Yes, it is possible, and in fact a 'semiconductor' that has no indirect gap in no longer a semiconductor, but a semi-metal. A typical example of a semiconductor whose indirect gap is smaller than its direct gap, is silicon in the diamond structure.

Ps. In general the value of the gap energy Eg, which may be direct or indirect, is easiest determined by considering the total (that is the k-integrated) density of one-particle states D(ε) and the value of the chemical potential μ at some small but non-vanishing value of the temperature. The distance (as measured in units of energy) between the two edges of D(ε) at both sides of μ is equal to the gap energy Eg. To establish whether this gap is direct or indirect, one will have to consider the partial density of one-particle states Δ(k,ε), for which one has D(ε) = Σk Δ(k;ε), where the summation is over the points of the first Brillouin zone (1BZ), and determine the values of k for which the distance between the lowest edge of Δ(k;ε) above μ and the highest edge of Δ(k;ε) below μ is equal to Eg. If the two values of k are equal (that is, if the absolute minimum of Δ(k;ε) above μ and the absolute maximum of Δ(k;ε) below μ are at the same k), then the gap is direct, otherwise it is indirect. 

Generally, "indirect bandgap" means that the narrowest gap in energy between the conduction and valence bands actually also require the interaction of a phonon (quantized lattice vibration) to absorb light of that energy or emit it.  (So it's very inefficient)  For example, when looking at an E-K (band structure) diagram, in an indirect semiconductor the highest point of the valence band and lowest part of the conduction band are at different momentum values.  For an electron to be excited from the valence band to the conduction band, it *must also* absorb a phonon (get hit by a lattice vibration).  Direct bandgap means that they are not offset and are directly above-below each other, so no phonons are required and the process can happen more readily.  

So unless you are just describing the shape of the band structure, it doesn't really make sense to say that both a direct and indirect bandgap to exist.  If you are just describing the band structure of Silicon for example,  the narrowest *Energy* is an *indirect* transition so its optical band gap is "indirect", but if you were talking about the "direct optical bandgap of silicon" that would imply that you mean the extra big energy gap at only zero momentum.  

So technically it's kinda possible, it just depends on what gap you're talking about, but physically a semiconductor can only be one or the other.  By definition the indirect bandgap exists *because* it is at a smaller energy, but also requires a change in momentum to occur.

https://en.wikipedia.org/wiki/Direct_and_indirect_band_gaps

https://en.wikipedia.org/wiki/File:Indirect_Bandgap.svg#/media/File:Indirect_Bandgap.svg

https://commons.wikimedia.org/wiki/File:Direct.svg#/media/File:Direct.svg

Not only is possible but necessarily, a semiconductor has both types of transitions, direct and indirect. Experimentally, a direct gap is observed in thin samples whereas the indect gap needs thick samples. The physical process for both transitions occurs as our collegues explain in the previous answers.

Best regards

Pedro 

    Dear Shahin,

    Is it possible for a semiconductor to have both direct and indirect optical band gaps?

    No.

    The direct band gap means that electron and hole have the same crystalline momentum and they can give or absorve a photon having conservation of energy and momentum.

    The indirect band means (as the photon hasn't crystalline momentum) that you cannot obtain or give to it a photon only. You need to add a phonon to have conservation of energy and momentum for such absortion or emittion of photon.

   In both cases the energy of reference is the Fermi level which cannot be simultaneously the same for such physical processes and you have different materials for each process. The width of the layer sample is related with the probability of founding a phonon through the path as it happens with some indirect gap materials. Thin layers always use the direct gap mechanism.

      can an indirect band gap energy be smaller than that of direct band gap?

       Yes.

      The reason is that the indirect gap can be negative, i.e. the energy of the valence band could be higher than the one of the conduction band. Notice that this cannot happen in a indirect gap.