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recent fabrication of single layers of graphite ( graphene monolayer)@xcite has posed the interesting question of charged particle energy loss in solid media where the conduction electrons behave distinctly different from those found in ordinary semiconductor heterostructures . in a two - dimensional electron gas ( 2deg ) realized in semiconductor heterostructures , the conduction electrons behave as ordinary electrons with a parabolic dispersion relation .
in contrast , the dispersion relation obeyed by conduction electrons in graphene monolayer is linear in momentum .
this occurs due to the unique crystal structure of graphene which is a two - dimensional ( 2d ) honeycomb lattice of carbon atoms .
quantum - mechanical hopping between the sublattices of graphene leads to the formation of  two energy bands , and their intersection near the edges of the brillioun zone yields the conical energy spectrum . as a consequence ,
the dispersion relation of electrons and holes bands is linear near @xmath0 points of the brillioun zone which is given by @xmath1 hence , the conduction electrons , known as dirac electrons , behave as massless particles with the effective speed of light @xmath2@xcite . in a graphene monolayer , electron transport is essentially governed by dirac s ( relativistic ) equation , rather than the usual schrodinger equation for nonrelativistic quantum particles .
the relativistic behavior of graphene was first predicted by p. r. wallace@xcite .
it thus provides a unique opportunity to study relativistic quantum dynamics in condensed - matter systems .
the dirac - like gapless energy spectrum was also confirmed recently by cyclotron resonance measurements in graphene monolayer@xcite .    in this work ,
we address energy loss experienced by charged particles as they traverse a non - local dynamic quantum plasma of dirac electrons , electrons that obey the dirac equation .
the energy loss calculation performed here requires that we determine the dynamic , non - local dielectric response function for the plasma of dirac electrons .
we determine the dielectric response function within the random - phase - approximation(rpa ) and employ it to calculate the dynamical structure function which yields the stopping power .    since the work of nozieres and pines@xcite , conduction electrons in solids have been treated as a solid state plasma , a gas of electrons in a neutralizing positive background .
moreover , electron energy loss spectroscopy has been an important probe of the dielectric response properties of solid state systems@xcite .
this was brought into clear focus by the pioneering work of ritchie@xcite where the energy loss of a charged particle traversing a bounded solid state plasma was considered .
dynamical processes in a solid state medium have been investigated with a variety of spectroscopic tools which employ an almost monochromatic beam of electrons , photons , neutral atoms or ions which scatter inelastically from the medium under study@xcite .
information concerning the electronic properties of the medium is obtained as a result of the interaction between the probes and the elementary excitations of the medium .
experimentally , this information is obtained by electron energy loss spectroscopy ( eels ) and inelastic light scattering studies .
recently , energy loss spectroscopy of free - standing graphene films was performed@xcite    the present paper is arranged as follows . in section
ii we give the formulation of the problem . the energy loss rate and stopping power of a 2d dirac plasma
is outlined .
this theoretical investigation is based on the evaluation of dynamical structure function and energy loss function . in section iii
, we present the results and discussion .
concluding remarks are made in section iv .
the system under consideration is a 2d quantum plasma of dirac electrons i.e. dirac electrons embedded in a uniform and rigid neutralizing background of positive charges .
we begin by considering the low - energy electronic eigenstates for this system , given by@xmath3 the dirac - like hamiltonian in eq(1 ) is@xcite@xmath4 where@xmath5 are pauli matrices and @xmath6  is the two - dimensional fermi velocity of dirac electrons ( with the characteristic velocity @xmath7@xcite .
conduction electrons in a dirac plasma behave as massless dirac particles with a linear dispersion relation @xmath8 @xcite where @xmath9 indicate the conduction @xmath10 and valence @xmath11 bands , respectively .
@xmath12 is the two - dimensional wave vector and is given by @xmath13 , here @xmath14 hence , electron dynamics is modelled by the following dirac equation @xmath15 from here onwards we take @xmath16 throughout this calculation .
the wavefunctions appearing in eq.([3 ] ) are given by@xcite@xmath17@xmath18 where @xmath19 is the area of the plasma sheet and@xmath20 and@xmath21 the wavevector and frequency dependent longitudinal dielectric response function within the random - phase - approximation(rpa ) can be expressed as@xcite@xmath22 where @xmath23 is the fourier transform of two dimensional coulomb interaction , @xmath24 is the background dielectric constant and @xmath25is the two - dimensional(2d ) polarizability .
the plasmon modes at finite wave vectors are given by the zeroes of the dielectric response function given by eq([8 ] ) .
since , in this work , we are primarily interested in the energy loss due to plasmons , we consider the dielectric response function in the high frequency , long wavelength limit @xmath26  which is given as @xcite@xmath27 here @xmath28 @xmath29is the plasma frequency with @xmath30 the fermi energy .
equivalently , in terms of dimensionless variables , we have@xmath31 in the above expression , we have used the dimensionless variables @xmath32.@xmath33is the wigner - seitz radius,@xmath34 is the background dielectric constant and @xmath35 @xmath36 being the spin and valley degeneracies respectively@xcite .
energy loss function is the basic parameter which accounts for the energy lost by the incident charged particle .
it is also the parameter which is of central importance in energy loss spectroscopy ( eels ) experiments .
energy loss function is defined as the imaginary part of the inverse dielectric response function . from the dielectric response function given by eq([9a ] ) and introducing the infinitesimally small parameter @xmath37 we obtain@xmath38 evaluating the coefficients @xmath39 and using dirac s prescription @xmath40 , in eq([10a ] ) and then expressing the variables in terms of dimensionless parameters @xmath41 @xmath42 and @xmath43 we obtain the energy loss function for graphene monolayer as @xmath44{c}\delta\left (   \frac{\nu}{\left (   1-\frac{\nu^{2}}{4}\right )   } -\sqrt { \frac{g_{s}g_{v}r_{s}x}{2}}\right ) \\
-\delta\left (   \frac{\nu}{\left (   1-\frac{\nu^{2}}{4}\right )   } + \sqrt { \frac{g_{s}g_{v}r_{s}x}{2}}\right ) \end{array } \right ]   .
\label{11}\ ] ] the interaction of charged particles with condensed matter system can be studied by means of the system s stopping power . the energy loss per unit path length is the stopping power @xmath45 and it accounts for the energy lost by an external charged projectile as it passes through matter . in the quantum mechanical framework , we consider an in - plane probe of charge @xmath46 mass @xmath47 and velocity @xmath48 interacting with the many particle system under consideration by treating the incident particle state as a plane wave state .
we calculate the probability that the point - like projectile loses an energy @xmath49 in a time interval @xmath50 in interaction with the 2d dirac plasma . during the interaction impulse transfer
is @xmath51 here @xmath49 and @xmath52 satisfy @xmath53 for a heavy incident particle , we neglect the recoil energy @xmath54 to obtain @xmath55the stopping power is given by summing over energy difference weighted by transition rate @xmath56 times the inverse projectile velocity , mathematically@xmath57@xmath58 we can expand energy loss in terms of @xmath59 @xmath54 to obtain @xcite@xmath60 here we also make use of the sum rule @xmath61 . to the lowest order , the stopping power for a 2d dirac plasma given by eq([11a ] ) can be expressed in terms of the dynamical structure functions @xmath62 as@xmath63 whereas the first order stopping power is given by@xmath64
d\omega . \label{11.1.2}\ ] ] the dynamical structure function is the auto - correlation function of the fourier components of the particle density and accounts for longitudinal charge oscillations of the electron density relative to the positive background . from the polarizability , we can obtain the dynamical structure factor at finite temperature for a 2d dirac plasma as@xmath65   \label{11.2}\ ] ] where @xmath66 is the boltzmann constant , @xmath67 is the absolute temperature , @xmath68 is the fermi wavevector and @xmath69is the electron density .
dynamical structure factor provides direct physical information about longitudinal excitations in the system .
it is the measure of the density - density correlations of the system .
calculations of dynamical structure function reveal that the electromagnetic response of a dirac plasma is substantially different from a 2deg system which is essentially due to the distinctly different dispersion relation of dirac electrons in a 2d dirac plasma as compared to ordinary electrons in a 2deg system .    for a solid state dirac plasma under consideration
, we can classify the rate of energy loss and stopping power into two categories , one due to the particle - hole excitations and the other due to the plasma oscillations .
so the total stopping power of the system is@xmath70 employing the following relation in eq([11.1 ] ) @xmath71   \right .
\nonumber\\ &   \left .
\times\frac{1}{\sqrt{v_{n}^{2}-(\varepsilon-\varepsilon_{n})^{2}}}\theta(\left\vert v_{n}\right\vert -\left\vert \varepsilon-\varepsilon
_ { n}\right\vert ) \right .
\label{12.1}\ ] ] for a single charged particle@xmath72 the lowest order stopping power , due to plasma oscillations , of a 2d degenerate dirac plasma is given by@xmath73 with @xmath74
the dynamic and static response properties of an electron system are all embodied in the structure of the dielectric response function .
we have employed the random - phase- approximation ( rpa ) based dielectric response function to determine the structure function and the energy loss function of a 2d dirac plasma .
the two types of excitations in the medium responsible for energy loss are the collective and single particle excitations .
collective excitations ( plasmons ) occur at a higher energy compared to single particle ( electron - hole ) excitations . at a frequency @xmath75 equal to @xmath76 collective excitations ( plasma oscillations )
dominate for @xmath77 and these exist as long as the wavevector remains less then a critical value @xmath78however , above this critical value plasmons undergo landau damping generating electron - hole pair excitations . therefore , energy loss rate and stopping power are affected by these two types of contributions , one due to the collective oscillations i.e. plasmons and the other due to the electron - hole excitations .
the energy loss function taking into account both these mechanisms of energy loss is presented here .
collective modes with the onset of landau damping change into electron - hole pairs .
we have plotted the loss function given by eq([11 ] ) due to collective excitations in fig.(1 ) . here
we essentially restrict the energy loss to plasmon excitations by plotting the result at @xmath79the parameters used are : electron density @xmath80 , @xmath81 @xmath82(using sio@xmath83 as the substrate material ) , @xmath84 .
note that the critical value of the wave vector @xmath85 and the corresponding critical value of @xmath41 is @xmath86 where the onset of landau damping of plasmons occur for the parameters considered here .
dynamical structure function shows delta function peaks at the plasma frequency .
the peaks in the figure are the manifestation of existence of plasmons in the system . in fig.(2 ) , in order to include the contribution of single particle excitations , we have plotted energy loss function against @xmath87 for following values of @xmath88 the peaks in the energy loss function seen in the figure can be interpreted as occurring due to single particle excitations of the system . in fig.(3 ) , we have plotted energy loss function versus @xmath89 for various values of @xmath90 the relationship between energy loss rate @xmath91 due to plasmons and velocity of the incident particle @xmath92 is shown graphically in figure(4 ) . we have plotted the energy loss rate versus the dimensionless velocity @xmath93 . stopping power
is shown in fig.(5 ) where the energy lost by the incident particle is due to plasma oscillations .
the stopping power decreases with the increase of the incident particle velocity .
furthermore , the lowest order energy loss rate and stopping power of the medium are found to be independent of the mass of the incident particle .
an analysis of energy loss suffered by fast particles in their interaction with matter requires the calculation of energy loss rate and stopping power . in this work
, we present a theoretical investigation of energy loss through the calculation of both the energy loss rate and the stopping power of a 2d solid state dirac plasma .
the motivation of this work is the recent realization of single layer of graphite , graphene monolayer .
the quasiparticles in graphene monolayer are found to obey the dirac equation with a linear in momentum dispersion relation .
the work presented here is based on the the dielectric response function and the dynamical structure function within the random - phase - approximation ( rpa ) .
the energy loss function taking into account the collective as well as single particle mechanisms of energy loss is presented here . in the collective excitation regime ,
energy loss function peaks at the plasma frequency , such peaks when detected by experiments , can also be used to identify energy at which plasmons occur in the medium .
furthermore , stopping power as a function of the incident particle velocity is also determined .
stopping power is found to decrease as the velocity of the incident particle is increased .
moreover , the lowest order energy loss rate and stopping power of the dirac plasma are found to be independent of the mass of the incident particle .
k. sabeeh ) would like to acknowledge the support of the pakistan science foundation ( psf ) through project no .
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the stopping power and energy loss rate of charged particles traversing a two - dimensional dirac plasma is investigated . 
 the dirac plasma considered here models a solid state system , recently realized graphene monolayer , where the conduction electrons obey the dirac - like equation and exhibit a linear in momentum dispersion relation . 
 theoretical work presented here is based on the the dielectric response function and the dynamical structure function within the random - phase - approximation ( rpa ) .