I am not sure if I understand your question correctly. However, I think I should tell the basic concept of using ln(R) vs 1/T plot to extract a band gap energy. Temperature dependent resistance for a semiconductor is strongly related to the change in carrier concentration which follows Arrhenius relationship. That is why we plot ln(R) as a function of 1/T. The measured resistance (R) involves the activation energy (Ea) of majority carriers which can be expressed by

R ~ exp[Ea / (2kBT)]

where kB is Boltzmann constant.

Let us consider an extrinsic semiconductor which is a general case. In the In(R) vs 1/T plot, it is possible that there are three linear regions, i.e., three slopes and thus three Ea values:

1. For low 1/T (high T) range, Ea1 can be seen as the band gap energy (which is what you want) because (nearly) all carriers are thermally excited across the band gap, and hence the electron and hole concentrations are equal.

2. For intermediate 1/T (intermediate T) range, Ea2 is related to the surface and defect states.

3. For high 1/T (low T) range, Ea3 is related to the ionized carrier states (e.g., dopants) where only a small activation energy can excite the carriers from such states.

Now, turn back to your question. If you see only a linear behavior over the wide temperature range in your ln(R) vs 1/T plot, it implies that your semiconductor is intrinsic. Honestly, I have never heard about an exponential behavior in ln(R) vs 1/T plot and used it to extract the band gap energy (as you stated in the first paragraph). As described above, the ln(R) vs 1/T plot should exhibit the linear behavior according to the Arrhenius relationship.

To obtain the band gap energy, the resistance measurement should be performed at high temperatures so that the other contributions from defect and impurity states are minimized. Please note that the extrinsic semiconductor behaves intrinsically at high temperatures and the criteria for this case is kBT >> |EF (intrinsic) - EF (n or p)|.

I am not sure if I understand your question correctly. However, I think I should tell the basic concept of using ln(R) vs 1/T plot to extract a band gap energy. Temperature dependent resistance for a semiconductor is strongly related to the change in carrier concentration which follows Arrhenius relationship. That is why we plot ln(R) as a function of 1/T. The measured resistance (R) involves the activation energy (Ea) of majority carriers which can be expressed by

R ~ exp[Ea / (2kBT)]

where kB is Boltzmann constant.

Let us consider an extrinsic semiconductor which is a general case. In the In(R) vs 1/T plot, it is possible that there are three linear regions, i.e., three slopes and thus three Ea values:

1. For low 1/T (high T) range, Ea1 can be seen as the band gap energy (which is what you want) because (nearly) all carriers are thermally excited across the band gap, and hence the electron and hole concentrations are equal.

2. For intermediate 1/T (intermediate T) range, Ea2 is related to the surface and defect states.

3. For high 1/T (low T) range, Ea3 is related to the ionized carrier states (e.g., dopants) where only a small activation energy can excite the carriers from such states.

Now, turn back to your question. If you see only a linear behavior over the wide temperature range in your ln(R) vs 1/T plot, it implies that your semiconductor is intrinsic. Honestly, I have never heard about an exponential behavior in ln(R) vs 1/T plot and used it to extract the band gap energy (as you stated in the first paragraph). As described above, the ln(R) vs 1/T plot should exhibit the linear behavior according to the Arrhenius relationship.

To obtain the band gap energy, the resistance measurement should be performed at high temperatures so that the other contributions from defect and impurity states are minimized. Please note that the extrinsic semiconductor behaves intrinsically at high temperatures and the criteria for this case is kBT >> |EF (intrinsic) - EF (n or p)|.

Thank you so much.

I was talking about the appearance of curve in whole range (low temp to high temp). In than case the curve is not linear in whole temperature range. As you said we extract band gap (2 Ea) from high temperature parts of curve.  (Fig 1)

In my experiment, the other measurements shows that I prepared semiconductor carbon nano structure.   But the ln R vs 1/T curve is linear in whole range (20 K to 300k), like Fig 2.

The question is : Can I extract Ea or (Eg/2) from this curve. (Fig. 2)

In addition, I don't think the material which I have is intrinsic!

exponential behaviour is related to R vs T. 

I will be happy to know your opinion.

Dear Sedi Salimian:

I am sorry to say that I have very little knowledge about carbon nanostructures. Therefore, I cannot answer your question. Nevertheless, please keep in mind that the temperature dependence of resistance (in particular, the case of nanostructures) is not only related to the activation process but also the other mechanisms such as variable range hopping (VRH) suggested by

1. Mott: Linear in the ln(R) vs T-1/4 plot,

2. Efros-Shklovskii (ES): Linear in the ln(R) vs T-1/2 plot.

What I would like to point out is that the temperature dependent resistance indicates the conduction mechanisms but is not necessary to be used for extracting the value of band gap energy if the conduction does not only rely on the activation process. Besides, there is a lot of unique transport phenomena occurring in nanostructures.

If we assume that the activation process is only responsible for your experimental result, the band gap energy can be extracted from Fig. 2. However, the obtained band gap energy should be compared with the value reported in literature or some calculations. If it is far more different, I think there might be other contributions in the conduction process which you need to clarify.

PS: Please wait for other experts to answer. You should add some technical terms related to Carbon in " TOPICS " to make your question more visible by experts in that field.

Thank you so much