they pass through the Fermi level as the magnetic field is increased for a fixed value of RSOI. The amplitude of the SdH type of oscillations is a monotonic function of magnetic field, when the Landau bandwidth is independent of the band index n . In a M2DEG in the presence of RSOI considered here, the Landau bandwidths oscillate as a function of the band index n. which significantly affects the plasmon spectrum of the intra-Landau band type. For the excitation spectrum, we need to numerically solve equation (12) for all vectors, energies, periodic modulation strength, temperature and magnetic field. We will consider the case of weak modulation ( V$_{0}$/E$_{F}$ << 1), constant RSOI strength and long wave length. In this case we are concerned with transitions within a Landau miniband, i.e. n = n ′ , ε ± n ′ − ε ± n = 0 and C$_{nn}$ ′ ( x ) → 1[13]. Hence, we are able to solve equation (12) analytically at finite temperature. ℏ 2 ∼ ω 2 = 16 e 2 πk ¯ q ℏ m ∗ ω$_{c}$ a × { sin $^{2}$( π a ( x ′$_{0}$ ) F ± n ( u ) } , (14) F ± ( u ) = − ∑ n V ± n × ∫ a/ 2 0 dx$_{0}$f ± ( ε ( n, x$_{0}$ )) cos( Kx$_{0}$ ) (15) The intra-Landau-band plasmon dispersion relation at finite temperature reduces to 1 = ∼ ω 2 ω 2 , where and f ± ( ε ( n, x$_{0}$ )) is the distribution function. Without RSOI ( α = 0) and at zero tempera- ture, this expression reduces to the result obtained in [17]. In the regime of weak modulation where f ′ ( x ) = d $_{dx}$f ( x ) is the derivative of the Fermi Dirac distribution function. After the substitution of this expansion in equations (14 & 15) and performing the integral over x$_{0}$ , the intra-Landau band spectrum is obtained here and f ± ( ε ( n, x$_{0}$ )) ≃ f ± ( ε$_{n}$ ) + f ±′ ( ε$_{n}$ ) { V ± n cos( Kx$_{0}$ ) } , (16) ℏ 2 ∼ ω 2 = 4 e $^{2}$m ∗ ω$_{c}$ πk ¯ q ℏ × { sin $^{2}$[ π a ( x ′$_{0}$ )] × [ G ++ + G −− ] } , (17) G ++ = ∑ n ∣ V + n ∣ ∣$^{2}$ × [ − f $^{+}$′ ( ε$_{n}$ )] = ∑ n ∣ V + n ∣ ∣$^{2}$ × β [ f $^{+}$( ε$_{n}$ ) { 1 − f $^{+}$( ε$_{n}$ ) } ] , G −− = ∑ n ∣ V − n ∣ ∣$^{2}$ × [ − f −′ ( ε$_{n}$ )] = ∑ n ∣ V − n ∣ ∣$^{2}$ × β [ f − ( ε$_{n}$ ) { 1 − f − ( ε$_{n}$ ) } ] , 8