is in the order of 2.5 eV and this is still in the range of semiconductors. The table gives a comparison of the band gap of various semiconductors. Organic semiconductors mostly fall in the category "wide-band-gap semiconductors".
 material  band gap
diamond C 5.47 eV
GaN 3.36 eV
polymer 2.5 eV
GaAs 1.42 eV
Si 1.12 eV
Ge 0.66 eV

With a bandage of 2.5 eV the polymer would not be conducting at all.  This distance between the valence band and conduction band is much too large to allow for free carriers to be generated at room temperature.  For that too happen, an electron would have to acquire huge thermal energy to make the jump to the conduction band. There are still some ways that we can have free carriers in the polymer:

Most organic semiconductors have a sufficient doping level to make them p-type (abundance of free holes), although for the most pure ones, the density of free holes is rather small and these can still be considered isolators.
After having answered the question where the conductance is coming from the next question is how the conductance looks. There are models with polarons, bipolarons, etc.
In classic semiconductors, the free electron and hole can bind to each other (on basis of their electrostatic [Coulomb] attraction) forming excitons which are hydrogen-like states of the electron around the hole. Due to the low mass of the hole, the binding energy of these excitons is very small (meV's). The exciton can also be bound to a defect and the energy of the system can then be further reduced. In classic semiconductors these energies are rather small, but in polymers, these energies can be huge and mid-gap states can result.

For the theory presented here it doesn't matter what shape the carriers take. We will assume that all carriers are free carriers. One important difference between classical semiconductors and polymers is worth to be pointed out here, though. In classical materials, the conduction is always 3-dimensional, while polymers are 1-dimensional conductors. Or, at least there is an anisotropy; the conduction in one dimension is better than in the other two. Hence, there exist (at least) two processes for conduction with two activation energies. One for conduction along the chain and one for "hopping" to near chains.

Device structures

From the semiconductors we can make several types of devices. The simplest one is a device with only ohmic (non-rectifying) contacts.  This can already give valuable information because we can measure the conduction model (for instance, number of conduction processes) or the mobility in Hall experiments. A little more complicated are Schottky barriers. They contain a single rectifying contact which can be used to probe the deep and shallow levels (via IV curves, CV curves, DLTS and TSC, for example). p-n junctions are very similar to Schottky barriers, but they can inject with ease electrons into the polymer. Finally we can make field-effect transistors (FETs) which will yield the mobility of the carriers.
ohmic contact IV curves
Hall measurement
transport model, shallow level
Schottky barriers
p-n junctions
admittance spectroscopy
transport model, barrier height
shallow level, barrier height
deep states
deep states
shallow level, deep levels, dielectric constant e
FETs IV mobility

Ohmic contacts, bulk samples

The simplest device structure is the bar of semiconductor with two electrodes connected via ohmic (= non-rectifying) contacts. Such a simple structure already has complicated laws of physics.
The conductivity s of a bar of p-type material is given by
s = e mp p
with e the elementary charge, mpthe (hole) mobility and  p the hole density. Because both the free-carrier density mp and the carrier mobility p are a function of temperature, the conductivity is strongly temperature dependent. In fact the hole density (given here without derivation, see Chapter 1 of Sze ) is equal to
p = 2-1/2 (NANV)1/2 exp(-EA/kT)    ~    T3/4 exp(-EA/kT)
Here NA is the acceptor concentration, NV the density of states at the top of valence band, EA the activation energy of the acceptor, k the Boltzmann constant and T the temperature.
The hole mobility depends on the limiting mechanism. We can have the following models:
acoustic phonons: mp ~ m*-5/2T-3/2
ionized impurities: mp ~ m*-1/2T3/2
optical phonons: mp ~ m*-3/2T1/2

In principle we can determine the activation energy for bulk conduction through the measurement of the conductivity as a function of temperature. Normally the resistivity is measured with a four-point probe.  Two points are applying a current and two other points are measuring the voltage.
The resistivity of the sample in the simple model does not depend on the voltage.  In more sophisticated models it does, but they will not be described here.

If we know the parameters of the device we can find the mobility directly from the DC measurements. Otherwise, we have to resort to FETs, Hall measurements, or Time-of-flight measurements, see the chapter on measuring mobility.

For insulating materials, the transport can be of yet another type. In general, it has a stronger field-dependence. Examples:

Fowler-Nordheim tunneling

J = J0 E2 exp(-E0/E)


J = J0 E exp(sqrt(qE/pe)/kT)
with E the electric field, E = V/d. In some cases, this is applicable to organic materials in LEDs. These are normally devoid of free carriers and, especially in view of the wide bandgap of some of these materials, can be considered insulators.

Schottky Barrier

A Schottky barrier is created by the intimate contact of a metal and a semiconductor surface. The Figure shows the situation of a Schottky barrier of a metal and a p-type semiconductor, for instance the polymers P3MeT or MEH-PPV used in our studies. The left drawing shows the situation before contact and the right drawing the resulting Schottky barrier after contact.
Schottky barrier before
                contact Schottky barrier after
A metal and a semiconductor before contact. Note
the different Fermi levels which will cause electrons
to flow to the semiconductor.
A Schottky barrier formed after contact of the
metal and the semiconductor. A region of un-
compensated charged acceptors results. This
"space charge" causes a voltage drop at the

At equilibrium, in the absence of externally applied voltages, the Fermi level must be constant throughout the sample, since otherwise a current would flow. In the metal the Fermi level is the top of the electron sea, while in the semiconductor, far from the interface, the Fermi level is determined ("pinned") by the impurity level. The Fermi level is matched in the following way: Before equilibrium, the Fermi level is lower in the semiconductor (when the work function of the polymer, Evac-EF = c + Vn, is larger than that of the metal, fm), therefore, electrons will flow from the metal to the polymer. This causes the build up of charges on both sides of the interface, resulting in an electric field and therefore a potential gradient according to Poisson's equation d2V/dx2 = r(x).  This is the so-called band bending. In this region, the electric field has caused the holes to move away from the interface; they drift to the top of the valence band. The result is that in this area -- of width W -- there is a surplus of negative charge caused by uncompensated charged acceptors, the "space charge region" or "depletion region", since there is an absence of majority carriers (holes in p-type semiconductors).

The parameters that describe the  Schottky barrier are:

For a Schottky barrier the forward bias is when a positive voltage is applied to the (p-type) semiconductor and a negative voltage to the metal. This will compensate the band bending and diminish the barrier.

Similar to the Schottky barrier we have a pn-junction (of two equal semiconductors with different doping details) or a hetero-junction (of two different semiconductors).  The theory of these devices is rather similar and will not be mentioned here further.

Calculation of the depletion width

of a Schottky barrier

(a) Space charge distribution, (b)
      electric field and (c) voltage as a function of xThe depletion width of a Schottky barrier can be calculated from Poisson's equation
d2V/dx2 = r(xEq.1
with r(x) the space charge density at position x. In a first approximation we can assume a rectangular distribution of the space charge. This is valid when the acceptors are all ionized and the free charges have moved out of the interface region leaving behind the now uncompensated ionized acceptors exactly up to a certain point W, see the first figure of the section with the Schottky barrier. In an equation:
r(x) = qNA   for   x <= W
r(x) = 0        for     x > W
(see the Figure).  In that case the electric field can be calculated by integrating  Eq. 1:
E(x) = dV/dx = (qNA/e) (x-W)
where the boundary condition E(W) = 0 was used. The voltage can then be found by integrating this equation to give:
V(x) = (qNA/2e) (x-W)2
(boundary condition V(W) = 0). At x=0 the voltage must be equal to the built-in voltage, Vbi, therefore:
Vbi = qNAW2 / 2e and Eq.2
W = (2eVbi / qNA)1/2
When bias is applied to the device the voltage at x=0 should be equal to Vbi-V, therefore:
W = [2e(Vbi-V)/ qNA]1/2 Eq.3

Capacitance of a Schottky barrier

When the bias voltage on the device is changed, a new equilibrium state will be reached with a new amount of space charge as described above.  The AC capacitance of a Schottky barrier can be defined as the incremental change of space charge Q in the depletion range upon an incremental change of voltage, C = dQ/dV. Note that this capacitance can depend on the applied voltage, in contrast to normal condensator which have a constant C = Q/V.  For the rectangular distribution described above, Q equals qNAWA, with A the active area of the device, and the potential was given by Equation 2, therefore
C = dQ/dV = d(qNAWA) / d(W 2qNA/2e) = eA/W
This relation is in fact a very important one.  The capacitance is determined completely by the width of the depletion region:

C = Ae/W

This shows that a Schottky barrier behaves like a condensator made of parallel plates with area A, spaced W and filled with a dielectric medium e. Substituting the calculated depletion width from Equation 3 also gives

C = A [qeNA / 2(Vbi-V)]1/2

Mott Schottky plotThis shows that a Mott-Schottky plot (C -2 vs. V) is a straight line and the slope reveals the acceptor concentration.
C -2 = 2(Vbi-V) / A2qeNA
In the same plot, extrapolating to C -2=0 reveals the built-in voltage and hence the barrier height (if the shallow level depth is known). It is important to note that NA stands for the density of ionized acceptors (and impurities). For normal shallow acceptor levels this is equal to the acceptor density, but for deeper levels it can be the case that they are only partially ionized. The capacitance (and depletion width) sees only the ionized defects. A capacitance transient for instance, monitors the change in density of these ionized defects.

The Figure shows an example of a system with an acceptor concentration of 4.28 1015 cm-3 and a built-in voltage of 0.75 V.

Capacitance in the case of non uniform doping

In case of a nonuniform doping profile, the relation Q = qNAWA no longer holds. Instead we have to use QqNA(x)Adx. The calculation of the depletion width is also different, but the relation C = eA/W is still valid. The end result is that the slope in a Mott-Schottky plot still reveals the acceptor concentration but locally at the end of the depletion width and the slope can therefore change with voltage:
dC -2 /dV = -2V / A2qeNA    (x = W = eA/C)
This enables the determination of the doping profile NA(x) through the CV measurement. The depth x is determined by the measured capacitance and the dopant density at that place is determined by the measured derivative.

Numerical calculation of the depletion layer

Alternatively we can describe the depletion layer in the following way. Instead of defining the origin x=0 at the interface, as
described above, it can also be defined as the start of the depletion layer. In that case,
  E(x) = dV/dx = (qNA/e)x
  V(x) = (qNA/2e)x2
At the interface (x=W), the potential should be equal to the band bending (VS = Vbi-V):
  V(W) = (qNA/2e)W 2 = VS
W = (2eVS / qNA)1/2
If V is increased with an amount DV, charge will flow into the interface and the depletion width will shrink. This happens at x=W (in this equation):
  C = dQ/dV|x=W   =   (dQ/dx)(dx/dV)|x=W
which should be evaluated at x=W. Note that dQ/dx equals Ar and dV/dx equals E(x), therefore
C = Ar/E(W) = A qNA / (qNA/2eVS)1/2
= A [qeNA / 2(Vbi-V)]1/2 Eq.4
as found before. This method only works for homogeneous distributions in space, when it is unimportant from which side you start your calculation. This is especially useful when you do numerical integration, since the values (V, E) at the starting point (x=0) do not depend on the end of the depletion width (x=W).  It is still very easy to calculate the capacitance, even when there is a complicated distribution of levels in energy. The computer can integrate until the cumulative band bending reaches the input parameter VS. At that point the electric field E and charge density r can be evaluated and the capacitance calculated
according to Equation 4.
Algorithm for the
                calculation of capacitance

The Figure above shows an algorithm used for calculating depletion widths and capacitances for Schottky barriers ("view image" to see it in more detail).  As an example, consider the situation of a system with two homogeneously distributed acceptor
levels, one 0.1 eV and the other 1.0 eV above the valence band with the Fermi level at EVB + 0.6 eV. Both have a concentration of 1016 cm-3. For e=5, a dot diameter of 2 mm, and a built-in voltage of 0.8 V, the Mott-Schottky plot is given in the Figure.
Mott-Schottky plot of a system with
        two levels, a shallow and a deep one (both responding)The Figure shows a simulation of a C-2V plot of a two-level system as described in the text. For voltages higher than 0.4 V (built-in voltage - ET1-EF the band bending is not large enough to put the deepest level below the Fermi level anywhere. Hence the space charge is only a result of the shallow level and we see a large slope (small concentration). For lower voltages, or reverse voltages, both levels contribute to the space charge and both are visible in the slope which is now much smaller. Note that the sharpness of the transition is due to the assumption that around the Fermi level the levels are either completely full or completely empty (emulating 0 K). The situation for T=300 K is shown with a dashed line. Note also that the two slopes do not point to the same built-in voltage Vbi. The slopes will still yield the correct acceptor concentrations, though.

DC: Current through a Schottky barrier

According to thermionic emission theory
J = A*T 2exp(-qfBp/kT ) [exp(qV/kT) - 1],
with A* the Richardson constant,
  A* = 4pqm*k2/ h3
(q is the elementary charge = 1.60217733 10-19 C, k is the Boltzmann constant = 1.380658 10-23 J/T, T is the absolute temperature in Kelvin, V the bias voltage in volt, fBp is the barrier height in volt, and h is Planck's constant = 6.6260755 10-34 Js).

Alternatively, the current can be following the diffusion theory:

J = { (q2DnNC / kT) [(2qND/e)(Vbi-V)]1/2 exp ( -qfBp/kT)}[exp(qV/kT) - 1],

Finally, a combination of the two theories, the thermionic emission-diffusion theory predicts a current following the equation

J = A**T 2exp(-qfBp/kT ) [exp(qV/kT) - 1],
with A** the effective Richardson constant.

The currents are all of the form

J = J0 [exp(qV/nkT) - 1]
In this n is the ideality factor (>1), parametrizing the deviation from theory. For classic semiconductors such as Si, Ge or GaAs the value of n is close to 1, while for polymers it is often in the order of 2.
thermionic-emission theory
diffusion theory
thermionic-emission theory IV
                curve diffusion-theory IV curve

From a DC current-voltage plot ("IV plot") we can determine

A final note: For high forward voltages, the current may be limited by the resistivity of the bulk. In that case, the current doesn't continue to rise exponentially with the bias, but only grows linearly. In the semi-log IV curves this is visible as a bending of the curve at strong forward bias.
Since the current is limited by the bulk conductivity we can again apply the theory for bulk samples.

Space-charge-limited current (SCLC)

A special case is when the currents are limited by the space charge. This means that the density of free carriers injected into the active region is larger than the number of acceptor levels. It is easy to see that we then go from situation of band bending caused by uncompensated ionized acceptor levels to the case of band bending caused by over-compensated acceptor levels. The band bending is then in the other direction (!). The local field the carriers feel when injected into the active region is then driving them out of there and back into the contact electrode. The entire current must come from the diffusion of carriers. In other words, the current is caused by the large gradient of density of free carriers.
Instead of the simple approximation with only the drift current
    I = qmenE
we have the more comples form including both the drift current and the diffusion current:
    I = qmenE + qDn dn/dx
In fact, the drift current is opposing the diffusion current as can be seen in the figure; the electric field will drive the holes back to the metal.

In general, we can expect space charge limited currents (SCLC) when the acceptor density is small (and the band-bending and E becomes negative for large amounts of injected carriers) or when the mobility of the carriers is small (so that the first term in the equation above becomes negligible compared to the second term), especially when and where the charge-density gradient dn/dx is large.

The Bell Labs group of Schön and Batlogg gives a nice summary of these measurements (PRB 58), which, in turn, is based on chapter 7.3.4. of Sze . The DC current of a device can be devided into the following types:
In the Ohmic regime, the current is proportional to the electric field. This is equal to a simple resistance.
The space-charge-limited-current regime (SCLC) occurs when the equilibrium charge concentration (before charge injection) is negligible compared to the injected charge concentration. This will form a space charge cloud neaar the injecting electrode; the concetration of the space charge rapidly dies out away from the electrode. In this regime, the current is proportional to the square of the electric field.
With the bias, the trap levels are filled. Above the trap-free voltage limit, the traps are filled and the device enters the trap-free SCLC.
The figure below summarizes this.
        J ~ mV

Space-Charge Limited Current (SCLC):
       J ~ mV 2

Trap-Free Voltage Limit (VTFL):
       VTFL ~ d2Nt

Trap-Free SCLC:
       J ~ mV 2

Note that this is the current through a Schottky barrier. That is, we assume an ohmic contact on one side and a Schottky barrier metal-semiconductor contact on the other side of the semiconductor. If we have two Schottky barriers (for instance because we use the same metal for both electrodes), we get so-called MSM (metal-semiconductor-metal) devices, which have different behavior (notably: the current is proportional to V instead of V 2 according to p. 616 of Sze. Not true, see p. 478).

Displacement current

As we have seen, the device can have a rather large capacitance.  This capacitance can give valuable information about the material.  In some cases the capacitance of the interface is obstructing the measurement, though. Imagine measuring the DC conductance of a condensator (theoretically zero). Every time the bias is changed the condensator has to reach the new equilibrium amount of charge stored. In other words, charge will flow in to (or out of) the device and this is the so-called displacement current. When the scanning speed is large (or the capacitance is large) this current can be substantial, even overshadowing the DC conductance of a device.
When the device has a capacitance C and we are scanning with a speed l = dV/dt, then the displacement current is equal to
= C
+ V dC = C dV + V dC dV

Note that the capacitance of the device also depends on the voltage (see the section on capacitance) and this also contributes to the displacement current.

The figure shows an example of a large displacement current compared to the DC conductance. (Sample: PMeT/Al, 2mm electrode diameter, dV=10mV, dt=100ms, T=300K).  The points where the current is zero have moved away from 0 V.  For upward scanning, the displacement current is positive and the situation of zero current is reached earlier, hence the crossing point is at negative voltage.  For the same reason, the zero-crossing point for downward scanning moves to the right.

The obvious remedy is lowering the scanning speed.

IV curve showing displacement

AC: Conductance of a Schottky barrier.

Spectra, Loss and Loss-tangent.

The currents in the previous section were static DC currents. For such currents, the resistance is defined as R = V/J and the conductance G = 1/R = J/V.  In AC measurements we apply a voltage that is a superposition of a DC voltage and an AC component v.
V = VDC + v sin(wt)
The resulting current can also be decomposed in a DC and an AC part:
J = JDC + j sin(wt)
JDC is the current of the previous section. The AC current, j, can easily be calculated if we assume that the AC voltage v is small. In that case, the current is proportional to the derivative of the DC current J(V) times the AC voltage. The conductance is then defined as the ratio of AC current and AC voltage
G = j/v = dJ/dV

In general G depends on the applied bias (VDC) as well as the frequency (w). Later, in the section on admittance spectroscopy we will see why it depends on the frequency. On the other hand, it is easy to show why G depends on the voltage. From the previous section we have

J = J0 [exp(qV/nkT) - 1]
and therefore the conductance is equal to
G = G0 exp(qV/nkT)
The unit of G is the reciprocal ohm (mho, 1/W) or siemens (S). In some cases the conductance  is expressed in terms of loss L:
L = G/w = L0 exp(qV/nkT)
whose units are farad (F). In either case, according to thermionic-emission theory a plot of the logarithm of conductance (or loss) vs. voltage is a straight line intersecting the vertical axis at G0 (or L0).
Conductance and Loss vs. Voltage
Finally, we can also define the loss-tangent, tand, as G/wC. The name loss-tangent stems from the fact that in a phase plot, the lossy current, comprised of wC and G, makes an angle d = tan-1 (G/wC) with the capacitance axis. For the simple model, the loss tangent shows no features in the spectra (tand vs. frequency) and is "flat" (~1/w). We will now show how deep states can produce peaks in the spectra.

Capacitance in the presence of a second (deep) level

When there are more levels in the forbidden gap the situation becomes more complex. Consider first the situation of two acceptor levels, one shallow and one deep.  In fact so deep that in the absence of bias it nowhere drops below the Fermi level.  This implies that all of the deep acceptor levels are filled (with holes) and are therefore neutral and do no contribute to the space charge and capacity; the slope in the Mott-Schottky plot represents then only the shallow acceptor concentration. For strong reverse biases the increased band bending can force the deep acceptor below the Fermi level at a region close to
the interface. Now if the level is shallow enough so that it can respond to the AC probing signal (of the order of 1 kHz) it will start contributing to the capacitance.  The apparent concentration is then the sum of the two concentrations of the acceptors. Energy band diagramThis is visible in a sudden drop and a reduction of the slope to 1/(NA1 + NA2) for biases smaller (more reverse) than Vbi- (EF -EA2)/q, as is easily seen in the Figure.  Also, note that the apparent built-in voltage seems to decrease; the intersect of the slope with the voltage axis is lowered to Vbi- NA2(EF-EA2)/q(NA1+NA2).

If the level is too deep and the probing frequency is too high compared to the characteristic level filling and emptying times
t, it does not respond anymore.  It is still visible in a certain reduction in the slope in the Mott-Schottky plot, but now without the sudden drop.  Important to note is that the slope is no longer linear and a straight-line fitting does not reveal the true acceptor concentration anymore; the Equation found for the derivative found before is no longer valid.  Also, any extrapolation of the local slope to 1/C2=0 increases to beyond Vbi.

In the presence of a donor level the situation is more complex, since the position of the electron (quasi) Fermi level is not so well known. Its position is determined by the number of free electrons in the conduction band and this depends on many factors such as the mobility, concentration and life time of these minority carriers. What we can say is that the Fermi level moves in the opposite direction compared to the hole (quasi) Fermi level. When the Fermi level doesn't cross the donor level anywhere (this is most often the case, especially for shallow donor levels), the donors just act as compensation for the acceptors and a reduced acceptor concentration is found in the plots, NA -ND.

The Figure shows two simulations. In both simulations there are two levels (one shallow and one deep). In the first simulation the deep level is responding (dotted line), while in the second it is non-responding (solid line). The parameters for the simulations are given in the Table.

Simulation with responding and
        non-responding second level
parameter value
EF 0.6 eV
Vbi 0.75 V
EA1 0.12 eV
NA1 4.28 1015 cm-3
EA2 1.35 eV
NA2 6.272 1016 cm-3
EVB 0 eV (defined)
As for the frequency response, we can say the following: For very low frequencies the deep states have adequate time to reach thermal equilibrium at all moments and the amount of charge transferred is proportional to the number of deep levels and is independent of the frequency. When the frequency is increased beyond the relaxation time of the levels (see section on emission and capture of carriers) the amount of charge transferred is diminished, simply because the levels have no time to reach thermal equilibrium. We will measure a much reduced capacitance and conductance.
To summarize: for deep levels we expect a bend or a drop in the CV plots and a peak in the loss-tangent (tand = G/wC) occurs when the radial frequency w is equal to the reciprocal relaxation time 1/t.

Interface states

In the above discussion the shallow and deep states were considered homogeneously distributed in space.  As a short excursion, non-homogeneously distributed shallow levels were described in the section on non-uniform doping. In the current section the other extreme will be described, namely levels that have a delta-function-distribution in space. The levels are only present at the interface. At first sight, nothing changes in the theory, but the effects are pronounced in the plots.

interface states at different

At forward bias, the Fermi level is under the interface states and they are therefore empty (contain no electrons). When the bias is lowered, there comes a moment where the Fermi level is resonant with the interface states and they are partly filled. In that case, when the Fermi level is modulated (by the external bias) charge will flow into and out of the interface (states). This is equivalent to a capacitance.  Because energy is lost in the process of moving the charges, part of the current is in-phase with the external voltage and is therefore observed as conductance. When the bias is further reduced, the Fermi level is completely above the interface states and modulating it will have no effect on the amount of charge present there; no current will exist that is associated with the movement of charge. We can therefore expect a peak in the capacitance-voltage  (CV) as well as a peak in conductance-voltage (GV), see the figure below.

C-2-V plot (left) and Log(G)-V plots (right) when interface states are present
As for the frequency response, this is identical to the deep states: For very low frequencies the interface states have adequate time to reach thermal equilibrium at all moments and the amount of charge transferred is proportional to the number of interface states and is independent of the frequency. When the frequency is increased beyond the relaxation time of the levels (see section on emission and capture of carriers) the amount of charge transferred is diminished, simply because the levels have no time to reach thermal equilibrium. We will measure a much reduced capacitance and conductance.
As a secondary effect, even when the interface states are not responding, they still change the CV plot. this is because charged interface states contribute to the space charge and this means that the depletion width shrinks and the capacitance increases. (Note: technically speaking this only happens when the levels are present in a non-infinitesimally thin region dW. Otherwise, if they are truly in a delta function at the interface they do not contribute to the voltage drop which is the double integral of the space charge). This is very similar to non-responding deep states from the previous section; even if the states are not responding, they still contribute the capacitance.

To summarize: for interface states we expect a peaked response in the CV and GV plots and flat Cw and Gw plots up to a certain frequency from where they will fall off rapidly. A maximum in loss-tangent (tand = G/wC) occurs when the radial frequency w is equal to the reciprocal relaxation time 1/t.

continous band of states

If, instead of a nice well-defined bunch of interface states - or, even better, a single interface state level as decribed above - we have a continous band of states in the forbidden gap of the semiconductor, we can still do the measurements. For every ac frequency we get different CV and GV plots. The states that respond best to the probing are the ones with trap-filling-and-emptying times in the order of the ac modulation period.
With the bias we can select which states we will measure (see Fig. 1 of this section). These states have a certain relaxation time (t). The deeper the level, the longer the relaxation time: t = exp(-Et/kT). If we now measure the loss as a function of the ac frequency we can expect a peak when the frequency hits the reciprokal relaxation time, wmax = 1/t. Then, if we increase the bias, this peak will move to higher frequencies. Such a movement is then immediately a proof that we are measuring interface states and not bulk states (Stallinga, 2001 ). Note that the relaxation time of bulk states is independent of where (in space) the fermi level crosses these levels and hence independent of the bias.

Summary of CV and GV measurements

Above we have shown how the simple picture of shallow levels resulting in a straight-line C-2V plot is distorted by the presence of deep levels or interface states. The table below summarizes the CV and GV measurements for Schottky barriers and p-n junctions
C-2-V plot
Log(G)-V plot
C, G/w        and           tand
E: single shallow level
x: homogeneous

straight line

straight line

"flat" (1/w)
E: + deep levels
x: homogeneous

+ bend or drop

straight line

E: + deep levels
x: interface (or local)

+ peak


peak (only at certain voltages)
E: + deep levels, 
x: interface (or local)

peak (pos. depending
on ac frequency)

peak (pos. depending
on ac frequency)

peak (pos. depending on bias)

Admittance spectroscopy, equivalent circuits

As we have seen above, the values of the measured capacitance (C) and conductance (G=1/R) depend on the probing frequency. The determination of this dependence, i.e., the shape of the spectra, is called admittance spectroscopy.

 Very often, the data can be described very well with equivalent circuits. The advantage of them is that a lot of things can be described in this way, but the disadvantage is that the physical meaning of the found parameters is often difficult to interpret. One example of an equivalent circuit that is very illustrative and that still has a strong link with the actual physical structure of the device is the following:
Equivalent circuit In the picture on the left the device is thought of as consisting of a bulk part (denoted with b) with its capacitance and resistance placed in series with the interface part (denoted with d).
This seems a very reasonable assumption. Even in such a simple circuit, the measured capacitance and conductance follow complicated equations. Even if we take the elements in the circuit to be frequency independent, the measured values are not. Some calculations can show that

C  = 
Rd2Cd + Rb2Cb + w2Rd2Rb2CdCb(Cd+Cb)
 (Rd+Rb)2 + w2Rd2Rb2(Cd+Cb)2

G  = 
Rd + Rb + w2RdRb(RdCd2+RbCb2)
(Rd+Rb)2 + w2Rd2Rb2(Cd+Cb)2

Under the approximations Rb << Rd and wRbRdCd2 >> 1 the loss tangent reduces to

tand  = 
1 + (wRb)2(Cd+Cb)Cb

capacitance and loss vs. frequencyloss tangent vs. frequency

The Figures show a simulation of the capacitance and loss (G/w) (left) and loss tangent (right) as a function of the probing frequency n (=w/2p) with the parameters as given in the Table. This resembles very much the spectra of deep levels and interface states as described in the previous sections. It is clear that for low frequencies the capacitance flattens out and saturates at the interface value Cd (assuming this is the larger of the two), while for higher frequencies it reaches the bulk value. This bulk value is in fact the so-called geometrical capacitance Cgeo which is the capacitance of the parallel plates with area A at mutual distance d filled with a dielectric medium e.
Cd 1.50 nF
Rd 4 MW
Cb 0.50 nF
Rb 2 kW


Some people prefer to present the admittance data in the form of so-called Cole-Cole plots. In that case we plot the loss versus the capacitance. The figure shows an example. For systems as described above it can be shown that the plot is a semicircle (with the center on the horizontal axis) and meeting the horizontal axis at C=Cgeo and C=Cd. If we know the dimensions of the device, A and d, we can calculate the permittivity e of the material:
e = dCgeo/A

In the figures of the loss-tangent we see a local maximum somewhere at a frequency

wmax  = 

Position of maximum in loss-tangent vs.
        temperatureThis gives a handle for the measurement of the activation energy of the bulk conductance. When the temperature is changed, the bulk resistance changes and - assuming that the capacitances are relatively independent of temperature - the position of wmax should follow the bulk resistance. With the assumption that the bulk conductance is singly activated, the resulting temperature dependence of wmax is

wmax = w0 exp(-Eb/kT)
Hence, an Arrhenius plot of the position of the peak in the loss-tangent graph, log(wmax) vs. 1/T, will yield the bulk activation energy Eb. The Figure gives an example for an MEH-PPV device (P. Stallinga et al., submitted to J. Appl. Phys., 1999 )

In the above theory, all parameters are assumed constants. This is not a correct picture. For instance, the capacitance of the depletion layer Cdis temperature and frequency dependent. A distribution of levels in energy will cause a frequency dependent capacitance of the form

Cd = C0 1 -
1 + w2t02 exp[ 2 (EV - EF + qVS)/ kT]
1 + w2t02 exp[ 2 (EV - EF)/ kT

For w=0 Cd reaches the low frequency value C0 (which is still voltage dependent), while for infinite frequencies the capacitance drops to zero. For deep levels, with t very slow, the most visible result is that the capacitance doesn't flatten out at lower frequencies. Otherwise not much changes. The position of the maximum of the loss tangent, for instance, is rather invariant to such effects.

capacitance and loss vs.
        frequencyloss tangent vs. frequency

param. value
C0 1.50 nF
T 300 K
Rd 4 MW
t0 100 ps
Cb 0.50 nF
EF-EV 0.2 eV
Rb 2 kW
VS 1 V

The Figures show a simulation with the same parameters as before but now with a frequency dependent capacitance. Note the non-saturating capacitance at lower frequencies.
Some remarks:

Measuring mobility

The (drift) mobility of the carriers is defined as the ratio of carrier velocity v and electric field E:
v = mE
That is, if we had any way of directly measuring the velocity of our carriers, assuming we know the electric field, we could directly find the mobility. In reality, measuring the velocity is difficult and we have to resort to more indirect methods.
To measure the mobility we have many tools to our disposal. The following sections describe the most common: Field-Effect Transistor (FET), Hall, Time of Flight (ToF). In some cases it is also possible to measure the mobility directly from an ohmic sample. For organic semiconductors the most common technique is via FETs. This is because it is a rather simple and cheap method. Moreover, the rather low mobilities of the carriers in organic materials make measuring in a Hall set up difficult. Since the results of the various measurements can be varying tremendously, we always have to specify wit which method the mobility was measured. Therefore, there exist mFET, mHall, mToF, etc.

Field Effect Transistor

(See also special pages on FETs for more detail)
A field effect transistor (FET) is worth to mention here because we can get additional information about the material. An FET is a structure in which the material under study is deposited on top of oxidized silicon. The silicon substrate is low ohmic and acts as the gate. On top of the polymer layer metal electrodes (source and drain) are deposited in a pattern with small inter-electrode distance. The voltage on the gate now controls the current through the source-drain channel. It either stimulates it (in an enhancement FET) or decreases it (in a depletion FET). In either case, we can determine the carrier mobility m from the data.
cross-section of an FET
symbol meaning
L channel length
Z channel width
d oxide thickness
VG gate voltage
VD drain voltage
ID drain current
m (hole) mobility
MOSFET (Metal Oxide Semiconductor Field Effect Transistor) structure. In this case the metal is replaced by low-resistive silicon.  The current from source (S) to drain (D) is controlled by the gate (G) voltage
The source-drain current, ID follows the equation

ID = (Z/L) m Ci [(VG-VT)VD-aVD2]

or for small VD:
LIN:           ID = (Z/L) m Ci [(VG-VT)VD]

The saturation current follows the equation

SAT:        IDSAT = (m Z/L) mCi (VG-VT)2
with m =1/2 for light doping (see p. 442 of Sze ). L is the channel length (distance between source and drain), Z is the channel width (length of source or drain electrode). The insulator thickness (d) or source-gate or drain-gate distance enters into the equation via the oxide capacitance Ci = e/d (capacitance per unit area), which can either be measured or calculated. For example, silicon oxide has an er of 3.9, with a d of 8000 Angstrom this gives Ci = 43.2 10-6 F/m2.

Here is an example: The picture on the right shows the IV plots (ID vs. VSD) of an FET for various gate-source voltages (VS=0), ranging from 0 V (top) to -35 V (bottom) in steps of 5 V. With a source (and drain) electrode length of 25 cm and distance of 25 mm and oxide thickness of 800 nm the mobility of the holes can be calculated to be 3 10-6 cm2/Vs.

Effects that we have to be aware of are
Velocity saturation. For high fields, the velocity of the carriers is not proportional to the field, but instead saturates to a certain value. This makes that the saturation current (Ids for Vds > Vgs-Vt) depends linearly on the gate voltage rather than quadratic (see p. 450 of Sze, )
Contact resistance. If we ignore the contact (or series) resistance at the electrodes of the device, we can underestimate the mobility, see Horowitz et al, JAP 1999).
Short-channel effects. These occur when the inter-electrode distance becomes comparable to the depletion layer widths. In this case, all sorts of deviations from standard theory can occur, see section 8.4 of Sze .
Other non-linear effects can make the mobility (gate) voltage dependent. In most cases the mobility drops for increased gate-voltage (Fig. 8.13 of Sze, ), but also the opposite can happen (Vissenberg, PRB 1998, ).

A good starting point for measuring organic FETs is the article of Horowitz, "Organic Field-Effect Transistors" .

Hall measurements

basic Hall measurement

When we take a bar of semiconductor without any rectifying contacts we can measure its resistivity r, or conductivity s=1/r which can be expressed in terms of mobility and free-carrier density:

s = e mpp
(e is elementary charge, mp is hole mobility, p is hole density). In a Hall experiment we can simultaneously measure the product ep and therefore we can determine the mobility of the carriers.  This works as follows. A voltage is applied along the x-direction and the conductivity is measured:
s = (Ix / Vx) (lx / Wydz)
mp = (Ix / Vx) (lx / Wydz) / ep           Eq.1
with Ix the current, Vx the voltage, lx the sample length, Wy the sample width and dz the sample thickness. At the same moment a magnetic field is applied along the z-axis. This creates a Lorentz force on the moving holes
FyB = e Bzvx
with Bz the magnetic field strength and vx the drift velocity of the holes. Holes will therefore move to the edge of the sample where the built-up of this charge will create an electric field that will compensate the magnetic forces
FyE = -eEy
At steady state the two forces are equal
Bzvx = Ey
The electric field Ey can be determined by measuring the voltage Vy on the sides of the bar and dividing by the sample width Wy. If we further realize that the velocity of the holes is equal to Jx/ep = Ix / (Wydzep) we can express the above equation as
Bz Ix / (Wydzep) = Vy/Wy
ep = Bz Ix / Vydz
If we apply this to Equation 1 we find that in a single experiment we can determine mp:
mp =
The mobility measured in this way is the so-called Hall-mobility.  It can deviate from the mobilities measured in other experiments. For instance, the assumption is made that all holes move with the same velocity vx. This is, in fact, not true. A Boltzmann distribution is a much better approximation. Such effects are normally accounted for in a correction factor r, which will not be discussed here. The factor lies in the range 1-2. See Chapter 3.2 of Blood and Orton .
Note that the sign of the voltage Vy depends on the type of the carriers. In this way we can unambiguously determine if the semiconductor is p-type or n-type.

Time-resolved Electroluminescence

This method is probably the most direct way of measuring the velocity and hence the mobility of the carriers. With an electric pulse, electrons and holes are injected from opposite sides into the active layer. When the carriers meet, somewhere in the active layer, they will recombine and produce light. The time delay (Dt) to the onset of luminescence directly gives you the speed of the fastest carriers (in most organic materials these are holes), assuming you know the thickness of the layer (d) through which the carriers must travel

m = v/E = (d/Dt) / (V/d) = d2 / VDt

Time of Flight (ToF)

This method is similar to time resolved electroluminescence, but instead of measuring the light we measure the time-resolved current on the other side of the voltage or light pulse. The results are the same.

Optical effects

Wherever there is a rectifying contact (be it a Schottky barrier or a hetero-junction) we can expect strong optical effects.  The device can either be used as a photo-detector or an LED (light-emitting diode).  The underlying mechanism is the same in both cases, namely the depletion zone which is void of free carriers in thermal equilibrium and has (for the same reason) a strong electrical field. Let us first look at the reason why a rectifying contact can emit light:

Light emitting diode:
Under forward bias, majority carriers (holes from the polymer and electrons from the metal) are injected into the depletion region. As the name implies, and as we have seen in the beginning of this document, in thermal equilibrium density of free carriers in this region is very low.  The injected carriers will therefore try to recombine in a constant way restoring equilibrium. The energy that is released in the process is carried of in the form of photons, which, if they are not reabsorbed in the material can be detected externally.
The energy of the photons, and hence their wavelength (read color) is determined by the difference in energy between the holes and the electrons. Holes are mostly available in the top of the valence band and electrons in the bottom of the conduction band. This energy difference is therefore equal to the bandgap. For classical semiconductors (e.g. Si) this energy is in the range of infra-red.  For wide bandgap semiconductors (such as GaN and most polymers) the bandgap is in the order of 2.5 eV and this implies that the photons are in the blue part of the spectrum. A decade ago, the existence of blue LEDs was still a thing of the future and research was focused on finding a suitable material. Candidates included SiC, GaN, ZnSe, etc.
In the ideal case, all injected electrons recombine with all injected holes (somewhere in the middle of the depletion region) and the quantum efficiency is then 1, namely one photon emitted for every electron-hole pair injected.

It has to be noted that for an LED it is not necessary that there exists a rectifying contact. The only prerequisite is a region of low equilibrium-free-carrier concentration into which free carriers are injected.  In fact, most polymer LEDs are made of ultra-pure (and hence highly depleted) semiconductor material sandwiched between two metals (often transparent ITO, indium-tin oxide) which act as electron injectors (cathodes) and hole injectors (anodes).  Because there is no space charge inside the polymer there exists a linear voltage drop, instead of the quadratic voltage drop of Schottky barriers, see the figures below.

There are two ways the light efficiency might be impaired:

  1. Unbalanced carriers. If only one of the types of free carriers are injected into the depletion region, there is obviously no recombination.  In that case, the excess energy is carried off in the form of heat.
  2. Non-radiative recombination. If there exist recombination paths without the emission of photons, the light efficiency can also be reduced. Many defects are non-radiative recombination centers. In a way, defects are needed to provide the material with free carriers in the neutral regions, but in another way, defects are unwanted because they can reduce the luminescent efficiency. On the other hand, for fast, non-optical electronics, recombination centers are needed
LED band diagram
a working LED
an LED hindered by un-
balanced injected carriers
an LED limited by non-
radiative recombination paths 


Photo-detector / solar cell:
Schottky barrier as optical detector
In a photo-detector or solar cell, the opposite process takes place.  The energy of photons is used in producing electron-hole pairs which are subsequently broken up by the strong electrical field in the interface. These free carriers arrive at the electrodes where they contribute to an external current. In this case, it is also not really necessary that there exists a rectifying contact. The only prerequisite is a strong electric field at the place where the electron-hole pair are formed. This field should be stronger than the binding energy of the e-h pair. At Schottky barriers such fields are easily present because the voltage drop occors over a very small length, but at sandwich structures large external voltages are needed. Moreover, no short-circuit currents are possible; The moment we short-circuit the device, the field is (nearly) gone and carriers are not separated or transported to the electrodes. Thus, while the sandwich structures are still good for photodetectors, they are not suitable for solar cells.

The parameters that characterize a solar cell are

The value for Voc is always positive and the value of Jsc is always negative; the current resembles a reverse-bias current.
The product of J and V has a minimum somewhere.  This is the maximum power Pmax that the solar cell can generate.
IV curves in dark and under
                illumination (log plot) IV curves in dark and under
                illumination (linear plot)

The figures show the IV curves in the dark (black) and under illumination (red) in logarithmic scale and in linear scale.

Emission and capture of carriers

For all the transient techniques of monitoring a system parameter over time we bring the system off-equilibrium in some way and observe how fast it recovers to thermal equilibrium.  The bottleneck process is in most cases the emission and capture of carriers on deep levels, although in some rare cases it is the diffusion of the free carriers out of the depletion zone. This section describes the relaxation process of reaching thermal equilibrium in the deep states.

Emission and capture basicsThe Figure shows the processes of capture and emission of holes from a deep acceptor level at ET with density NT. The number nT stands for the number of levels filled (with holes). The rest of the holes, NT-nT, must then be in the valence band (in this simple model which neglects a contribution of the intrinsic holes originating from the conduction band). The other numbers, ep and cp stand for the emission and capture rates (in units number-per-second-per-available-hole). The number of holes actually being emitted and captured is then proportional to these values and the number of available holes in the states: cp(NT-nT) holes per second will be captured onto the trap and epnT will be emitted per second. In equilibrium there is no net transfer of holes. This requires

epnT = cp(NT-nT) Eq.1
The holes must also obey the Fermi-Dirac distribution (note that this is the distribution for holes! Levels are full when they are above the Fermi level):
nT / NT =1 / [1 + exp{(EF-ET)/kT}] Eq.2
and therefore we can calculate the ratio of emission and capture:
ep / cp = exp{(EF-ET)/kT}    Eq.3
To verify if this is correct: imagine a situation with the trap far below the Fermi level. In that case, the above equation tells us that the emission rate is much larger than the capture rate and the levels will be empty (contain no holes). This is exactly what we expected on basis of the starting assumption of the position of the Fermi level.

As seen above, the ratio of emission and capture depends on the Fermi level. We now make the assumption that the emission of holes from a trap level to the valence band is a property of the trap1. The capture of holes from the valence band is proportional to 1: the capture cross-section, sp. 2: the number of free holes in the valence band, p. 3: the average thermal velocity of these free holes, <vp>.  In Equation 1:

cp(NT-nT)  = sp <vp> p
Also we know that the density of free holes is governed by the Fermi level:
(NT-nT) = p = NV exp{-(EF-EV)/kT}
Further we need an approximation that most of the trap levels are empty. Equation 2 then becomes
nT = NT exp{(ET-EF)/kT}
And with these three terms, the emission rate of  Equation 1 becomes:
e = sp <vth> NVexp{-(EF-EV)/kT}/ NT exp{(ET-EF)/kT}
= sp <vth> [NV / NT] exp{-(ET-EV)/kT}
If we substitute the absolute expressions for the parameters
sp = s0 exp[ -DEs/kT ]
<vp> = [3kT / m*]1/2
NV = 2MV [ (2pm*kT)/(h2) ]3/2
we can calculate the explicit temperature dependence of the emission rate

e = gT2spa exp(-Epa/kT)

This defines the pre-factors g and spa, which are not so interesting for the experiment. On the other hand, a plot of ln(ep/T 2) vs. 1/T will show a straight line and reveals the activation energyEpa of the trap.  In the case when there is no capture (for instance when all the free holes are immediately swept away by the electric field), the thermalization time constant is equal to the reciprocal of ep and can be measured in the experiment. This gives a possibility of determining the trap activation energy directly, as we will see later in the section on DLTS.

1: we make here the assumption that the number of available "destination" states is infinite so that is not causing a bottleneck in the emission process

Capacitance-Transient spectroscopy (predecessor of DLTS)

band diagram at 0 V band diagram directly after
                switching the bias

In DLTS (deep level transient spectroscopy) the capacitance is monitored over time after a sudden change of bias. The above figures clarify what happens to the capacitance after a switch of bias. Easiest is it to understand if we remember that the capacitance is directly linked to the depletion width:

C = Ae/W

So, if for instance for some reason the depletion width increases, the capacitance decreases.
Assume that the device is completely in thermal equilibrium at 0 V bias (left Figure). All the levels, even the deep ones, have their thermal equilibrium populations and a certain depletion width and capacitance is attained. (See the section on Calculation of the depletion width if you want to read again how to find the depletion width via double integration of Poisson's Equation d2V/dx2 = r). The depletion width W is determined by
r(x)d2x = Vbb
The integration is done from x=W to x=0. Vbb is the total band bending or voltage drop, Vbb = Vbi-V. Note that in the above Figures, the charge density is not constant; inside the depletion width there exists a smaller region where the deep level is below the Fermi level and where the space charge is larger.

When the voltage is switched to reverse bias many things will happen. A different voltage drop Vbb occurs at the interface. This implies a new depletion width and a new capacitance according to the above equations. The new equilibrium capacitance takes time to reach. The following things will happen:

Capacitance transient for a majority
      carrier trapTo summarize: after a switch of the voltage to reverse bias we first observe an immediate decrease of the capacitance (1) and after that a slow (partly) recovery of the capacitance at time scale t (3). The explicit form of the time dependence of the capacitance is
C(t) = C0 + DCexp[-t/t]

To find the trap activation energy, Epa, we can use the equation for the emission rate ep (=1/t) found in the section Emission and capture of carriers. e = gT2spa exp(-Epa/kT). If we monitor the characteristic decay times of the transients as a function of temperature and plot this in the form ln(tT 2) vs. 1000/T, the resulting straight line will yield the activation energy.

Note: to truly measure the capacitance of the interface, Cd, we must chose a frequency low enough to be below the cut-off frequency of the bulk, see the section on admittance spectroscopy. For higher frequencies we would measure only the bulk properties, which have no transient effects.

capacitance transient of minority traps

band diagram at 0 Vband diagram imediately after switching the bias
For minority traps the idea is similar. We have to keep in mind, though, that a minority trap is a trap that more readily communicates with the minority-carrier band than with the majority carrier band. For instance, an electron trap will thermalize with the conduction band rather than the valence band as described in the previous section.  In other words, we have to draw the picture with the minority quasi Fermi level EFn rather than the majority quasi Fermi level EFp. As explained before, the minority quasi Fermi level moves in the opposite direction compared to the majority quasi Fermi level.

When the voltage is switched to reverse bias, the following things happen (the differences for minority carrier traps are indicated in a bold font):

Capacitance transient of a
              minority carrier trap To summarize: after a switch of the voltage to reverse bias we first observe an immediate decrease of the capacitance (1) and after that a slow further decrease of the capacitance at time scale t (3). This is very similar to majority carrier traps, but the sign of the transient is reversed; DC is positive here. We can still determine the trap activation energy by monitoring the characteristic transient decay time t as a function of temperature.
As an example the following graph of energy levels in MEH-PPV on silicon substrates as determined by capacitance-transient spectroscopy (taken from P. Stallinga et al., Synthetic Metals 1999; see reference list). The blue dots are majority-carrier traps with upward-trend transients and the red dots are minority-carrier traps with downward-trend transients. The size of a dot represents the transient amplitude (in the right panel it is multiplied by 5 for better visibility). The plot reveals several trap levels and shows the result of the first successful capacitance transient experiment in MEH-PPV.
transient times as a
                function of temperature revealing the activation

DLTS (deep level transient spectroscopy1)

Historical background
Deep level transient spectroscopy is probably the most used technique in the electrical characterization of semiconductors. This is based on the fact that it is fast, cheap in terms of equipment, and very sensitive. Commercially sold set ups exist in abundance.
The inventor of DLTS is D.V. Lang who wrote an article how to analyze capacitance transients in a systematic way , at that time enabled by the advent of modern computers with their data-crunching and experiment-steering powers. Looking back at it now, it can be said that "it was time" for such a technique to be invented. Lang did it and is now one of the most cited people in literature. His first article in the Journal of Applied Physics is still the best source for understanding the technique. Most things are described very elegantly there. I will do my best to describe it here.
When the computers became more powerful more modern versions of DLTS were invented that use more of the data and use more and more-complicated calculations. The result is an increased resolution and sensitivity. The Laplace-DLTS technique is a good example. This uses the entire transient in a Laplace transformation to yield a much higher resolution. Obviously, for this a lot more calculating power is needed.

Basic principles
How DLTS works is the following. Capacitance-transients are recorded in the same way as described above. The device is placed in the emptying voltage VE and the system is allowed to reach thermal equilibrium. For a short time the device is placed under the filling voltage VF. In this times the traps will fill with charges. When the voltage is switched back to VE a transient is observed as described in the previous sections. From this transient two samples are taken at times t1 and t2 after switching the bias. The DLTS spectrum is now the difference in capacitance at these two times as a function of temperature:

S(T) = C(t1) -C(t2)

This means (substituting the explicit form of the transient, see the section on capacitance-transient spectroscopy)

S(T) = [exp(-t1/t) - exp(-t2/t)]
The maximum of this signal occurs when the relaxation time t reaches the value tmax which can be found by differentiating S(T) with respect to t:
tmax = (t2-t1) / [ln(t1/t2)]

The explicit temperature dependence of t is (see section on emission rates)

t(T) = t0T-2exp(Epa/kT)
From this it is clear that the temperature at which the maximum DLTS signal occurs does not reveal the activation energy directly, unless t0 is known. If not, at least two measurements have to be made with different time windows (t1, t2) and the two points (Tmax, tmax) entered into the above equation will yield the trap activation energy Epa.

Parameters of DLTS are:

Capacitance transients at
                several temperatures resulting DLTS signal for
                majority (red) and minority (blue) traps
 Capacitance transients at several temperatures  Resulting DLTS signal for majority traps (down)
 and minority traps (up)

The Figures show a simulation of DLTS. The left figure shows simulations of transients caused by a 0.2 eV deep majority carrier trap for various temperatures (44 K to 52 K in 2 K steps). For the lowest temperatures the transients are very slow (bottom trace) and the difference between the capacitance at t1 and at t2 is small, hence the DLTS signal S(T) is very small. When the temperature is increased the transients become faster and the DLTS signal increases. Note that the signal has negative sign; for majority carrier traps, the transient has an upward trend and therefore C(t1) is smaller than C(t2). When the temperature is further increased, the transients become so fast that they already died out before the time window (t1-t2) and the DLTS signal has vanished again. The figure on the right summarizes this for the majority carrier trap (red dots; lower trace). In the same figure a comparison is given with a minority carrier trap (blue dots; upper trace) whose DLTS signal has the opposite sign because the related transients have downward trends. Also, in this simulation the minority-carrier trap is chosen to be a little more shallow (0.16 eV). Assuming that the pre-factor t0 is identical (1.3 10-18 s in both cases) it means that the peak has shifted towards lower temperatures.

The advantage of this scheme is that it can be done completely automatically with a minimum of data processing. The disadvantage is that of the entire transient only two points are used; most of the data are thrown away and the duty cycle of the measurement is very low. In Laplace DLTS the duty cycle is much higher because the entire transient is used in a Laplace transformation. The cost is a more elaborate calculation; only modern computers can be used.
In reality, DLTS is more often used not to determine the trap activation energies but more to show the presence of certain impurities through their fingerprint spectra and to determine their densities through the intensity of the DLTS spectra and through the underlying amplitude of the transients. This is based on the fact that the amplitude of the transient is linearly proportional to the density of the deep defect relative to the density of the dopant (assuming that this ratio is small).

DC/C = N / 2(NA-ND)

1: To be precise, the word spectroscopy is misplaced, because in a spectroscopy experiment an experimental value is monitored while the frequency is scanned; a spectrum is the visualization of an observable as a function of frequency. In DLTS no frequency is scanned, so the name is slightly inadequate. In DLTS (normally) the temperature is scanned and the DLTS-signal intensity is plotted. The end plot resembles a spectrum in that it shows peaks and that it is a "fingerprint" of the defect in the material. It would have been better if "scanning" was used for the S in the acronym DLTS.

TSC (Thermally Stimulated Current)

With temperature stimulated current experiments trap level depths and concentrations can be found. The underlying idea is that the current that is needed for restoring thermal equilibrium can be monitored. In practice this means that the sample is cooled down and the thermal equilibrium is disturbed in one way or another. This can be done either by applying a strong bias (forward or reverse, while cooling or afterwards) or by illumination. At low temperatures, the relaxation time of the sample is so large that it cannot relax back to its ground state. When the sample is warmed up, the charge emission rates become appreciable. The emitted carriers flow out of the interface region and combine into a minute, but observable, external current, to so-called TSC. When the temperature is further increased, the emission rates become very fast and the current increases. When thermal equilibrium is reached again, the charges are no longer emitted from the traps and the related current drops back to zero.
Band diagram at zero bias Band diagram at forward bias
Band diagram at 0 V (cooled at
                forward bias) Process of emission of holes
The Figures above show the procedure of Temperature Stimulated Current: a) The device in zero-bias at room temperature, b) The device is placed in forward bias to fill all the traps. c) The sample is cooled down and the bias is removed; all the traps are still frozen out. The dotted circles show the place where the traps are off-equilibrium. (In this simple picture, the Fermi level is considered to be independent of temperature; the shallow acceptors are not frozen out yet). d) The temperature is increased and at a certain temperature the charges (holes) are emitted from the traps. These charges are rapidly pulled out of the depletion zone and a current is monitored  This continues until all the traps (within a certain region indicated by the dotted circles) are emptied.

From TSC, the activation energy of the trap can be determined in the following way (p-type material):
In the depletion zone, there are no free carriers and the capture of carriers does not take place. Holes are emitted from the traps at a rate ep and the number of filled traps nt therefore changes with a speed

dnt / dt = -epnt Eq.1
If we assume that all the charges that are emitted immediately drift towards the bulk, without being recaptured, all contribute to the current:
J = epnt
When we make a temperature scan of this current, one important parameter is the temperature at which the maximum of the current occurs. At this place, the derivative of the current, dJ/dT=0. According to the above equation, the derivative of the current is equal to
dJ / dT = nt (dep/dT) + ep (dnt/dT)   Eq.3
Note that
dnt / dT = (dnt/dt)(dt/dT) = -epnt/b
where Equation 1 was used and the reciprokal scanning rate b = dt/dT was defined. This can be put into Equation 3 and to find the maximum, the result should be set to zero.
nt (dep/dT -ep2/b) = 0
One trivial solution is nt=0. This occurs at the end of the scan, when thermal equilibrium is restored. The other temperature, Tm, can be found by substituting the emission rate found before (see the section on emission rates), resulting in
(sgTm4/b) exp(-Ea/kTm) = (2Tm + Ea/k)
For kTm << Ea
ln(Tm4/b) = Ea/kTm + ln(Ea/sgk)
To determine the trap activation energy Ea, a set of scans should be made with different scanning speeds b. Each time the temperature Tm at which maximum current occurs is noted and the slope of a plot of ln(Tm4/b) vs.1/Tm is then proportional to Ea/k.

The activation energy can also be found directly from a single TSC scan. A peak in a TSC scan follows the equation

I  =
A exp(-Q)
 [ 1 + B exp(-Q)Q-2 ]2
with A and B parameters depending on the properties of the carriers and the trap and the scanning rate b, and Q = EA/kT  (see Karg et al., 1999 ). A is independent of scanning rate, while B ~ 1/b. The interdependence of the parameters makes the fitting very difficult, though.

In either case, the integrated current over time reveals the number of defects emptying their charge and this can then be related to the defect density if we know the dimensions (depletion width W *and electrode area A) of the active region.

NA = (1/eWA) I dt
The figure shows an example of a TSC measurement with parameters: Ea = 0.5318 eV,  A = -0.269 A, B = 9.49 1012 (b=1x), 4.74 1012 (b=2x), 2.37 1012 (b=4x). Note that the integrated curve (over time!) is equal in all three cases. The biggest curve looks larger, but it is scanned at higher speed.
I dt   =  I dT dt/dT    =   (1/b) I dT
* We make here the assumption that all defects in the depletion width   release their charge in a TSC experiment. More precise would be to use a value DW instead of W, where DW is the part of the depletion width where the deep level is forced above the Fermi level with the bias (and is below is without the bias!), see the figures a and b in the beginning of this section.

Experimental setup

Items in an experimental set up are: The electrical schemes are shown below. For temperature dependence measurements, the device should be mounted in a cryostat.
DC measurements
Setup for DC conductance (G)

Setup for current transient 

              measurements Setup for AC measurements
of conductance (G), loss (G/w)
and capacitance (C)

Setup for capacitance transient
measurements and DLTS

How does the RCL bridge work? In most cases it works like a lock-in detector that monitors the current when an AC voltage it applied to the device. All the current that is in phase with the voltage is interpreted as conductance (G=1/R) and the current that is out-of-phase is linked to the capacitance. This is based on the properties of resistors and capacitors, IR = V/R (Ohms law) and IC = C dV/dt. When a sine-wave voltage Vin is applied to the device, the currents for the capacitor and resistor can be calculated:

 Vin = Vsin(wt)
IR = GVsin(wt) = I0
IC = wCVcos(wt) = I90

From this, the conductance can be calculated as G = I0/V and the capacitance as C = I90/wV, with I0 the amplitude of the in-phase current and I90 the amplitude of the out-of-phase current. Here we made use of the fact that sin and cos are 90 degrees out of phase. Now the trick is how to decompose the current into the in-phase current and the out-of-phase current parts. For this we can make use of a lock-in detector (a.k.a. "phase-sensitive detector"). What happens is the following:
phase sensitive detection
  1. The signal S (proportional to the current I) is multiplied by a square wave signal with the same period as Vin, the so-called reference signal Vref. When the phase of this reference is equal to the phase of the signal S, the result is a rectification of the signal (see the Figure).
  2. After rectification, the signal is put through a low-pass filter (RC-filter). The result is a DC signal whose amplitude is proportional to the amplitude of that part of the input signal S that was in-phase with the reference. If the starting signal was out of phase with the reference, the result would have been zero after the RC-filter, see the right part of the Figure. The output signal is proportional to the conductance only.
  3. In this way, simultaneously, two channels are used. One channel with the reference signal in phase with the applied voltage Vin and one channel with the reference 90 degrees out of phase. The first will monitor the conductance and the latter will monitor the capacitance.

Measurement summary

The following table summarizes the applications of the various measurement techniques described in this text:

difficulty /
IV Schottky/pn
CV Schottky/pn  
IV FET        
Hall bulk        
TSC Schottky/pn    
DLTS Schottky/pn    
ToF bulk        


In the text the following variables and constants were used:
variable description unit
A area of the interface m2
b temperature scanning speed K/s
C capacitance F
C0 background capacitance
or DC capacitance
DC capacitance-transient amplitude F
cp hole capture rate 1/s
e electrical permittivity F/m
ep hole emission rate 1/s
E electric field V/m
Epa, EA,
EA1, EA2,
activation energies J
EF Fermi level J
EVB, ECB valence-band level and
conduction-band level
fm (metal) work function J
G conductance (1/R) 1/W
I current A
J, J0 current density A/m2
NT, NA, ND impurity densities 1/m3
Q total space charge C
R, Rb, Rd resistance W
r charge density C/m3
S DLTS signal F
variable description unit
T temperature K
t, t1, t2 time s
t, t0 life time or decay time s
Vbi built-in voltage V
Vbb total band bending V
V, VE, VF applied external voltage V
Vn Fermi level depth in semiconductor
relative to conduction band
W depletion width m
w, w0 radial frequency rad/s
x space coordinate m
c electronaffinity V

constant description value unit
e0 permittivity of vacuum 8.854187817 ....10-12 F/m
h Planck constant 6.6260755(40) 10-34 Js
k Boltzmann constant 1.380658(12) 10-23 J/K
q elementary charge 1.60217733(49) 10-19 C

Note: the units presented here are according to S.I.
In reality often different units are encountered. For
instance, energies are nearly always given in eV
(1.6 10-19 J) , depletion widths in nm or Å
and impurity levels in cm-3

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