Laplace DOS is said to be more suitable to characterize charge conduction in the grain boundaries of a polycrystalline semiconductor, predominantly when explaining conduction in terms of diode current. This is because it is easier to study the density of states with energy in a diode as a large number of energy levels are inspected. The application of voltage at the metal in forward bias causes almost all of the voltage to drop across the semiconductor. The higher the application of voltage, the deeper the investigation of energy levels is carried out. Through the ohmic contact large voltages may be applied thereby allowing a free flow of carriers into the semiconducting material. This consequently allows the study of a much wider distribution of carriers within the semiconductor, which is unlike in MOS capacitors and OTFTs, where only a small voltage is used to vary the carriers on the surface as most of the voltage is across the oxide layer.
3eV 37meV Log I
45 meV
V
Figure 4.10. A sketch of the log-linear Current-Voltage characteristics of a Schottky diode in forward bias. The voltage drop across the e xponential region is much more than the Meyer Neldel Energy of approximately 45meV for polycrystalline materials.
When a small forward bias is applied to the ohmic metal, the current rises exponentially across the barrier into the Schottky metal as indicated in Fig.4.10. Although thinning of the neutral
region can cause the expansion of the depletion region due to the expansion of the exponential voltage range, the conduction is limited by the contraction of the depletion region and the expansion of the neutral region. At higher voltages, the current is limited by the resistivity of the neutral region and therefore it becomes harder for carriers to pass through to the metal. This is shown in Fig.4.10 as the decrease in current at higher voltages from exponential to SCL currents. The difference between Gaussian and Laplace distribution as in Fig.2.5 is that the prediction of current density for a higher range of energy levels is possible with Laplace than the Gaussian distribution to correspond to the rise of the Fermi level with applied bias in the depletion region. The carrier concentration at the peak of the barrier can be explained by the product of both the carrier density defined by the Laplace DOS and the Maxwell Boltzmann approximation from the tail of Fermi Dirac statistics as in Eq. (4.9). These are both exponential functions and can be used to define the exponential rise of carrier concentration at the peak of the barrier. The Maxwell Boltzmann statistics above the Fermi level is defined in Section 2.3, Eq. (2.3).
Vg
Vg
very small voltage drop (~37meV)
all voltage (>>37meV) M I S
M S
Figure 4.11. Metal Insulator Semiconductor and Schottky diode structure describing the amount of voltage drop across the semiconductor.
Detailed analysis on the Meyer Neldel energy is conducted in Chapter 3. The Meyer Neldel rule is defined in terms of the width of the carrier distribution. The Meyer Neldel energy in the range of
30-40meV is produced in the exponential region of a Schottky diode, representing the density of DOS with energy [6, 17]. Polycrystalline semiconductor based Schottky diodes provide a high MN energy of approximately 45meV from Fig.3.8. But this is only a small portion of the exponential region when compared to the experimental data obtained. In ideal diodes, the applied bias at the metal is completely dropped across the semiconductor (Fig.4.11). The applied bias can scan a wider range of energy levels (>> 37meV) unlike when there exists a dielectric layer between the metal and the semiconductor like in MIS capacitors and TFTs (Fig.4.11) (as in Chapter 5).Gaussian DOS gives a good fit for the energy range measured in TFTs (37meV) but becomes a doubtful distribution when scanning a larger number of states (around 3eV) as in diodes.
Section 2.4.3, Eq. (2.7), gives the carrier density characterised in terms of L1 distribution. The carrier density at E=0 is NL1(0) and can be written as
4.34 where 4.35
and where is defined as the characteristic temperature corresponding to the Laplace DOS.
4.36
To define the whole distribution,
4.37 The rate of change of energy NL1’ (E) becomes
where is similar to defined in terms of exponential DOS. Using Eq. (4.9), the carrier concentration in terms of Laplace DOS is written as
4.39
Assuming that trap density is significantly less below the Fermi level and are almost completely full such that the hopping rate is significantly reduced, the trap density can be ignored. The carrier density is also very small at the peak of the distribution and therefore can be used as an integration limit.
4.40
where T0incorporates both absolute temperature T and characteristic temperature due to structural disorder in the Laplace distribution, , given by
4.41
The current density above the Fermi level, not considering the conduction mechanisms involved, is proportional to the carrier concentration
4.42
If the current is limited by the grain boundaries, the above mentioned phenomenon is true. However, if the current is limited by the conduction in grains, the neutral region has a higher mobility and the curve defining the current turns on much more quickly. The ideality factor obtained in such a case, if high, is due to the poor back contact. This causes a voltage drop at the back metal contact which is visible at lower voltages (such as in the exponential region) and at higher voltages, this voltage drop becomes insignificant. The current in the ordered grains is expected to be much higher than the total disordered grain boundaries and thus higher mobility values can be expected. The Laplace can be compared against the Gaussian distribution if the variance is kept constant and is known as the standardised classical Laplace distribution (Fig.2.5).
The distributions show good similarity, except that Laplace has a thicker tail than the Gaussian distribution and has a higher pointed peak at the centre [18].