Silicon transistor has lower Gain-Bandwidth,because germanium has higher mobility so it improve the current,due to this gain will improve.

Dear Biwajet, 

The gain band width product is expressed by

GBP=  gm/ 2 pi( Cbe +C bc) assuming bipolar transistor as you described above.

By relating the small signal parameters to the dc operating point and the material parameters one can predict how the GBP changes with the material parameters.

It is so that the trans conductance gm= dIC/ dVBE= Ic/Vt with IC the dc collector current and Vt is the thermal voltage= kT/q.  The base to emitter capacitance Cbe= T  IB/ Vt, where T is the transit time of minority carriers across the base region, IB the bae current. Cbc is the junction capacitance of the base to collector=  epsilon Ac/ wscrc,

where epsilon is the dielectric constant, Ac the collector area and and wscrc is the width of the space charge region of the collector to base junction. The transit time T can be expressed by

T= Wb^2/  2D, with Wb the base width and D the diffusion coefficient.

The material parameters are the diffusion constant and the dielectric constant. Assuming the same doping the wscrc will be same for agiven collector voltage.

One can see now the GBP:

- Increases with increasing the diffusion coefficient

In order to decrease the transit time more,one builds the so called drift transistor where the built in electric field assists the diffusion of carriers across the base shortening the Transit time.

- Increases with decreasing the dielectric constant,

The diffusion coefficient increases the mobility mu according to the relation D= muVt . The result is in agreement with Rajev.

It is so that the dielectric constant of the common semiconductors do not greatly differ, then remains the mobility which is main material factor in limiting the speed of the transistor.  Ga As has much higher speed than Si because of its much higher electron mobility.

Best wishes