Charge stripes in the graphene-based materials

We present an analytical model of the charge density wave instability in graphene sheets within the intercalated graphite CaC6 compound. The instability yields the experimentally observed uniaxial charge stripes of periodically modulated electron density, coupled to the softest phonon mode of the superlattice consisting of the Ca atoms intercalated between graphene planes. The Fermi surface of the chemically doped graphene undergoes the novel type of instability driven by the mechanism that gains the condensation energy of the stripe state by the topological reconstruction of the Fermi surface. This mechanism appears to be entirely different from the one based on the Fermi surface nesting, which has been considered a paradigm in the present literature concerning the onset of charge density waves.

The initial graphene band ε(k) = ℏv F |k|, where k is the electron wave vector and v F is the Fermi velocity, with the origin of the reciprocal space (k x , k y ) shifted to the edge of the new Brillouin zone imposed by the periodicity vector leads to the reconstructed band after lifting the degeneracy in the origin by a finite gap parameter ∆.We define the dimensionless variables where ϵ 0 ≡ ℏv F Q/2 is the energy scale, in which the above expressions read Since only the lower band e − (κ, δ) ≡ e(κ, δ) contribution takes part in the final result (will be shown later), for simplicity of presentation we present only the corresponding expressions further on.Dividing the κ x ∈ (0, 1) domain by an arbitrary κ 0 obeying only the condition δ ≪ κ 0 ≪ 1, for the lower band contribution at the energy close to ϵ 0 (see Fig. 2 in the main text and SFig. 1 here), we perform the Taylor expansion within each interval with respect to the small parameter characteristic for that particular interval.
Supp.Fig. 1 Schematic presentation of the division of the κx ∈ (0, 1) domain by κ 0 for which δ ≪ κ 0 ≪ 1. κy(κx) is approximated by κ < y (κx) within κx ∈ (0, κ 0 ) and by κ > y (κx) within κx ∈ (κ 0 , 1) subintervals.a) κ < y : In this limit we use expansion in small (κ x , κ y ) ≪ 1, yielding from which we finally obtain (5) b) κ > y : In this limit we use expansion in small δ ≪ (κ x , κ y ), yielding for which, after one more expansion in additional small parameter of the order of δ, i.e. ∆e ≡ e − 1 ≪ 1, we get and, keeping accuracy up to ∼ δ 2 (later shown to be sufficient), we finally obtain 2 Area of the reconstructed Fermi surface; integrals We define the integral (area of the reconstructed Fermi surface) where a) I < : In the Eq. ( 9), the indefinite integral evaluates to where F(ϕ, m) and E(ϕ, m) are the incomplete elliptic integrals of the first and second kind, respectively.Inserting the limits of integration and performing the asymptotic expansion in r ≫ 1, we obtain where K(η) and E(η) are the complete elliptic integrals of the first and second kind, respectively.Using the fact that η ≡ (a − 1)/(a + 1) ≪ 1 is small, we expand the complete elliptic integrals where we used a = ∆e/δ and (∆e − δ)/δ ≪ 1.Using the above, we finally obtain the integral (9), up to the leading terms, b) I > : We express the integral as where To evaluate the integrals ( 11), first we take the corresponding indefinite integrals.The first integral is in which we insert limits of integration and expand in terms of which finally yields the value of the first integral (11) up to the leading terms in which we insert limits of integration and expand in terms of which finally yields the value of the second integral (11) up to the leading terms Collecting all the terms (8), we obtain the integral ( 7) where One should note that the final result does not contain contributions containing the arbitrary parameter κ 0 , which cancel each other.That also comes as a test of the controlled expansion of the same function in the different regime of parameters.
3 Optimization of the reconstructed band energy with respect to ε F 0 , ε F and Q We start with general expressions for the number of particles (electrons) N (ε) and the band energy E band (ε) at energy ε, in terms of the density of states ν(ε).For those we can use an alternative formulation in terms of the 2D equienergy surface S(ε) (i.e. the Fermi surface for ε = ε F ), where k is the 2D electron momentum and s = 2 and Z CaC6 = 6 are the spin and CaC 6 degeneracies.The band contribution to the condensation energy is defined as where band is the initial band energy, and band is the reconstructed band energy.Using the expressions above it reads where S 0 and S r are the initial and reconstructed Fermi surface, N 0 and N r are the numbers of electrons in the initial and reconstructed system, ε F 0 and ε F are the Fermi energies of the initial and reconstructed system.S r and ε F depend on Q and ∆, but in this consideration we keep ∆ fixed (it is optimized later in the total condensation energy).E (c) band needs to be optimized (i.e.maximized) with respect to ε F and Q with the condition of conservation of the number of particles in the reconstruction process.In that respect we define the Lagrange function where µ is the Lagrange multiplier and ∆N = N r (ε F ) − N 0 (ε F 0 ).Performing the standard optimization procedure, we obtain the following three conditions: The system (15) shows that the optimal value of the Fermi energy of the reconstructed system is equal to the initial one, i.e. ε F = ε F 0 as long as there is Q = Q opt for which ∂S r (Q)/∂Q| Q=Qopt = 0, i.e. S r (Q) has a maximum (it is checked by taking the second derivative).It happens only if this maximum is achieved in the lower band before the upper band is reached.The presentation of the Fermi surface reconstruction process as Q is varied is shown in SFig.2, here presented schematically before the calculation for the sake of clarity of presentation.There, it is evident that, taking only the lower band (red contour) into account, as Q is reduced the enclosed area S(Q) is first unchanged, then it starts to increase due to finite gap parameter, and then starts to decrease due to the sole fact that the size of the Brillouin zone (−Q/2, Q/2) is reduced by reduction of Q.This competition apparently creates a maximum of area S max r at Q = Q opt , as it is schematically presented in SFig.3. It is important to notice that the scenario described above is valid as long as the maximum S max r (Q opt ) appears, by reducing Q, before the second (upper) band is formed.If the second band is formed before reaching Q opt , i.e. for Q > Q opt , then E (c) band immediately starts to decrease, forming a maximum at that value of Q.Therefore, the reconstruction process goes on by reducing Q until ε F = ε F 0 where S r (ε F ) is maximal unless the upper band is reached before.In that case the reconstruction stops at Q = Q opt for which the upper band would be formed and ε F < ε F 0 .
Using the conservation of the number of particles N 0 (ε F 0) = N r (ε F ), i.e. S 0 (ε F 0) = S r (ε F ) in dimensionless variables (3), and integral (12), we get πe 2 F 0 = 4I, i.e. ∆e F = e F − 1, which, in the original (non-scaled) variables, reads as presented in the main text.Now, we apply the new scaling (natural scaling, normalization to the initial Fermi energy , and write the expression ( 16) in the form We test if ẽF = ẽF 0 is achieved before the upper band is reached using Eq. ( 17).Using perturbation theory for q opt = q (0) opt + q (1) opt , q (1) opt ≪ 1, we get q opt ≈ 1 − δ − α 2π δ3/2 which means that the overlap of the initial FS is larger than the critical q opt = 1 − δ for which the upper band is reached.This means that the reduction of q stops at q opt = 1 − δ.For that value, using perturbation theory in ẽF = ẽ(0) F ≪ 1, we obtain ẽF ≈ 1 − α 2π δ2 .It leads to the result presented in the main text.
4 Optimization of the total condensation energy with respect to ∆

The reconstructed band energy
We calculate the band energy of the reconstructed system, first for an arbitrary Q and ∆, as where s = 2 and Z CaC6 = 6 are the spin and CaC 6 degeneracy respectively, k 2) (and also the main text).Using the scaling (3), it reads where e(κ x , κ y , δ) = e − (κ x , κ y , δ) from Eq. ( 4).Inserting the optimal value of the wave vector that was found in the previous section, where q opt ≡ 1 − δ and To evaluate the integral J, as a function od δ, we develop the following procedure: for which, using the conservation of the number of particles S r (e F , δ) = S 0 (e F 0 ), where S 0 (e F 0) = πe 2 F 0 , after taking the derivative, we finally get Utilizing this result (after changing derivatives) we calculate the first term in Eq. ( 20) The second term in Eq. ( 20) is Again, using the conservation of the number of particles, S r (e F , δ) = S 0 (e F 0 ) = const, we evaluate the first term of the above expression to zero.The second term is expanded in powers of δ, where a linear term ∼ δ appears from δ in the numerator of the expression (setting it to zero everywhere else).
The other contributions (from the denominator and the boundaries of the integral) are of higher order i.e. ∼ δ 3 .The linear term is sufficient for the final result, therefore we take Collecting the terms above, we assemble Eq. ( 20) from which we calculate J from Eq. ( 19) simply by integrating with respect to δ, i.e.
The energy of the reconstructed band ( 18) is now easily calculated using e where is the energy of the initial, unreconstructed band and the constants α and β are defined in the expressions ( 13) and ( 21), respectively.

"Elastic" energy of the deformed Ca lattice
The last term in the mean-field Hamiltonian of the main text (taken per unit area), usually called "the elastic energy", describes the gain in the energy of the mechanically deformed crystal lattice due to the established CDW ground state coupled to the phonon mode with wave vector Q and frequency ω Q .It reads where λ ≡ ν F 0 g 2 /(ℏω Q ) is the dimensionless electron-phonon coupling constant for the phonon mode ω Q and ν F 0 is the initial DOS at the Fermi energy.

The condensation energy of the CDW state
The condensation energy of the CDW ground state is defined as a difference between the initial and the final energy of the system, i.e.
Using expressions ( 23) and ( 24), we obtain the result where and α and β are defined in expressions ( 13) and ( 21), respectively.In order to stabilize the CDW ground state, the condensation energy should be positive for a nonvanishing value of the order parameter.In this case, E CDW has a maximum for ∆ ̸ = 0 if λ > λ c , i.e. ∂E CDW /∂∆ = 0 gives the value of the order parameter for a given electron-phonon coupling λ.For the matter of presentation, we denote the initial band energy E (0) band by E 0 in the main text.

Supp. Fig. 2
Schematic presentation of the 2D Fermi surface reconstruction process at the same Fermi energy (characterized by the Fermi wave number k F ) while Q is varied.Reduction of Q controls the overlap of the initial Fermi surfaces (dashed curve) and size of the new Brillouin zone.The full red curve represents the contour of the reconstructed Fermi surface as Q is reduced, i.e. a) Q > 2k F , b) Q = 2k F , c) Q < 2k F , d) Q < 2k F and the second (upper) band is formed (blue).

Fig. 3
Schematic presentation of the change of area of the reconstructed Fermi surface Sr(Q), as Q is varied, of the lower band (only) -see also SFig. 2. For Q significantly larger than 2k F , gap parameter has no contribution, yielding the Sr(Q) being equal to πk 2 F .As Q is reduced to Q ∼ 2k F , lifting of degeneracy takes place by the (pseudo)gap opening and Sr(Q) increases.By further reduction of Q, Sr(Q) finally starts to decrease because size of the Brillouin zone (−Q/2, Q/2) is decreased by reduction of Q.Therefore, Sr(Q) has a maximum S max r at Q = Qopt.
) Previously, we have found ε F = ε F 0 1 − α 2π δ2 , i.e. e F = e F 0 1 − α F 0 , i.e. we can change the derivative for the more convenient one.The terms in the expression (20) are calculated below.First, we find