Dear Ajaya Kavala, I'd want to stress again that things are not as simple as you'd like to have them. You have to define precisely what you mean by "behaves like semiconductors". Which quantity, which experiment do you refer to or want to refer to?

1) there is a very simple argument due to Kubo, I believe. It was mentioned in another recent thread (search for Kubo). In terms of single particle (independent electron approximation) energies you may roughly consider the band width of the conduction band(s) crossing the Fermi level (in the infinite solid). In Au, this is the 6s,p band, with one electron per atom. The total band width is probably of the order of 20 eV or so. Divide this by the number of atoms in the particle to get a zero order (!!!) estimate of the energetic distance between electronic states around the Fermi level.

[this estimate is not quite accurate since it neglects degeneracies / symmetry related features of all kinds].

In a particle of 5nm diameter you end up with approx. 4000 atoms, which directs you towards a 'gap' in the single particle spectrum of about 5meV and accordingly smaller for larger particles.

2a) How to measure this experimentally? Well, cou could try to do a transport experiment, maybe (resonant) tunneling, differential tunneling conductivity or the like. You will not find in this experiment the above value, since the intermediate (resonant) states are such with an electron added to the particle, it is therefore (temporarily, if you like) in a charged state. This extra Coulomb energy is known as Coulomb blockade in the pertinent literature.

2b) An alternative experiment is to do photoelectron spectroscopy on negatively charged particles (cluster anions). The final states of the process are charge neutral since one electron is removed by the photoelectric effect. If the particle was in the electronic ground state before the emission process then we may try to estimate the energy difference between the two topmost features (peaks at highest and 2nd highest kinetic energy resp. lowest and 2nd lowest ionisation potential) which usually are interpreted as emission of the HOMO+1 electron and the HOMO electron, respectively (again, this language tacitly assumes that it is permissible to discuss the features within the approximation of independent electrons). HOMO is then the highest occupied state of the neutral particle. [Complications may arise with this experiment (resp. the assignments of electronic states) if the equilibrium atomic structures of the neutral and negatively charged particles are not the same. In veryjavascript:void(0); small particles this is not infrequent.]

This is maybe the best estimate of the single particle gap you can hope for. I do not know up to what sizes such experiments have been done. there is a large bunch of experimental literature of gas phas anion photoelectron spectroscopy which you should consult for this purpose. I recommed to start out with the recent work of the group of Bernd von Issendorff.

3) these experiments will highlight to you another simultaneous feature: The nature of these states is perfectly understandable as to result from *metallic* bonding. I.e. the single electron states are delocalized and approximately well described in terms of a refined jellium picture. This is very different from semiconductors which form predominantly localized covalent bonds to first order, delocalization of these states being adequately understood in the tight binding picture of band formation.

The fundamental gap in semiconductors therefore is primarily a hybridization gap. the gap in metallic particles is 'merely' a result of the discretization of the single particle (electron) energy spectrum in finite objects. In experiments, you will have to take into account the precise electronic nature of those states which determine the outcome of the experiment (e.g. charged vs. non charged or whatever else may apply).

In order to give an answer that hits your need, you need to know what really you are asking for. It may sound trivial but it isn't. Also, it depends on the size range you're interested.

You may define the band gap in terms of electron affinity and ionisation potential, or, alternatively through charge-neutral electron hole excitations. In general this is not equivalent either if excitonic effects are important and/or when electronic correlations are strong. In small particles, the EA and IP are additionally modified by the charge localisation energies (electrostatic Coulomb energy related to the fact that the ionizing charge is localized to a small volume).

What is relevant to you depends on what you need the value for. You will have to tell.

@ Kai Fauth, Thank you for your great reply. But at certain range of nano particles behaves like a semiconductors, so that means the band gap is depends upon the size of nano particle.Here I would like to know the different size that means 5,10,20,30,40,50...100nm and their band gap value (experimentally proved).

Dear Ajaya Kavala, I'd want to stress again that things are not as simple as you'd like to have them. You have to define precisely what you mean by "behaves like semiconductors". Which quantity, which experiment do you refer to or want to refer to?

1) there is a very simple argument due to Kubo, I believe. It was mentioned in another recent thread (search for Kubo). In terms of single particle (independent electron approximation) energies you may roughly consider the band width of the conduction band(s) crossing the Fermi level (in the infinite solid). In Au, this is the 6s,p band, with one electron per atom. The total band width is probably of the order of 20 eV or so. Divide this by the number of atoms in the particle to get a zero order (!!!) estimate of the energetic distance between electronic states around the Fermi level.

[this estimate is not quite accurate since it neglects degeneracies / symmetry related features of all kinds].

In a particle of 5nm diameter you end up with approx. 4000 atoms, which directs you towards a 'gap' in the single particle spectrum of about 5meV and accordingly smaller for larger particles.

2a) How to measure this experimentally? Well, cou could try to do a transport experiment, maybe (resonant) tunneling, differential tunneling conductivity or the like. You will not find in this experiment the above value, since the intermediate (resonant) states are such with an electron added to the particle, it is therefore (temporarily, if you like) in a charged state. This extra Coulomb energy is known as Coulomb blockade in the pertinent literature.

2b) An alternative experiment is to do photoelectron spectroscopy on negatively charged particles (cluster anions). The final states of the process are charge neutral since one electron is removed by the photoelectric effect. If the particle was in the electronic ground state before the emission process then we may try to estimate the energy difference between the two topmost features (peaks at highest and 2nd highest kinetic energy resp. lowest and 2nd lowest ionisation potential) which usually are interpreted as emission of the HOMO+1 electron and the HOMO electron, respectively (again, this language tacitly assumes that it is permissible to discuss the features within the approximation of independent electrons). HOMO is then the highest occupied state of the neutral particle. [Complications may arise with this experiment (resp. the assignments of electronic states) if the equilibrium atomic structures of the neutral and negatively charged particles are not the same. In veryjavascript:void(0); small particles this is not infrequent.]

This is maybe the best estimate of the single particle gap you can hope for. I do not know up to what sizes such experiments have been done. there is a large bunch of experimental literature of gas phas anion photoelectron spectroscopy which you should consult for this purpose. I recommed to start out with the recent work of the group of Bernd von Issendorff.

3) these experiments will highlight to you another simultaneous feature: The nature of these states is perfectly understandable as to result from *metallic* bonding. I.e. the single electron states are delocalized and approximately well described in terms of a refined jellium picture. This is very different from semiconductors which form predominantly localized covalent bonds to first order, delocalization of these states being adequately understood in the tight binding picture of band formation.

The fundamental gap in semiconductors therefore is primarily a hybridization gap. the gap in metallic particles is 'merely' a result of the discretization of the single particle (electron) energy spectrum in finite objects. In experiments, you will have to take into account the precise electronic nature of those states which determine the outcome of the experiment (e.g. charged vs. non charged or whatever else may apply).

what is band gap for gold particle ? I  know they are good conductor but  the band gap should be = 0 eV or 1 ev? 

Have you gone through the preceding post? What exactly is the quantity you're asking for and what is the context?

Gold nanoparticles are plasmonic particles. Although metal doesn't have a band gap due to the continuous structure of its conduction band, for nanoparticle, the structure might not be continuous. Despite of having discrete, structure, bandgap of Au NP determination is not that straightforward. If you are looking for the absorption of AuNP, then the peak will appear at the plasmon peak, although there are intraband transitions also occur at the higher energy.

Gold nano-particles do not have a band gap. However they alter the band structure properties on deposition.

When the partical size decreased increased spacings of the energy levels and to the formation of energy gaps. The

electronic properties can be thus changed from a metallic to

semiconductor or even insulator depending on size also depended on the structure of material.as show in paper

https://journals.aps.org/prb/abstract/10.1103/PhysRevB.72.155430