Casimir-Polder Force on Atoms or Nanoparticles from the Gapped and Doped Graphene: Asymptotic Behavior at Large Separations

The Casimir-Polder force acting on atoms and nanoparticles spaced at large separations from real graphene sheet possessing some energy gap and chemical potential is investigated in the framework of the Lifshitz theory. The reflection coefficients expressed via the polarization tensor of graphene found based on the first principles of thermal quantum field theory are used. It is shown that for graphene the separation distances starting from which the zero-frequency term of the Lifshitz formula contributes more than 99\% of the total Casimir-Polder force are less than the standard thermal length. According to our results, however, the classical limit for graphene, where the force becomes independent on the Planck constant, may be reached at much larger separations than the limit of large separations determined by the zero-frequency term of the Lifshitz formula depending on the values of the energy gap and chemical potential. The analytic asymptotic expressions for the zero-frequency term of the Lifshitz formula at large separations are derived. These asymptotic expressions agree up to 1\% with the results of numerical computations starting from some separation distance which increases with increasing energy gap and decreases with increasing chemical potential. Possible applications of the obtained results are discussed.


Introduction
The Casimir-Polder force [1] acts between electrically neutral small bodies (atoms, nanoparticles) and material surfaces. This force is induced by the zero-point and thermal fluctuations of the electromagnetic field which have their origin in the microscopic charges and currents occurring inside all material bodies. It is a generalization of the van der Waals force to separation distances where the relativistic effects already make a pronounced impact on the force value. This typically happens at separations exceeding several nanometers.
An important question is how soon the Casimir-Polder force from graphene approaches its limiting form given by the zero-frequency term in the Lifshitz formula, which is reached at large separations (high temperatures). In [73], the asymptotic behavior of the Casimir-Polder interaction was investigated in the case of an undoped graphene sheet possessing zero chemical potential. However, real graphene sheets are characterized not only by the energy gap in the spectrum of quasiparticles ∆ = 2mv 2 F , where m is small but nonzero mass of quasiparticles [46,74,75], but they are also doped, i.e., their crystal lattice contains some fraction of foreign atoms. This can be described by a nonzero value of the chemical potential µ depending on the doping concentration [76].
In this article, we examine the behavior of the Casimir-Polder force between atoms (nanoparticles) and real graphene sheets in the limit of large separations (high temperatures) as a function of the atom-plate separation a, the energy gap ∆, and the chemical potential µ. First, we demonstrate that the term of the Lifshitz formula at zero Matsubara frequency contributes more than 99% of the force magnitude at separations exceeding some value a 0 , which is distinctly less than the standard thermal lengthhc/k B T) ≈ 7.6µm at room temperature T = 300K (here k B is the Boltzmann constant). The value of a 0 decreases with increasing ∆. For sufficiently small ∆, a 0 increases with increasing µ, but for a larger ∆ the dependence of a 0 on µ becomes nonmonotonic.
Then we compare the large-separation Casimir-Polder force from graphene, given by the zero-frequency term of the Lifshitz formula, with that from an ideal metal plane. It is shown that for a fixed energy gap an agreement between these two quantities becomes better with increasing chemical potential of a graphene sheet.
Next, we derive simple asymptotic expressions for the zero-frequency contribution to the Lifshitz formula at large separations and find how it agrees with the results of numerical computations. For this purpose, we use the zero-frequency term of the Lifshitz formula with reflection coefficients expressed via the polarization tensor of graphene. The polarization tensor is calculated using several small parameters. The analytic asymptotic expressions for the large-separation Casimir-Polder force are derived for any values of the energy gap and chemical potential of a graphene sheet.
The derived asymptotic expressions are compared with numerical computations of the Casimir-Polder force at large separations. The application region of the analytic asymptotic results is determined. We show that with increasing energy gap an agreement between the asymptotic and computational results becomes worse, whereas, at the same separation, an increase of the chemical potential brings the asymptotic results in better agreement with the results of numerical computations.
The article is organized as follows. In Section 2, we present the Lifshitz formula for the Casimir-Polder force and reflection coefficients for the case of gapped and doped graphene in terms of the polarization tensor. Section 3 contains the exact expression and numerical computations of the Casimir-Polder force at large separations. In Section 4, the analytic asymptotic expressions for the Casimir-Polder force are derived. In Section 5, the asymptotic results for the Casimir-Polder force are compared with the results of numerical computations. Section 6 contains a discussion and Section 7 -our conclusions.

The Lifshitz Formula and Reflection Coefficients for Gapped and Doped Graphene
The Casimir-Polder force between an atom or a nanoparticle and any plane surface is expressed by the following Lifshitz formula, which we present in terms of dimensionless variables [54,77] Here, the prime on the sum in l means that the term with l = 0 is divided by 2, ζ l = ξ l /ω c , where ξ l = 2πk B Tl/h (l = 0, 1, 2, . . .) are the Matsubara frequencies, ω c = c/(2a) is the characteristic frequency, and α l = α(iζ l ω c ). The dimensionless variable y is defined as where k ⊥ is the magnitude of the wave vector projection on the plane of graphene, and r TM,TE are the reflection coefficients on graphene for the transverse magnetic (p) and transverse electric (s) polarizations of the electromagnetic field. Note that both the dynamic polarizability α l and the reflection coefficients r TM,TE are calculated at the pure imaginary frequencies iζ l . The reflection coefficients in (1) are expressed via the dimensionless polarization tensor of graphene [54] r TM (iζ l , y) = y Π 00,l y Π 00,l + 2(y 2 − ζ 2 l ) , where the components of the dimensionless Π βγ and dimensional Π βγ tensors (β, γ = 0, 1, 2) are connected by The dimensionless quantity Π l in (2) is defined as with a summation over β. The arguments of the polarization tensor components are omitted for brevity. As mentioned in Introduction, the polarization tensor of graphene is equivalent to the spatially nonlocal transverse and longitudinal dielectric functions defined in twodimensional space [78,79]. Using the dimensionless variables, one obtains The explicit expression for Π 00,l in terms of the dimensionless variables ζ l and y is presented in [54]. After identical transformations it can be put in a more convenient form Here, α = e 2 /(hc) is the fine structure constant, and, finally, In a similar way, the combination of the components of the polarization tensor Π l entering Equation (3) can be written in the form [54] By using Equations (1), (3), (7), and(11) one can compute the Casimir-Polder force between an atom or a nanoparticle and a real graphene sheet characterized by some energy gap and chemical potential.

The Casimir-Polder Force at Large Separations
It is well known that at large separations or, equivalently, at high temperatures the dominant contribution to the Casimir-Polder force is given by the term of (1) with l = 0 [77,80]. For atoms and nanoparticles interacting with the three-dimensional plates made of ordinary materials the zero-frequency term of the Lifshitz formula is approximately equal to the total force already at the thermal lengthhc/(k B T) ≈ 7.6 µm at room temperature. Below we demonstrate that for graphene the zero-frequency term determines the total force value at even smaller separations depending on the energy gap ∆ and chemical potential µ.
The zero-frequency term of the Lifshitz formula is obtained by separating the component with l = 0 from (1) where the reflection coefficient from (3) simplifies to r TM (0, y) = Π 00,0 (y) Π 00,0 (y) + 2y . (13) Note that the TE reflection coefficient does not contribute to (12) because in (1) taken at l = 0 it is multiplied by zero. Here we have explicitly indicated the argument y of the polarization tensor. The component of the dimensionless polarization tensor Π 00,0 is obtained from (7) by putting ζ 0 = 0 and w 0 (u, y) is defined in (9) with l = 0 and B 0 from (15). Now we perform numerical computations in order to find such separation a 0 that at all separations a ⩾ a 0 the quantity F 0 from (12) contributes no less than 99% of the total Casimir-Polder force (1). This is done for heavy atoms, for instance, Rb and for nanoparticles. It is apparent that the value of a 0 depends on the energy gap and chemical potential of the specific graphene sheet, so that a 0 = a 0 (∆, µ). For this purpose, first we compute F 0 from (12) as a function of separation using Equations (13)- (15). All computations here and below are performed at room temperature, T = 300 K, in the range of ∆ from 0.001 eV to 0.2 eV with a step of 0.01 eV for 4 values of µ = 0, 25, 75, and 150 meV. Similar computations of the total Casimir-Polder force F from (1) were performed by Equations (1), (3), (7), and (11) at separations exceeding 1 µm where, without the loss of accuracy, one can use an approximation of the static atomic polarizability α l ≈ α(0) = α 0 [77]. The point is that computations of the Casimir-Polder force from graphene by (1) using the frequencydependent polarizability α(ω) show [54] that even for light atoms like He * the value of a 0 is above 1 µm. In so doing, for heavy atoms like Rb and for nanoparticles all α l giving contribution to the result (i.e., with l ≤ 6 at a = 1µm, l ≤ 3 at a = 2µm, and l ≤ 2 at a = 3µm) are approximately equal to α 0 .
Using the obtained computational results, in Figure 1 we plot a 0 as a function of the energy gap of graphene ∆ by the four lines (black, red, blue, and brown) for the chemical potential µ equal to 0, 25, 75, and 150 meV, respectively. From Figure 1 it is seen that the value of a 0 decreases with increasing energy gap. For ∆ < 0.15 eV the value of a 0 increases with increasing µ, but for larger ∆ this already not the case. Specifically, the value of a 0 for a graphene sheet with µ = 0 may become larger than for sheets with µ = 25 and 75 meV. Intuitively it is clear that increasing µ brings graphene closer to an ideal metal and, thus, leads to an increase of a 0 . To the contrary, an increase of ∆ results in decreasing a 0 due to the suppressed impact of the thermal effects on the Matsubara terms in (1) with l ≥ 1. The actual value of a 0 for small µ and large ∆ results from the interplay between these two effects. By and large, the value of a 0 for gapped and doped graphene is distinctly less than the thermal length for ordinary materials equal to 7.6 µm. It is interesting also to compare the Casimir-Polder force from gapped and doped graphene at large separations F 0 with that from an ideal metal plane given by [77] This is so-called classical limit because the force does not depend on the Planck constant. To understand where the large-separation Casimir-Polder forces from gpaphene and from an ideal metal plane come together, we compute the relative quantity .
The computational results for a graphene sheet with ∆ = 0.2 eV are shown in Figure 2 at T = 300 K as a function of separation by the four lines (black, red, blue, and brown) from bottom to top for the chemical potential µ equal to 0, 25, 75, and 150 meV, respectively. In the inset, the behavior of blue and brown lines at short separations (µ = 75 and 150 eV) is shown on an enlarged scale with better resolution. The dashed lines indicate the border of the 1-percent relative deviation between the large-separation behavior of the Casimir-Polder forces from a graphene sheet and an ideal metal plane. The separation region a ⩾ 3 µm is considered where, according to Figure 1, F 0 represents the large-separation behavior of the Casimir-Polder force.
As is seen in Figure 2, for ∆ = 0.2 eV, µ = 150 meV (the top line) the Casimir-Polder forces from graphene and from an ideal metal plane agree within 1% at all separations considered. With decreasing µ to 75, 25, and 0 meV the agreement within 1% occurs at separations exceeding 7, 36, and 54 µm, respectively. This result is physically natural if to take into account that larger µ correspond to larger doping concentration, i.e., make graphene more akin to an ideal metal plane. Thus, for graphene sheets with relatively low chemical potential the classical limit is reached only at rather large separation distances.
In the next section, we obtain simple asymptotic expressions for the quantity F 0 from (12), which allow to calculate the Casimir-Polder force from gapped and doped graphene at large separations do not using complicated expressions for the polarization tensor.

Asymptotic Expressions for the Casimir-Polder Force
We consider the Casimir-Polder force (12) where the reflection coefficient r TM is given by (13) and the polarization tensor is expressed by (14) with the notations (15). We seek for the asymptotic expression of (14) and (12) under the condition At T = 300 K this condition is well satisfied for a > 0.2 µm, i.e., is not restrictive. The reflection coefficient (13) can be identically rewritten in the form r TM (0, y) = 1 − 2y Π 00,0 (y) + 2y .
As is seen from (14), the parameter (18) stands in front of the second contribution to Π 00,0 by making it much larger than unity. Note that this contribution does not depend on y. Simultaneously, the main contribution to (12) is given by y ∼ 1. Because of this, one can replace y with unity in the denominator of (19) and neglect by 2 in comparison with Π 00,0 . As a result, (19) takes the form Substituting (20) in (12) and integrating with respect to y, one obtains the asymptotic expression where F IM 0 is the Casimir-Polder force from an ideal metal plane at large separations defined in (17). Note that F as 0 (a, T) depends on the Planck constanth through the polarization tensor of graphene Π 00,0 (1). Now we deal with the asymptotic expressions for the polarization tensor Π 00,0 (1) and start from the case ∆ = 0, µ ̸ = 0. In this case, we have from (15) D 0 = 0 and from (8) Ψ(0) = π. Because of this (14) simplifies to Owing to the condition (18), the first contribution on the right-hand side of (22) is much less than the second and can be neglected. Owing to the same condition, according to (15), Because of this, it holds B 0 u ≪ 1 and one can put exp(B 0 u) ≈ 1 in (22). As a result, (22) takes the form Calculating the integral, we find an expression where, thanks to (18), the second term is much less than the first. As a result, one obtains In the special case of a pristine graphene ∆ = µ = 0, (26) reduces to which agrees with [81]. We are coming now to the case of arbitrary, but not too small, values of ∆ and any value of µ. In fact we assume that D 0 defined in (15) with y = 1 is much larger than unity The assumption (28) is not too restrictive. The point is that we consider the Casimir-Polder force at large separations a > 2 µm, i.e.,hω c < 0.05 eV. This means that the condition (28) is satisfied for all ∆ > 0.001 eV.
Let us consider the first term in the polarization tensor (14) with y = 1. Using the definition of Ψ in (8) and expanding arctan(D −1 0 ) in powers of small parameter D −1 0 , we obtain The maximum value of the latter quantity in (29) [i.e., of the first term in (14)] for our values of parameters is unity and it decreases with increasing ∆. Thus, thanks to (18), the first term in (14) is much less that the second one containing the logarithm function. We turn our attention to the third term in (14). Due to (28), the lower and upper integration limits are very close and one can replace B 0 u with B 0 D 0 in the powers of exponents entering w 0 (u, y) defined in (9). Taking into account that, according to (15), B 0 D 0 = ∆/(2k B T), we can rewrite (14) with y = 1 in the form The integral in (30) is easily calculated Under the condition (28), we find from (31) I ≈ −D 0 and (30) leads to After making identical transformations in the first and second terms of this expression, we bring it to the form [see (A7) in Appendix A for details] By putting µ = 0 in (33), one finds Under the additional condition ∆ ≪ 4k B T, we can neglect by the second term in (34), as compared to the first one, and obtain This result coincides with that obtained earlier in [81] if to take into account that [81] uses the notation∆ = ∆/(hω c ), where ∆ is equal to ∆/2 in our current notations, i.e., to one half of the total energy gap. Note that at T = 300 K the application region of (35) reduces to 0.001 eV < ∆ < 0.01 eV, i.e., it is rather narrow. In the Appendix A, using the condition opposite to (28), we prove, however, that (35) remains valid for arbitrary small values of ∆ [see Equation (A8) with any µ including µ = 0]. Now we finalize the asymptotic expression F as 0 for the Casimir-Polder force from gapped and doped graphene with not too small energy gap ∆. For this purpose, we substitute (33) to (21). The obtained expression F as 0 is valid under the condition (18). In the next section, we find how close would the asymptotic Casimir-Polder force be to the numerical values of the force at large separations F 0 .

Comparison Between Asymptotic and Numerical Results
Here, we compare the analytic asymptotic expressions for the large-separation Casimir-Polder force F as 0 obtained in Section 4 with numerical computations of F 0 for different values of the energy gap and chemical potential.
We begin with the case of an undoped graphene sheet, µ = 0, and calculate the ratio F 0 /F as 0 for different values of the energy gap ∆. In doing so, F 0 is computed by (12)-(15) and F as 0 by (21) and (34). All computations are performed at T = 300 K. In Figure 3, the ratio F 0 /F as 0 is shown as a function of separation between an atom (nanoparticle) and a graphene sheet by the three lines counted from top to bottom for the energy gap ∆=0.1, 0.15, 0.2 eV, respectively. The case of large separations up to 100 µm is shown in the inset for ∆=0.15 and 0.2 eV.  As is seen in Figure 3, the best agreement between the asymptotic and computed Casimir-Polder forces holds for the smallest ∆=0.1 eV. In this case, F as 0 agrees with F 0 in the limits of 1% at any a > 3 µm. With increasing ∆, an agreement between F as 0 and F 0 gets worse. Thus, for a graphene sheet with ∆=0.15 eV the 1% agreement is reached at a=14 µm. As to graphene with ∆=0.2 eV, the 2% agreement is reached only at a = 50 µm. Now we consider an impact of the chemical potential on the measure of agreement between F as 0 and F 0 . For this purpose, we consider the graphene sheets with ∆=0.2 eV (the case of the worst agreement in Figure 3) but various values of the chemical potential. Computations of F as 0 are performed by (21) and (33). In Figure 4, the ratio F 0 /F as 0 is again shown as a function of separation by the three lines counted from top to bottom for the chemical potential µ = 150, 75, and 25 meV, respectively (brown, blue, and red lines). In the inset, the lines for a graphene sheet with µ = 75 and 25 meV are shown in the region of large separations up to 100 µm.
From Figure 4, one can conclude that an increase in the value of the chemical potential makes an agreement between F as 0 and F 0 better. Thus, for µ = 150 meV the 1% agreement occurs at all separations a > 3 µm, whereas for µ = 75 meV at a > 5.5 µm. For a graphene sheet with µ = 25 meV the 1% agreement is reached only at a ≈ 34 µm. We can say that an increase in the values of ∆ and µ acts on an agreement between F as 0 and F 0 in the opposite directions by making it worse and better, respectively, at the same separation distance.
The above results allow to determine the region of distances where the large-separation Casimr-Polder force F 0 can be replaced with its asymptotic behavior F as 0 depending on the values of the energy gap and chemical potential of the specific graphene sheet. These results are valid for both light and heavy atoms and for spherical nanoparticles.

Discussion
As discussed in Section 1, the Casimir-Polder force on atoms and nanoparticles from different surfaces including graphene is the subject of topical investigations in the interests of both fundamental physics and its applications. The Casimir-Polder force from graphene attracts an especial attention because graphene is the novel material of high promise due to its unusual mechanical and electrical properties.
From the theoretical point of view, graphene offers major advantages over the more conventional materials because its response functions to the electromagnetic field can be found on the basis of first principles of thermal quantum field theory without resort to phenomenological models. This is not the case for real metals whose response to the lowfrequency electromagnetic field is described by the phenomenological Drude model, which lacks an experimental confirmation in the area of s-polarized evanescent waves giving an important contribution to the Casimir effect [82,83]. As a result, there are contradictions between the predictions of the Lifshitz theory and measurements of the Casimir force between metallic surfaces (see [77,80,84,85] for a review).
Although the Casimir-Polder force from graphene is not yet measured, the already performed measurements of the Casimir force between a graphene-coated plate and an Au-coated sphere demonstrate an excellent agreement between theoretical predictions of the Lifshitz theory using the polarization tensor of graphene and the measurement data [86,87]. Because of this, the above results for the Casimir-Polder force from gapped and doped graphene at large separations, obtained here using the formalism of the polarization tensor, are of high degree of reliability.

Conclusions
To conclude, in the foregoing we investigated the Casimir-Polder force acting on atoms and nanoparticles from the gapped and doped graphene sheet at large separations. We have found separation distances starting from which the zero-frequency term of the Lifshitz formula coincides with the total Casimir-Polder force acting on heavy atoms or spherical nanoparticles in the limits of 1%. It was shown that, depending on the values of the energy gap and chemical potential of graphene, the classical limit may be reached at much larger distances than the limit of large separations.
Furthermore, we derived the analytic asymptotic expressions for the zero-frequency term of the Lifshitz formula at large separations with the reflection coefficient expressed via the polarization tensor of graphene. These expressions are valid for light and heavy atoms and nanoparticles of spherical shape. The obtained asymptotic expressions were compared with numerical computations of the zero-frequency term. According to our results, with increasing energy gap of graphene, the separation distance ensuring a better than 1% agreement between the asymptotic and numerically computed forces also increases. By contrast, an increase of the chemical potential of graphene leads to a 1% agreement between the asymptotic and numerical results at shorter separations.
The obtained results make it possible to easily calculate the large-separation Casimir-Polder force from the gapped and doped graphene sheets and to control it by varying the values of the energy gap and chemical potential. This can be used in precision experiments on quantum reflection and Bose-Einstein condensation near the surfaces of graphene, as well as in various technological applications. In future it would be interesting to investigate the large-separation Casimir-Polder force from the graphene-coated substrates made of different materials.

Appendix A. Asymptotic Expression for Graphene with Small Energy Gap
The asymptotic expression for the Casimir-Polder force from gapped and doped graphene obtained in Section 4 is valid for graphene, satisfying the condition (28), i.e., having not too small energy gap. Now we consider the separation region where the condition (18) is again satisfied but the energy gap satisfies the condition which is just the opposite to (28).
Owing the condition (18), the inequality (23) preserves its validity and the first contribution to the polarization tensor (14) with y = 1 is much less than the second and can be omitted.
First, we evaluate the third contribution to (14) given by By introducing the new integration variable, v = u − D 0 , this term takes the form Using (A1), we conclude that the upper integration limit in (A3) 1 + D 2 0 − D 0 ∼ 1. Then, because of (23), one can put exp(B 0 v) ≈ 1 and rewrite (A3) as Calculating the integral in (A4), we obtain With account of (18), it is seen that the magnitude of I(1) is much less than the second term in the polarization tensor (14) and can be omitted. Thus, we are left with only the second term in (14) which can be transformed similar to (32). Here, we present this transformation in greater detail Owing to condition (A1), the last term in (A7) can be neglected and, as a result, For µ = 0, (A8) reduces to (35). Thus, (35) is really valid for arbitrary small ∆ satisfying the condition (A1).