(Abridged) We present the results of a systematic GBT and GMRT survey for 21-cm absorption in a sample of 10 DLAs at 2<z_abs<3.4. Analysis of L-band VLBA images of the background QSOs are also presented. We detect 21-cm absorption in only one DLA (at z_abs = 3.1745 towards J1337+3152). Combining our data with the data from the literature (a sample of 28 DLAs) and assuming the measured core fraction at milliarcsecond scale to represent the gas covering factor, we find that the HI gas in DLAs at z> 2 is predominantly constituted by WNM. The detection rate of 21-cm absorption seems to be higher for systems with higher N(HI) or metallicity. However, no clear correlation is found between the integrated 21-cm optical depth (or spin temperature) and either N(HI), metallicity or velocity spread of the low ionization species. There are 13 DLAs in our sample for which high resolution optical spectra covering the expected wavelength range of H_2 absorption are available. We report the detection of H_2 molecules in the z_abs = 3.3871 21-cm absorber towards J0203+1134 (PKS 0201+113). In 8 cases, neither H_2 nor 21-cm absorption are detected. The lack of 21-cm and H_2 absorption in these systems can be explained if most of the HI in these DLAs originate from low density high temperature gas. In one case we have a DLA with 21-cm absorption not showing H_2 absorption. In two cases, both species are detected but do not originate from the same velocity component. In the remaining 2 cases 21-cm absorption is not detected despite the presence of H_2 with evidence for the presence of cold gas. All this is consistent with the idea that the H_2 components seen in DLAs are compact (with sizes of < 15 pc) and contain only a small fraction (i.e typically <10%) of the total N(HI) measured in the DLAs.
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