On the interpretation of single and bilayer graphene minimal conductivity

: Within the Boltzmann kinetic theory minimal dc conductivity of single and bilayer graphene is studied. It is shown that the uncertainty principle plays a key role in graphene band structure. In the paper a new alternative interpretation for minimal universal conductivity of single and bilayer graphene is proposed. Minimal conductivity is determined by degenerate electrons and holes of the overlap range of electron and hole bands. It is established that the minimal conductivity of single and bilayer graphene can be explained in the framework of one definite physical approach based on the uncertainty relations. Within the proposed theoretical approach minimal conductivity of both single-layer and bilayer graphene equals h e / 4 2 . It is found that in crystals with parabolic dispersion quantum expansion of an energy level E is asymmetric with respect to E ; in a crystal with linear dispersion an energy level quantum expansion is symmetric.

Experimental observation shows that practically there is no difference between the minimal conductivity of the single and the bilayer graphene.Minimal conductivity of both single and bilayer graphene is universal value and equals h e / 4 2 [5][6][7][8][9][10].In the literature the existence of nonzero minimal conductivity at low temperatures ( K T 0 ~) is presented as a unique, unusual, amazing or surprising feature of graphene.Authors believe that in the limit of zero temperature there are no scattering and no current carriers, i.e., 0 min   even at zero carrier concentration.Characterization of minimal conductivity as "carrier free" and "scattering free" conductivity it seems not a true interpretation of experimental data.
Parameter  is the factor chracterising the difference between the theoretical results and experimental data.Note there is essential disagreement with the experiment, 08 0 1 0 ~. .  .Theoretical conductivity min  for both single and bilayer graphene is a non-universal quantity and it is inferior to the experimental conductivity for several times.In contrast to experimental data, theoretical works predict that minimal conductivities of single and bilayer graphene are different.
Generalizing the results of the proposed theoretical approaches the following conclusions can be expressed: -different theoretical approaches and calculation methods lead to different values of min


for the same graphene (single or bilayer) system; -same theoretical approaches lead to different values of min


for the single and bilayer graphene; -theoretical results are nonuniversal and practically are not in agreement with the measured data; -nature of graphene minimal conductivity is not established finally.The explanation of minimal conductivity of graphene systems within one definite physical approach still remains a theoretical interest.At discussing the task for single and bilayer graphene, as a rule, a model of zerogap semiconductor with linear and quadratic low-energy dispersions is used, respectively.It is assumed that the conduction and valence bands touch at the K and K' critical points (Dirac points) of the Brillouin zone.This model underlies almost all theoretical approaches proposed so far (see, e.g., theoretical papers cited above).On the other hand, the value of the bandgap energy is one of the basic characteristics of a crystal.The concept of graphene as a crystal with a zero ( 0 = g E ) bandgap has a principle nature.Note that especially in crystals with a zero (or rather small) bandgap, influence of various well-known effects (for instance, temperature effects, spatial and temporal fluctuations of g E due to the lattice chaotic vibrations, effects conditioned by uncertainty principle etc.) on g E become decisive.Therefore, there are real physical reasons to study electronic, partly conductivity features of graphene whithin the uncertainty principle.Such investigations and detailed analyses practicaly have not been carried out yet.

Task outline
For better understanding graphene properties in the present work, "minimal conductivity-uncertainty principle" relationship is considered.Stating the task in this way is motivated by the view that graphene minimal conductivity is a fundamental phenomenon.Below dc conductivity of intrinsic (perfect) single and bilayer graphene at low temperatures ( K T 0 ~) is analyzed.Consideration is carried out on the base of Boltzmann semiclassical kinetic theory with taking into account well-known general conclusions of the solid state band theory and the key principle of uncertainty (including energy-time as well as momentum-position uncertainties).As a result of the discussion a new alternative interpretation for minimal universal conductivity of single and bilayer intrinsic graphene is proposed.

Materials and methods
For the transparent physical understanding graphene conducting properties a semiclassical approach can be used.Below low-temperature conductivity of single and bilayer intrinsic (defect-free) graphene are considered on the base of Boltzmann's transport theory.i) Single-layer graphene.According to Boltzmann's kinetic equation one valley electron conductivity n  of single-layer graphene within the  -approximation [29,30] is presented (taking into account spin) as follows: Here, k is the magnitude of the electron wave vector,  is the electron transport scattering (or relaxation) time, E is the conduction electron energy, F v is the Fermi group velocity of the Dirac electron and 0 f is the equilibrium (Fermi) distribution function of the conduction electrons.In Eq.( 1) and below, electron and hole conductivities are denoted by n  and p  , correspondingly.
The low-energy dispersion law of single-layer graphene is linear, , where the  signs refer to the conduction and valence bands, respectively.Then converting integral over k to an integral over energy Eq.
(1) can be written in the form where c E is the conduction band edge.
Such important features as magnitude and temperature dependence of Many physical properties of condensed matter are explained by band structure.At theoretical studies the following classical model of the band structure of single-layer intrinsic graphene is widely used.It is assumed that c E and v E edges of the conduction *  (or electron) and valence  (or hole) bands touch at the Dirac point, . According to classic approach, a single isolated graphene sheet is a zero-gap semiconductor (or zero overlap semimetal).On the other hand, each model as a rule has certain limits of applicability.Therefore, before further estimation of the low-temperature conductivity, let us first clarify some important details about the low-energy spectrum of single-layer graphene.In a real crystal with finite sizes, motion of an electron is limited.As it is well-known, quantum confinement leads to an electron non-zero minimal energy.Quasimomentum and energy of an electron in finite crystal space are quantized.The effect of quantization on the physical properties is primarily determined by the sizes and temperature of the crystal.In the case of confinement by space with macroscopic sizes the separation between adjacent energy levels is extremely small.The adjacent values of allowed energy and quasimomentum are separated by albeit small but finite intervals.In Figure 1a low-energy spectrum of single layer graphene crystal with finite sizes is plotted near Dirac point.From a discrete set of energy levels in Figure 1a only the first (lowest) E allowed levels of electron and hole bands, are represented, respectively.Note, that holes energy axis is directed opposite the direction of conduction electron energy axis k E (Figure 1 and 2).
For the convenience of further discussion, in Figure 1a formally presents also the conduction   Taking into account that in Figure 1a the minimal energy of a conduction electron is denoted by ). Note, that this feature has a general character and does not depend on the graphene crystal size and temperature.Similarly the low-energy spectrum of the hole band can be considered and it can be stated that the hole minimal energy . Thus, if one takes into account quasimomentum uncertainty only, one can state that the single-layer graphene is a zero-gap semiconductor, Therefore, it can be stated that within a more accurate approach single-layer intrinsic graphene is a semimetal with a non-standard (direct band) band structure (or a semiconductor with a negative band-gap ).
In the given case, the Fermi level F E is in conduction and valence bands simultaneously, Here it should be noted that the uncertainty principle has a significant meaning and is manifested in various physical phenomena.Particularly, the well-known effect of temperature dependence of a semiconductor band-gap width is explained by the energy uncertainty.
Let us now calculate the graphene electron one valley conductivity n  on the base of the band structure plotted in Figure 1c.For the calculation, the following equation should be used instead of Eq.( 2) where Using the relation Since , where for the minimal conductivity of one valley electrons (taking into account spin) the following expression is obtained: Minimal width of energy level E is determined as As one can see from Eqs.( 8)-( 9), + E and − E edges are asymmetric with respect to E .Now let us this peculiarity apply to the lowest energy level If one denotes the width , where ), then one can state that the level This ratio of asymmetric expansion has a general nature.The expansion , respectively.Thus, in the case of parabolic dispersion the total uncertainty In particular, the second relation of Eq.( 14) can be represented as follows Using the standard condition In similar fashion, hole band edge 1 , v E should be considered.For the edge ) is located inside the conduction band, at the distance . Therefore, intrinsic bilayer graphene is a semimetal also whose conduction and valence bands are overlapped by magnitude τ /  .Now based on the band structure presented in Figure 2b

Results and Discussion
Minimal conductivity and other important transport or electronic properties of a crystal are primarily related to the energy spectrum.According to the above-presented approach, if one ignores uncertainties quantum effects, then conduction and valence bands of a single and bilayer graphene are separated by forbidden range, 0  g E , see Figure 1a and   is differ from the expressions obtained in other theoretical papers by using various calculation methods.Second, which is very important, the value min  is in complete agreement with the experimental data and it is not "carrier free" conductivity.The proposed theoretical approach predicts that the minimal conductivity of bilayer graphene is not different from that of single layer graphene.
Note that evaluations which were carried out on the base of uncertainty relations usually are considered as qualitative.On the other hand, the result h e / 4 2 min =  , which is also obtained on the base of the uncertainty principle, directly coincides with the experimental data.Therefore, it can also be considered as quantitative result.In solid state physics, there are cases when results of estimations based on the uncertainties coincide with the experimental data and/or with the results of more accurate calculations.Other important conclusion of the presented approach is the following.In crystals with parabolic dispersion, quantum expansion of an energy level E is asymmetric with respect to E .In particular, expansion of the lowest energy level of the conduction and hole bands are distributed as follows -1/4 part downward and 3/4 part upward.In the case of linear dispersion, the lowest energy level expansion E  2 is symmetric -1/2 part downward and 1/2 part upward.

Conclusions
So, the main conclusions of the paper are the following:

Conflict of interest
There is no conflict of interest for this study.

Appendix
Rate of electron intravalley scattering off iLA phonons is given by [30,31] Here k and q are the electron and phonon wave vectors (boldface type is used to represent vectors), respectively,  is the angle between k and ' k vectors (scattering angle), q N is the phonons equilibrium distribution function,  is the mass density of graphene, A is the area of graphene sheet, A simple summation over ' k is carried out using Kronecker delta q k , k' −  .For further calculations it is taken into account that iLA phonons dispersion law of single-layer graphene is linear, , where ac v is the acoustic phonon velocity [13,30,32].Then, from Eq.(A2) one obtains It is more convenient that the polar coordinate axis is directed along vector k and the following transition from sum to integration is used where  is the angle between vectors k and q .
Converting the sum (A3) to an integral one have where  -function expresses the energy conservation law.Limits of the integration over q are determined by the energy conservation as follows: , where

n,
are mainly determined by the relaxation time  .At low ( characteristic time  is determined by the scattering of electrons off crystal boundaries ( s  ) and off iLA (in-plane longitudinal acoustic) phonons ( L is the size of a 2D crystal.Calculation of electron-iLA phonon scattering rate ac  / 1 in graphene at low temperature range is presented in Appendix.

E
band edges, which are not allowed levels.

Figure 1 . 2 .
Figure 1.Low-energy band structure of single-layer intrinsic graphene K valley in the case of: a) -ignored energy uncertainty, b)presence of energy uncertainty

E
1b.However, this conclusion cannot be considered as final yet; it is an intermediate conclusion.Presented in Figure1bband structure significantly changes if one now takes also into account the second component of energy uncertainty  E  , which is related to energy-time uncertainty relation .As shown in Figure1c, the actual edge of the conduction band, which denoted by ' c E , is located now inside the valence band.The actual edge of valence band, denoted by ' v E , is located inside the conduction band.Thus, due to total uncertainty  valence bands overlap.

).
In similar fashion, hole band should be considered and the minimal conductivity of holes min , p  should be calculated.According to Figure1cband structure the Fermi level F E is located inside the valence band at the distance For the minimal conductivity of one valley holes (taking-taken into account spin) Thus, for the minimal value of total conductivity of single-layer graphene the following result is obtained (taking into account spin and two valleys)

Figure 2 .E
Figure 2. Band structure of an intrinsic bilayer graphene K valley at energy uncertainty ignored (a) and presence (b) cases In Figure 2 shows a low-energy spectrum of an intrinsic bilayer graphene in the cases of ignored (a) and presence (b) of energy uncertainty.As shown in Figure 2b, the actual edge ' c E of the conduction band (i.e., the

electron conductivity n 
is calculated.According to Boltzmann's theory, one valley electron dc conductivity of a bilayer graphene can be represented by the following relation (taking into account the spin) obtained.In a similar way for the hole minimal conductivity of a one valley (taking into account the spin) conductivity of intrinsic bilayer graphene (taking into account spin and two valleys) one has

. 8 / 4 2.
If one take into account the quasimomentum-position uncertainty only, then conduction and valence bands are touching each other, 0 = g E .If one takes into account the energy-time uncertainty as well, then conduction and valence bands are overlapping, 0  g E .Bands overlap is determined by the energy-time uncertainty  E  and consequently by characteristic time  and the crystal sizes L (see, Eq. (A9)).Single-layer graphene bands overlap is  / 2 ; bilayer graphene bands overlap is  /  .So, for example, the overlap of single-layer graphene with 200 = L nm is ~ 6 meV.For comparison, note the overlap of threedimensional graphite bands is ~ 40 meV (experiment) or about 30 meV (theory).Thus, intrinsic single and bilayer graphene crystals are semimetals with a slightly overlap bands.Therefore, graphene must have and has the semimetallic behavior.Semimetals are characterized by a weak bands overlap; conductivity is always finite and non-zero; in the minimal conductivity electrons as well as holes have contributions.Within the above-proposed Boltzmann's quasi-classical E  -model, minimal conductivity min  of both single layer and bilayer graphene is equal to h e It is a universal quantity and does not depend on the characteristic time  and the 2D crystal sizes L .Note, first, the expression h

.
is the iLA phonon frequency, ac D is the acoustic phonon intravalley deformation potential.In expression (A1) The first term on the r.h.s. of the sum (A1) describes the electron scattering with phonon induced absorption ( ab ac,).The second term on the r.h.s. of the sum (A1) describes the electron scattering with induced and spontaneous emission of phonon ( phonon scattering is due to phonon spontaneous emission only.Therefore, at low temperatures from Eq. (A1) for the relaxation rate ac  Using the following relationship between angles  and  .(A7) is curried out first over  (  -integration) and then over q .As result, taking into basic peculiarities of low-temperature electron-iLA phonon scattering rate in single-layer graphene.Thus, in the low-energy range the electron total scattering rate can be presented as Like single-layer, within the classical model an intrinsic bilayer graphene is a zerogap semiconductor, but the conduction and hole bands of which are parabolic touching.Now let us apply presented in the previous section i)E -approach to study bilayer graphene minimal conductivity.First, let us analyze the energy spectrum of bilayer graphene on the base of uncertainty principle.Energy uncertainty E is important.In the case of linear dispersion, average value E coincides with the energy E .Consequently, the energy level expansion in single-layer graphene is symmetric with respect to E as well as to E 1.in crystals with parabolic dispersion, quantum expansion of an energy level is asymmetric; 2. in crystals with linear dispersion, quantum expansion of an energy level is symmetric; 3. single and bilayer intrinsic graphene are semimetals whose bands overlap determined by the uncertainties relations and equal of single and bilayer intrinsic graphene is determined by the degenerated electrons and holes of the overlap range of allowed bands. 2