I try a lot of calculation method (inc GW-LDA, PBE0, HSE), however I am still fail to reproduce correct band order of Ge. I found a paper that discussed about core polarization potential GW. They reported that the band order of Ge is successfully obtained within this method. Here is the link : http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.69.2955
Could anyone tell me how to do this cal in VASP?
I don't think Vasp has this feature, you could download berkeley GW and try with that code. You also need quantum espresso to do the ground state calculation.
Hybrid functionals
http://cms.mpi.univie.ac.at/wiki/index.php/Bandgap_of_Si_using_different_DFT%2BHF_methods
Bandgap of Si in GW
http://cms.mpi.univie.ac.at/wiki/index.php/Bandgap_of_Si_in_GW
I think GW@HSE03 should do a proper enough job (in particular with one of the newest "_GW" POTCARs) -- because somehow with the PAW implementation core-valence interactions should be described well enough (the paper you mentioned still dealt with pseudo potentials). Maybe just one hint: If you use hybrids or GW (anything that involves "exchange integrals") you should consider the INCAR flag LMAXFOCKAE which should be set to 4 (or even 6 for f-elements). This can also sometimes influence the results a bit ... .
The really crucial thing is (what I've heard) the lattice constant!! Deformation potentials of the direct and indirect gaps are so high that already a very small change in the lattice constant (one percent) can change the gaps by several tenths of an eV(!) -- and since the changes go in opposite directions you can achieve any gap ordering you like or you could even close one of the gaps (direct semiconductor, indirect semiconductor, metal/half-metal). It's a mess. I think very close to the experimental lattice constant you should have a chance to somehow reproduce the "correct" band structure of Ge (which lattice constant did you use? -- maybe try some few different lattice constants and see what happens to your direct and indirect gaps ...).
Hence, the real challenge seems to be to determine a *theoretical* lattice constant for Ge that is close enough to the experimental one. PBEsol or AM05 are already much better in this respect than LDA or PBE ("disastrous deviations" in view of your problem), in the worst case you have to try various hybrids (maybe even HSE06 with a modified AEXX -- it seems to me AEXX=0.3 gives often rather improved lattice constants).
I think GW should serve the purpose. Second choice may the PBE0 if bandgap is to be obtained.
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