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the Fermi energy remains fixed as the bias increase, the carrier density decreases and the intra- (inter-) band con- tribution at large (small) q becomes less sensitive to the bias and decreases (increases) with the bias. 

The plasmon modes are obtained by solving the zeros of the real part of the dielectric function ε$_{r}$ ( q, ω ) = 0 and the corresponding imaginary part ε$_{i}$ represents the damping rate of the plasmon modes. In Fig. 3, we plot the typical spectrum of plasmon modes (solid and dashed curves) in doped BLG at zero temperature under po- tential bias (a) U = 30 meV and (b) U = 60 meV. The upper light shadow is the interband SPEC edged at ω = √ U 2 + k 4 $_{F}$/m ∗ $^{2}$/ 2 + √ U 2 + ( k$_{F}$ − q ) $^{4}$/m ∗ 2 and the lower dark shadow is the intraband SPEC edged at ω = √ U 2 + ( k$_{F}$ + q ) $^{4}$/m ∗ 2 − U 2 + k 4 $_{F}$/m ∗ $^{2}$/ 2. One undamped mode is located in the SPEC gap due to fi- nite Fermi energy. In the long wavelength limit, its dis- persion is the same as that of Fermi 2D gas of two val- ley, ω 0 = √ ( e $^{2}$/ 4 πϵ 0 ) / ( Nπq/m $^{∗}$). However, compared to the plasmon dispersion in BLG without bias which is just slightly modified from that of Fermi 2D gas, this dispersion is greatly softened for finite q as also shown in Fig. 4. Our numerical analysis shows that this is a re- sult of the deformation near the bottom and top of the energy bands. Note that the lowered plasmon group ve- locity may be helpful for making a stimulated plasmon oscillator [18]. A Landau damped mode is located just below the intraband SPEC edge as usually happens in traditional 2D Fermi gas but is pushed to lower energy at larger q . The undamped mode can enter into the in- 

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terband SPEC and becomes a slightly damped mode in some cases as shown in Fig. 3(a) under U = 30 meV or merges with the damped mode and disappears near the cross of intra- and inter-band SPEC edges as shown in Fig. 3(b) under U = 60 meV. 

With the plasmon spectrum in mind, we now explore how U affects the energy and damping properties of the modes. In the left panels of Fig. 4, we show ω versus U at several typical q when keeping the Fermi energy con- stant. As in Fig. 3, the light shadow indicates the inter- band SPEC and the dark shadow for the intraband one. At small q as illustrated in (a), there is one undamped plasmon mode with energy located inside the SPEC gap of which the width is about 2 E$_{F}$ and one damped mode of low frequency. When U reaches and passes 2 E$_{F}$ , the Fermi level drops below the conduction bottom and the two plasmon modes merge and disappear. At larger q , the intraband SPEC edge shifts up and the interband one shifts down for U < 2 E$_{F}$ and ω increases as shown in (b) and also in Fig. 3. Then the two SPECs will merge and the previous undamped plasmon mode enter the inter- band SPEC and become slightly damped. In this case, we may open the SPEC gap again by applying a stronger bias and transfer the slightly damped plasmon mode into a undamped as shown in (c). The ω versus U curve forms a shoulder when it meets the interband SPEC reflecting the strong coupling between the single particle and col- lective excitations as also shown in other cases [2, 15, 20]. 

If N remains constant as shown in the right panels, the Fermi vector is also constant but the E$_{F}$ shifts up with U . This is clearly shown in (d) by the interband SPEC edge of small q which is located near ω = 2 E$_{F}$ . 

