Acoustic phonons and spin relaxation in graphene nanoribbons
AAcoustic phonons and spin relaxation in graphene nanoribbons
Matthias Droth and Guido Burkard
Department of Physics, University of Konstanz, 78457 Konstanz, Germany
Phonons are responsible for limiting both the electron mobility and the spin relaxation time insolids and provide a mechanism for thermal transport. In view of a possible transistor function aswell as spintronics applications in graphene nanoribbons, we present a theoretical study of acousticphonons in these nanostructures. Using a two-dimensional continuum model which takes into ac-count the monatomic thickness of graphene, we derive Hermitian wave equations and infer phononcreation and annihilation operators. We elaborate on two types of boundary configuration, whichwe believe can be realized in experiment: (i) fixed and (ii) free boundaries. The former leads to agapped phonon dispersion relation, which is beneficial for high electron mobilites and long spin life-times. The latter exhibits an ungapped dispersion and a finite sound velocity of out-of-plane modesat the center of the Brillouin zone. In the limit of negligible boundary effects, bulk-like behavioris restored. We also discuss the deformation potential, which in some cases gives the dominantcontribution to the spin relaxation rate T − . PACS numbers: 63.22.Rc, 72.20.Dp, 76.60.Es, 81.07.Oj
I. INTRODUCTION
Its interesting electronic, mechanical, and thermalproperties have made graphene a promising candidatefor a wide range of applications, including ballistic tran-sistors as well as spintronics and nanoelectromechani-cal devices and heat management.
There are, how-ever, a number of challenges: (i) for epitaxial graphene,which is desirable for a controlled, large-scale production,the strong coupling to a substrate compromises theseproperties, (ii) graphene has no band gap, a handicapfor typical semiconductor applications, and (iii) acousticphonons limit the carrier mobility relevant for transistorfunctions.
The first issue can be overcome by removing substratematerial from underneath the carbon layer such that atrench is formed and the electronic properties of free-standing graphene are restored.
The second chal-lenge is met by graphene nanoribbons (GNRs), graphenestrips with a width at the nanometer scale (e.g., L ∼ µ m, W ∼
30 nm) which can exhibit a band gap.
Combining these advantages, the free-standing GNR ob-tained from epitaxial graphene on a trenched substrateis a very interesting design that deserves a detailed dis-cussion of its phonons.In this paper, we use a continuum model to studythe acoustic phonon properties and displacement fields u ( r ) = ( u x , u y , u z ) (out-of-plane modes shown in Fig. 1)of two different types of GNR that we think can be re-alized in experiment: (i) extended graphene that coversa thin trench, resulting in a GNR parallel to the trenchand with fixed lateral boundaries, Fig. 2 (a); (ii) a stripof graphene that stretches over a wide trench, leading toa GNR perpendicular to the trench and with free lateralboundaries, Fig. 2 (b). For both setups, we derive thelow-energy acoustic phonon spectra from a continuummodel that respects the monatomic structure of grapheneand write down the quantum mechanical form of these FIG. 1: (Color online) Displacement field u z of out-of-planemodes. The dimensionless coordinates are ¯ x = x/W , ¯ y = y/W , and ¯ z = z/W . (a),(b) Fixed boundaries. (c),(d) Freeboundaries. (a),(c) Fundamental mode. (b),(d) First over-tone. phonons.Our results can be probed experimentally via estab-lished techniques like electron energy loss spectroscopy orBrillouin light scattering. In addition to the electronmobility, the phononic behavior is essential for carbon-based nanoelectromechanical systems.
A recent ex- a r X i v : . [ c ond - m a t . m e s - h a ll ] O c t (a) fixed ( boundaries (b) free ( boundariesW WL Lxy yx FIG. 2: (Color online) Two nanoribbon configurations wheregraphene (blue) is spanned over a trenched substrate (orange):(a) fixed and (b) free boundaries. The coordinate systemis chosen in such a way that the undeformed ribbon lies inthe xy plane and that the y and ribbon axes coincide. Weassume the ribbon length to be much larger than the ribbonwidth ( L (cid:29) W ) and parallel lateral boundaries (dark blue)at x = ± W/ ample where the electron-phonon coupling has been ob-served experimentally is the Franck-Condon blockade insuspended carbon nanotube quantum dots. Phononsalso give rise to spin relaxation within a time T , whichis important for spintronics devices . The spin-orbitinteraction admixes different spin states ( ↑ , ↓ ) and elec-tron orbits k [see Eq. (18) below]. As a consequence,the electron-phonon coupling H EPC can mediate the Zee-man energy gµ B B = ¯ hω , where g is the electron g -factorin graphene and µ B denotes Bohr’s magneton, to thephonon bath with phonon numbers n ω . The rate forspin relaxation via emission of a phonon with energy ¯ hω is given by1 T = 2 π ¯ h |(cid:104) k ↓ , n ω + 1 | H EPC | k ↑ , n ω (cid:105)| ρ states (¯ hω ) , (1)with an explicit dependence on the phonon density ofstates ρ states . Several mechanisms contribute to T − andin some cases the deformation potential gives thedominant contribution. If the Zeeman energy lies withinthe energy gap of GNR phonons with fixed boundariesand if the temperature is sufficiently low, the spin life-time obtained from Eq. (1) diverges due to a vanishingdensity of states. II. CONTINUUM MODEL IN 2D
Low-energy acoustic phonons at the center of the Bril-louin zone have a wavelength much larger than atomicdistances and thus can be derived from continuum me-chanics. The carbon atoms in graphene lie within a two-dimensional surface and this property is conserved upondeformations, making graphene a quasi-two-dimensionalmaterial in three-dimensional (3D) real space. Conse-quently, all components of the displacement field u ( r ) can be nonzero but the components u iz of the strain ten-sor u ik = ( ∂ i u k + ∂ k u i ) / u xz and u yz are known to vanish for thin plates in the xy plane in general, the monatomic thickness of grapheneimplies that u zz must vanish as well. With u iz ≡
0, theelastic Lagrangian density of monolayer graphene is givenby L = T − V = ρ u − κ u z ) − λ u ii − µu ik , (2)where ∆ = ∂ x + ∂ y , the sum convention with u ii = u xx + u yy + u zz and u ik = u xx + u xy + · · · has been used, ρ is thesurface mass density, and κ is the bending rigidity. Note that the 3D bulk elastic constants have been re-placed by their 2D analogs λ = 2 hµ λ / (2 µ + λ )and µ = hµ where h is the plate thickness. The bulkand shear moduli are then given as B = λ + µ and µ ,respectively.Application of the Euler-Lagrange formalism to thefunctional (2) leads to the coupled set of differential equa-tions for in-plane modes ρ ¨ u x = ( B + µ ) ∂ x u x + µ ∂ y u x + B ∂ x ∂ y u y ,ρ ¨ u y = ( B + µ ) ∂ y u y + µ ∂ x u y + B ∂ x ∂ y u x , (3)which are decoupled from the differential equation for theout-of-plane modes, ρ ¨ u z = − κ (cid:0) ∂ x + ∂ y (cid:1) u z . (4)Assuming nanoribbon alignment with the y axis, fixedboundaries are described by u x = u y = 0 (in plane), (5) u z = ∂ x u z = 0 (out of plane) (6)at x = ± W/
2; see Fig. 2 (a). While these boundary con-ditions hold for both 2D and 3D lattices, we emphasizethat lattice dimensionality does affect free boundaries.For free edges in 2D it is required that, at x = ± W/ ∂ x u x + σ∂ y u y = 0 ∂ x u y + ∂ y u x = 0 (cid:27) (in plane), (7) ∂ x u z + (2 − σ ) ∂ x ∂ y u z = 0 ∂ x u z + σ∂ y u z = 0 (cid:27) (out of plane) , (8)where the quantity σ denotes Poisson’s ratio, Fig. 2 (b).Together with Young’s modulus E = h E , σ relates tothe bulk and shear moduli as B = E − σ ) , µ = E σ ) . (9) III. CLASSICAL SOLUTION
Typically, the length of a graphene nanoribbon exceedsits width many times, L (cid:29) W , thus allowing for FIG. 3: (Color online) Dispersion relations obtained from theprocedure described in Sec. III. The wavenumber q is givenby ¯ q = qW and the frequency ω of in-plane (out-of-plane)phonons by ¯ ω xy = ω (cid:112) ρ/ E W (¯ ω z = ω (cid:112) ρ/κ W ). (a) In-plane modes with fixed boundaries. (b) In-plane modes withfree boundaries. (c) Out-of-plane modes with fixed bound-aries. (d) Out-of-plane modes with free boundaries. (e)-(h) Dispersion relations (a)-(d) at the center of the Brillouinzone. (a),(c),(e),(g) Modes with fixed boundaries exhibit agap. (b),(d),(f),(h) Modes with free boundaries are gapless.(a),(b) Despite the coupling of transverse and longitudinalmodes, we find predominantly longitudinal and transversemodes on lines which we label LA (dashed blue line) andTA (dash-dotted red line), respectively. (c),(d) Independentof the boundaries, out-of-plane modes disperse quadraticallyfor large wave numbers (dashed green line). (h) Free out-of-plane modes feature a branch with linear dispersion at thezone center (dashed orange line). a plane wave ansatz along the y direction with periodicboundaries. Due to their decoupling, in-plane modes u x/y ( x, y, t ) = f x/y ( x )exp[ i ( qy − ωt )] and out-of-planemodes u z ( x, y, t ) = f z ( x )exp[ i ( qy − ωt )] can be treatedseparately. Exploiting the plane wave ansatz and denoting the i -th derivative of f as f ( i ) , Eq. (3) can be written as M xy ( f x , f y ) = − ρω ( f x , f y ), where M xy : (cid:18) f x f y (cid:19) (cid:55)→ (cid:32) ( B + µ ) f (2) x − µq f x + iBqf (1) y − ( B + µ ) q f y + µf (2) y + iBqf (1) x (cid:33) . (10)The general solution of this eigenvalue problem is( f x , f y ) = (cid:80) i =1 c i a i exp[ λ i x ], with a = (1 , iq/λ ), FIG. 4: (Color online) Displacement vector field ( u x , u y ) ofin-plane modes. Size and color of the arrows indicate the mag-nitude of the local deformation. We use the dimensionless co-ordinates ¯ x = x/W and ¯ y = y/W . (a),(b) Fixed boundaries.(c),(d) Free boundaries. (a),(c) Predominantly longitudinalmodes. (b),(d) Predominantly transverse modes. a = (1 , iq/λ ), a = (1 , iλ /q ), a = (1 , iλ /q ), and λ , = ± (cid:112) q − ρω / ( B + µ ), λ , = ± (cid:112) q − ρω /µ .Fixed boundaries are characterized by f x ( ± W/
2) = f y ( ± W/
2) = 0 and by virtue of the λ i , the set of linearequations deriving from these boundary conditions de-pends on the parameters q and ω . A numerical treatmentof this linear system yields the dispersion relation [Figs.3(a) and 3(e)] as well as the coefficients c i for the explicitform of the in-plane mode with fixed boundaries [Figs.4(a) and 4(b)]. Other boundary conditions and the out-of-plane modes can be treated likewise. The eigenvalueproblem obtained from (4) is M z f z = ( ρω /κ − q ) f z ,where the map M z and its eigenfunctions and eigenval-ues are given by M z : f z (cid:55)→ f (4) z − q f (2) z (11)and f z = (cid:80) i =1 d i e λ i x with λ i = ± (cid:113) q ± ω (cid:112) ρ/κ . IV. MODE ORTHONORMALITY ANDQUANTIZATION
In order to quantize the vibrational spectrum of thegraphene nanoribbon in terms of phonon creation andannihilation operators, the eigenfunctions of the origi-nal differential operators [Eqs. (3) and (4)] must be or-thogonal. While orthogonality of eigenmodes with differ-ent wavenumbers q follows from the plane wave ansatz,eigenmodes with same q require orthogonal functions( f ( α,q ) ,x , f ( α,q ) ,y ) and f ( α,q ) ,z . The index ( α, q ) labelsthe phonon branch α and the wavenumber q of a spe-cific eigenmode.The map (10) is Hermitian and hence has orthog-onal eigenfunctions if and only if the scalar product (cid:82) + W/ − W/ d x (cid:0) f ∗ x , f ∗ y (cid:1) M xy ( f x , f y ) T is real for all vectorfunctions ( f x , f y ) in the domain of M xy . One easilyshows via partial integration that M xy is Hermitian ifand only if the boundary terms satisfy( B + µ ) f ∗ x f (1) x + µf ∗ y f (1) y + iBqf ∗ x f y (cid:12)(cid:12)(cid:12) + W − W ∈ R (12)and that both fixed and free boundaries do indeed satisfythis condition.The general in-plane displacement field is u (cid:107) = (cid:88) α,q r ( α,q ) (cid:0) f ( α,q ) ,x e x + f ( α,q ) ,y e y (cid:1) e iqy , (13)where the harmonic time dependence has been absorbedin the normal coordinate. Using the orthogonality rela-tions mentioned above, one can resolve the normal co-ordinate and derive the Lagrangian and the canonicalmomentum. The identification r ( α,q ) = (cid:113) ¯ h/ ρLW ω ( α,q ) ( b ( α,q ) + b † ( α, − q ) ) , (14)where b † ( α,q ) ( b ( α,q ) ) creates (annihilates) an ( α, q )-phonon, complies with coordinate-momentum commuta-tion relations, and allows for a quantum mechanical for-mulation of (13). Quantization of the out-of-plane modesis achieved in the very same way. The Hermiticity of M z follows from f ∗ z f (3) z − f ∗ (1) z f (2) z − q f ∗ z f (1) z (cid:12)(cid:12)(cid:12) + W − W ∈ R (15)and, as above, fixed as well as free boundaries do satisfythis condition. The general out-of-plane displacement isgiven by u ⊥ = (cid:88) α,q r ( α,q ) f ( α,q ) ,z e z e iqy . (16) V. DISCUSSION OF PHONON SPECTRA
As specific values for sound velocities, etc., depend onthe elastic constants, we shall first discuss these constantsbefore turning to the properties of acoustic phonons. Dueto their decoupling, in-plane and out-of-plane phononscan be treated separately. For each case we will considerfixed and free boundaries.
A. Elastic constants
For graphene, most elastic constants remain to be set-tled by experiment and some seem to exhibit a tempera-ture dependence, which we do not take into account here.Moreover, a consistent set of constants must respect Eq.(9). The Zeeman energy for typical laboratory magneticfields ( ∼ ∼ σ of graphene rangefrom σ = 0 . Young’s modu-lus of a quasi-two-dimensional material, E = E h , fol-lows from its corresponding 3D bulk value and its as-sociated thickness h . While the most common litera-ture value of E for graphene is 1 TPa, a muchsmaller value, 0 . We use E = 3 . . σ and E into Eq. (9), we find B = 12 . and µ = 9 . for the bulk andshear moduli, respectively, in agreement with literaturevalues. All these values are in agreement with resultsof simulations for zero temperature. The bending rigidity of graphene, κ , is mainly deter-mined by the out-of-plane p z orbitals such that it can-not be inferred from other elastic constants. It has beenshown that κ decreases with increasing temperature. Literature values for zero temperature rangefrom 0 .
85 to 1 .
22 eV and we choose κ = 1 . ρ = 7 . × − kg / m ,follows directly from the atomic weight of natural carbon,12 .
01 u, and the interatomic distance in graphene, 1 .
42 ˚A.
B. In-plane phonons
The dispersion relation of in-plane modes with fixedboundaries is gapped and features infinitely manybranches with different energies originating from the zonecenter, Figs. 3(a) and 3(e). The gap relates to theenergy necessary for fixing the boundaries and is givenby 2 . h (cid:112) E /ρ /W . For W = 30 nm, this gap will be1 . . v LA = 22 km / s and v TA = 14 km / s, indepen-dent of the ribbon width. These values and the ratio v LA /v TA = 1 . (19 . / s, 12 . / s) andcarbon nanotubes (19 . / s, 12 . / s). Neverthe-less, we point out that our sound velocities are propor-tional to (cid:112) E /ρ , a value that is still under discussion forgraphene. The approach to linear, bulk-like behavior isexpected for large wave number, where the finite ribbonwidth appears like bulk for short-wavelength phonons.For free boundaries, the dispersion relation of in-planemodes is ungapped and the two branches that start atzero energy converge slightly below the TA line, Figs.3(b) and 3(f). The sound velocities and linear behav-ior for large wave number do not depend on boundaryconditions, as one would expect from the same argumentas above. Predominantly transverse and predominantlylongitudinal modes are shown in Figs. 4(c) and 4(d). Thetypical zero-point motion amplitude of in-plane modes is40 fm. C. Out-of-plane phonons
The dispersion relation of out-of-plane modes withfixed boundaries is shown in Figs. 3(c) and 3(g). Thegap due to the fixed boundary conditions is given by22 . h (cid:112) κ/ρ /W , which yields 7 . µ eV for W = 30 nm.The corresponding magnetic field is 68 mT. There are in-finitely many branches that correspond to different trans-verse excitations, Figs. 1(a) and 1(b). Again, away fromthe zone center, all branches approach bulk behavior,that is, a quadratic dispersion for out-of-plane modes. Similarly, the out-of-plane modes with free boundariesdisperse quadratically as in the bulk, for large wave num-bers, Figs. 3(d) and 3(h). The dispersion relation is gap-less and one branch exhibits a finite sound velocity atthe zone center. This sound velocity amounts to about70 m / s for W = 30 nm, is proportional to (cid:112) κ/ρ /W , andhence goes to zero for large W , again in agreement withbulk graphene. The typical zero-point motion amplitudeof out-of-plane modes is 0 . VI. DEFORMATION POTENTIAL AND SPINRELAXATION
Several mechanisms contribute to spin relaxation: out-of-plane modes via direct spin-phonon coupling and in-plane phonons via the deformation potential and bond-length change . Due to inversion symmetry, piezoelec-tric coupling does not occur in graphene. Here, we dis-cuss the deformation potential, which gives the dominantcontribution to T − under certain conditions.We find that any given in-plane phonon branch α cou-ples either via bond-length change or via the deformationpotential, depending on whether its displacement field iseven or odd in the x coordinate. The branch that origi-nates from ¯ ω xy = 3 . α , has a flatdispersion at the zone center and couples to the spin onlyvia the deformation potential. As a consequence of thedensity of states in Eq. (1), this mechanism will give thedominant contribution if the magnetic field is tuned to a value where the Zeeman energy is close to the Van Hovesingularity of α and coupling to out-of-plane modes isweak. Van Hove singularities also occur for out-of-planemodes at different values of ¯ ω z , Figs. 3(c) and 3(g). How-ever, ¯ ω xy and ¯ ω z scale differently with W , which allowsus to choose a ribbon width where there is a singularityfor in-plane modes ( α ) but not for out-of-plane modes.This situation will be discussed below.For the branch labeled α in Fig. 3 (f), which is linearnear the zone center, the spin couples to phonons onlyvia the deformation potential, as well. Even though itsdensity of states is finite, we discuss its contribution toEq. (1) as it is in accordance with previous results forsemiconductor quantum dots. In leading order, the deformation potential dependsonly on in-plane phonons, H EPC = g D ∇ · u (cid:107) ( x, y ) , (17)where g D ≈
30 eV is the coupling strength and ∇ = ( ∂ x , ∂ y ). The deformation potential is independentof the electron spin ( ↑ , ↓ ) but it does couple differentelectron orbits ( k ). As a consequence, H EPC couples tospin indirectly when Rashba-type spin-orbit interaction, H SO , is taken into account. In lowest order, the spin-orbit-perturbed electronic states are given by | k ↑(cid:105) = | k ↑(cid:105) (0) + (cid:88) k (cid:48) (cid:54) = k | k (cid:48) ↓(cid:105) (0) (0) (cid:104) k (cid:48) ↓ | H SO | k ↑(cid:105) (0) E k − E k (cid:48) + gµ B B , (18)where the superscript (0) indicates unperturbed productstates. Using these spin-orbit admixed states, we find (cid:104) k ↓ | H EPC | k ↑(cid:105) (19)= (cid:88) k (cid:48) (cid:54) = k (cid:34) ( H EPC ) kk (cid:48) ( H SO ) ↓↑ k (cid:48) k E k − E k (cid:48) + gµ B B + ( H EPC ) k (cid:48) k ( H SO ) ↓↑ kk (cid:48) E k − E k (cid:48) − gµ B B (cid:35) , where we denote the numerator in Eq. (18) as ( H SO ) ↓↑ k (cid:48) k and the spin-conserving transitions of H EPC accordingly.This is the matrix element required to calculate the re-laxation rate in Eq. (1).We find that for a given k (cid:48) the two terms in Eq. (19) ex-actly cancel each other at B = 0. This effect is known asVan Vleck cancellation and is expected for time-reversal-symmetric systems. Moreover, ( H SO ) ↓↑ k (cid:48) k vanishes if both k and k (cid:48) are even or odd at the same time.For fixed GNR edges, the phonon spectrum is gapped.In the range 3 . ≤ ¯ ω xy ≤ .
3, the branch α shows analmost flat dispersion. Its sound velocity increases as v α ∝ q such that the corresponding density of statesbehaves as ( ρ states ) α ∝ q − . The matrix element (19)varies as q B . : the dipole approximation gives rise toone order in q and Van Vleck cancellation to oneorder in B , reduced by ω − . ∝ B − . due to the pref-actor in Eq. (14). In total, the contribution to the spinrelaxation rate (1) is proportional to the magnetic field.Due to the Van Hove singularity of α at ¯ ω xy = 3 . T − ∝ B is the dominant behavior inthe range 3 . ≤ B ≤ . . ≤ B ≤ . W = 100 nm ( W = 30 nm), where the density of statesof out-of-plane modes is relatively small. These are ac-cessible laboratory magnetic fields and hence allow forexperimental examination of our results.If the magnetic field is tuned to a value where theZeeman energy lies within the gap of both in-plane andout-of-plane phonons [Figs. 3(e) and 3(g)], the electronspin cannot flip due to phonon emission. Then, multiple-phonon processes, where the Zeeman energy correspondsto the difference between an absorbed and an emittedphonon, become important. Again due to the gap, theseprocesses can be frozen out if the temperature T is lowenough. As discussed in Sec. V, the very soft out-of-plane modes have a much smaller gap, which thereforeimposes a tighter condition and which scales as W − .Assuming W = 30 nm, the spin lifetime inferred fromEq. (1) diverges for B <
68 mT and T (cid:28)
90 mK. Verynarrow GNRs with W = 10 nm are studied experimen-tally, as well. Accordingly, the requirements for such aribbon would be
B < .
61 T and T (cid:28) . B < ∼
100 mT). The branch α couples onlyvia the deformation potential and the other branch is apure shear mode. Due to its linear dispersion, we find B ∝ ω ∝ q and a constant density of states for α . Thematrix element in Eq. (19) scales as B . : one order in B arising from each of the Van Vleck cancellation, dipoleapproximation, and the gradient in Eq. (17), againreduced by the prefactor ω − . ∝ B − . in Eq. (14).Consequently, for low magnetic fields, the contributionof deformation potential and spin-orbit coupling to thespin relaxation rate (1) scales with B . In semiconduc-tors, T − ∝ B holds, as well. VII. CONCLUSION
Acoustic phonons are relevant for many GNR applica-tions and can be probed with established techniques.
Using a continuum model that accounts for themonatomic thickness of graphene, we derive boundaryconditions that lead to Hermitian wave equations. Wefocus on two types of boundary configurations: fixedand free boundaries. We explicitly give the correspond-ing classical solutions and, ensuring Hermiticity, infer aquantum theory with ribbon phonon creation and an-nihilation operators. Free boundaries lead to ungappeddispersion relations. In contrast, fixed boundaries leadto a gapped phonon dispersion of both in-plane and out-of-plane modes, which is most suitable for achieving highmobilities as well as long spin lifetimes. Regardless ofthe boundary configuration, all dispersion relations ap-proach bulk behavior for wavelengths small compared tothe ribbon width. Sound velocities that relate to trans-verse and longitudinal acoustical in-plane ribbon modesare in good accordance with values for bulk graphene.We also study phonon-induced spin relaxation in GNRs.We find that, if the Zeeman energy is tuned close to aVan Hove singularity of the density of states of in-planephonons, the deformation potential can be the dominanteffect for spin relaxation. In this case, it should be possi-ble to probe our predicted behavior for T experimentally.If the Zeeman energy lies within the gap of both in-planeand out-of-plane phonons with fixed boundaries and forlow enough temperatures, coupling to the lattice is in-hibited such that the spin lifetime obtained form Eq. (1)diverges. VIII. ACKNOWLEDGEMENTS
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