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The optical phonon resonance scattering with spin-conserving and spin-flip processes between Landau levels in graphene
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 J. Phys.: Condens. Matter 26 395302 (http://iopscience.iop.org/0953-8984/26/39/395302) View the table of contents for this issue, or go to the journal homepage for more
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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 395302 (9pp)
doi:10.1088/0953-8984/26/39/395302
The optical phonon resonance scattering with spin-conserving and spin-flip processes between Landau levels in graphene Zi-Wu Wang1 , Zhi-Qing Li1 and Shu-Shen Li2 1
Tianjin Key Laboratory of Low Dimensional Materials Physics and Preparing Technology, Department of Physics, Tianjin University, Tianjin 300072, People’s Republic of China 2 Institute of Semiconductor, CAS, Beijing 100083, People’s Republic of China E-mail: [email protected] Received 8 May 2014, revised 22 July 2014 Accepted for publication 24 July 2014 Published 5 September 2014 Abstract
In the frame of Huang–Rhys’s lattice relaxation model, we theoretically investigate the electron relaxation assisted by optical phonon resonance scattering among Landau levels with spin-conserving and spin-flip processes in graphene. We not only consider the longitudinal optical (LO) phonon scattering, but also the surface optical (SO) phonon scattering induced by the polar substrate under the graphene. The relaxation rate displays a Gaussian distribution by considering the effect of lattice relaxation that arises from the electron-deformation potential acoustic phonon interaction. We find that the relaxation rate of the spin-conserving process is three orders of magnitude larger than that of the spin-flip process for the same phonon mode. Moreover, the discrepancy of relaxation rates between the SO and LO phonon scattering is at two orders of magnitude for the same process. The opposite temperature dependence of the relaxation rates are also obtained in the resonance energy regime in the present model. In addition, the influences of the strength of Rashba spin–orbital coupling, the dielectric constant of different polar substrates and the distance between the graphene and substrate on the relaxation rates are also discussed quantitatively for the SO phonon scattering. The obtained results could be useful for the graphene-based applications on the mid-infrared and terahertz modulation and spintronic devices. Keywords: graphene, optical phonon scattering, spin-flip process (Some figures may appear in colour only in the online journal)
properties such as the giant optical nonlinearity and very high rate of photon production in the mid-infrared and THz range [4, 5]. These properties may be used for the development of novel electronic devices for the mid-infrared, THz and optoelectronic applications proposed very recently [6, 7]. Therefore, the detailed studies of relaxation processes between LLs in graphene are crucial for the performance of these devices. Optical phonon scattering between the LLs is a very important relaxation process, which belongs to the non-radiation channels and hinders seriously the radiation of the infrared and THz photons. It is also known as magnetophonon resonance
1. Introduction
The basic properties of graphene in the strong magnetic field have been extensively discussed in literatures [1–3]. The unique feature is that the Landau levels (LLs) in graphene are unequal between them due to its linear band structure and this has been observed in early experiments [2, 3], which differs from the conventional two-dimensional materials and structures. Transitions between LLs in graphene fall into the mid-infrared to terahertz (THz) range for a magnetic field in the range 0.01–10 Tesla, which results in a series of novel 0953-8984/14/395302+09$33.00
1
© 2014 IOP Publishing Ltd
Printed in the UK
J. Phys.: Condens. Matter 26 (2014) 395302
Z-W Wang et al
(MPR) [3] where the optical phonon energy matches the energy separation between LLs. The MPR has been widely used to explore the effective mass of electron, the frequency of optical phonon, the strength of electron-optical phonon coupling in conventional semiconductor and low-dimensional structures. In fact, the MPR for the longitudinal optical (LO) phonon in monolayer and mutilayer graphene has been observed from the magneto-Raman scattering and magneto-optical conductivity experiments [8–10]. Several theoretical models have also been presented to explain these experimental results [1, 3]. In recent years, the SO phonon scattering, which is induced by the polar substrate of the graphene as shown in figure 1(a), aroused immense attention as it gives rise to a series of special properties, such as the saturation of the current, the change of the resistivity at room temperature, and the polaronic shifts of the absorption peaks in magneto-optical conductivity. Recently, the MPR for surface optical (SO) phonon modes were also studied theoretically in n-doped graphene using the balance-equation scheme for nonlinear magnetotransport [11]. Although the LO and SO phonon scattering have been extensively studied in various experiments, the detailed studies for the relaxation rates of the LO and SO phonon scattering and the comparisons between them have not yet been mentioned until now. The knowledge of this aspect is significant not only for the potential applications for mid-infrared and THz radiation, but also the electrical and thermal transport studies in graphene. In addition, most of the studies for the optical phonon scattering between LLs in graphene considered the spin-conserving processes and the spin-flip processes are neglected. In the present paper, we calculate numerically the relaxation rates of the LO and SO phonon resonance scattering with spin-conserving and spin-flip processes between LLs of the graphene based on the Huang–Rhys lattice relaxation model. The relaxation rates display a Gaussian distribution with the line-width of several milli-electron volts (meV) by taking into account the effect of lattice relaxation that arises from the electron-deformation potential acoustic phonon interaction. Through the numerical calculation, we find that the relaxation rate of the spin-conserving process is three orders of magnitude larger than that of the spin-flip process for the same phonon mode. Moreover, the SO phonon scattering is two orders of magnitude larger than the LO phonon scattering in the same process. We also discuss the temperature dependence of the relaxation rates, the opposite temperature dependence are obtained in the resonance energy regime. In addition, the dependencies of relaxation rates on the strength of Rashba-type spin–orbital coupling (SOC), the dielectric constant of different polar substrates and the distance between the graphene and substrate are also analyzed quantitatively for the SO phonon scattering. The obtained theoretical results related to the possible experimental phenomenon are also discussed. We hope that our results can stimulate and highlight related experimental work to identify these conclusions.
Figure 1. (a) The schematic diagram of the sandwich structure (air/graphene/substrate), d denotes the distance between the graphene and substrate, the magnetic field B exerts in the perpendicular direction. (b) The schematic diagram for the distribution of the Landau levels with Zeeman spin splitting and the optical phonon scattering from the LL n = 1 to n = 0+ with the spin-conserving and spin-flip processes, 2G is the opened gap of graphene, ↑ (↓) stands for the spin up (down) state.
in figure 1(a)). d denotes the distance of the graphene above the substrate, the air has the permittivity of free space ε0 . The existing surface optical phonon modes are described in detail in section 4. In the presence of a uniform static perpendicular magnetic field B the total Hamiltonian is given by H = He + Hz + Hph + Hep
(1)
with the first Hamiltonian He = H0 + Hso . H0 is the single free carrier Hamiltonian without the SOC [12]: H0 = vF (p + eA) · σ + τ Gσz ,
(2)
where the first term is the well-known Dirac Hamiltonian for graphene. The second term represents the opened band gap 2G which can be induced, e.g. by the substrate or a monolayer of intercalated structure [13, 14]. τ = ±1 denote the K and K’ valleys. We can restrict the scattering in a single valley in the absence of valley scattering, e.g. τ = +1. Hsoc is the term of SOC, which is the key element to spin-flip relaxation. Hz = 21 gμB B·s describes the Zeeman coupling of the electron † spin to the magnetic field. Hph = ¯ ωqλ aqλ aqλ is the qλ h phonon Hamiltonian with ωqλ standing for the phonon energy spectrum of branch λ and momentum q. Hep is the Hamiltonian of electron–phonon interaction. The single free carrier Hamiltonian in equation (2) can be solved analytically. In the frame of symmetric gauge for A = (B/2)(−y, x, 0) , the eigenstate and eigenenergy are given by [1, 12] T C m−1 φ (r)ei(m−1)θ χs , (3) |n, s(0) = ψn (r, θ )χs = n An m A iTn CBn φB (r)eimθ En = Kn (|n|¯hωB )2 + G2 , (4) with
2. Theoretical model
m−1 = CAn
The graphene monolayer we considered is sandwiched between the substrate and the air (see the schematic diagram
n!/2π lB2 (n + |m| − 1)!,
φA (r) = ξ |m|−1 e−ξ 2
2
/2
|m|−1
Ln
,
J. Phys.: Condens. Matter 26 (2014) 395302
and m CBn =
Z-W Wang et al
where Γ is the line-width that arises from the mixing of many mechanisms, e.g. the impurity, disorder, electron–phonon interaction. There were many researches for the line-width Γ in bulk and low-dimensional systems. In general, the line-width Γ is set to a fixed value to discuss the theoretical and experimental results quantitatively. However, in the present study, the delta function is replaced by the Fourier expansion and we consider the contribution of the electron-acoustic phonon interaction to the line-width due to the effect of lattice relaxation, which is convenient to investigate the temperature dependence of the line-width in the relaxation and transition processes. For the coupled electron lattice system, the LL state in graphene can be decomposed into the Born–Oppenheimer approximation as Φi (, Qiq ) = ψi (R, Qiq ) i (Qiq ), where R(r, θ ) is the electron position coordinate, Qiq models the normal coordinate to describe the lattice vibration. ψi and
i are the electron and lattice wave functions, respectively. In the electron wave function, Qiq is only a parameter. The lattice wave function i is a direct product of states of harmonic oscillators whose oscillating equilibrium origins are influenced by the electron state, especially in the presence of a strong magnetic field, which can be expressed as:
i (Qiq ) =
nq (Qiq ) =
nq (Qq + Δiq ) (9)
n!/2πlB2 (n + |m|)!,
φB (r) = ξ |m| e−ξ
2
/2
|m|
Ln .
√ Here Tn = 1 or (1/ 2) are for the cases of n = 0 or (n = 0); |m| n = n + (|m| + m)/2, (n = 0, ±1, ..., m = 0, ±1, √...). Ln is the √ Laguerre polynomial. The parameter ξ = r/√2lB with ¯ /eB denotes the magnetic length; h ¯ ωB = 2¯hvF / lB lB = h is the effective magnetic energy; vF = 106 m/s is the Fermi velocity near the Dirac point; χs is the spin state with s = ↑, ↓, ±1. Furthermore, Kn = 1 (−1) is for the electrons (holes) when n > 0 (n < 0), and Kn = 0 for n = 0± . The schematic diagram of the distributions for the zero and first excited LLs with spin states are shown in figure 1(b). In general, the SOC is composed of the Dresselhaus-type and Rashba-type D R Hsoc = Hsoc + Hsoc = ΔD τ σz νz + ΔR (τ σx νy − σy νx ),
(5)
where ΔD denotes the Dresselhaus coupling strength, which is very weak due to the symmetrical lattice structure of graphene and is disregarded in the present paper. ΔR is the Rashba coupling strength that we mainly considered, which can be modulated by an external factor, such as the electrical field or the substrate [15]. The SOC can be treated as a small perturbation to the energy of LLs in equation (4). However, the effect of the SOC is to weakly mix the eigenstates equation (3), e.g. the state |n, ↑(0) mixes with the other states |n , ↓(0) (n = n ). In the first-order perturbation, the SOC corrects the electron spin product states |n, ↑ = |n, ↑(0) and |n, ↑ = |n, ↑(0) to |n, ↑ = |n, ↑(0) + |n, ↓ = |n, ↓(0) +
n , ↓ |Hsoc |n, ↑(0) |n , ↓, En − En − 21 gμB B
(6a)
n , ↑ |Hsoc |n, ↓(0) |n , ↑, En − En − 21 gμB B
(6b)
q
q
ωq Qiq h ¯
LA where Δiq = ψi |HEPI |ψi /ωq2 describes the shift of the lattice normal oscillator originated before and after the electron transiLA denotes the electron-longitudinal tion. The Hamiltonian HEPI acoustic (LA) phonon interaction and can be written as [16]: Dq i 0 † LA HEPI = )eiq·r , (10) (aq + a−q 0 i ρω /¯ h q q
(0)
(0)
2π op | Φi | HEPI | Φf |2 δ(Ef i − h ¯ ωop ), h ¯ f
21
ωq = Hiq √ n q 2 nq ! π h ¯ ωq Q2iq × exp − , 2¯h
respectively. We illustrate the zero (n = 0+ ) and first excited (n = 1) LLs mixing with the adjacent LLs for the Rashba-type SOC in appendix A. According to the selection rule L|n|+1 → L|n| in graphene for the transitions between LLs [4], based on the Fermi golden rule, the relaxation rate assisted by the optical phonon scattering can be written as: W =
√
† ) is the creation (annihilation) operator for the where aq (a−q LA phonon with the wave vector q, D denotes the deformation potential constant, whose value is in 10–50 eV intervals [16]. ωq = υq is the frequency of the LA phonon and satisfies the linear dispersion relation and υ = 2 × 104 m s−1 is the acoustic velocity. We emphasize that different electron-acoustic phonon interactions correspond to different changes in the lattice induced by phonons. Dominant electron–phonon interaction for long wavelength acoustic mode is LA phonon through the deformation potential mechanism, which is caused by an area change in the unit cell and gives the main contributions to the effect of lattice relaxation and the line-width. Hiq is the Hermitian polynomial, nq is the LA phonon number. Considering the effect of lattice relaxation, equation (7) can be rewritten as [17]: 2π op W = | Φi | HEPI | Φf |2 δ(Ef i − h ¯ ωop ) (11) h ¯ f
(7)
where Φi (Φf ) is the initial (final) electron state of the LLs in graphene as described in equation (1). Ef i is the energy separation between the initial and final LLs with spin states, h ¯ ωop is the optical phonon energy with dispersionless relation. For usual cases, in order to evaluate the relaxation rate of equation (7), the delta function is replaced by a Lorentzian line-shape function: Γ /2 1 δ(Ef i − h (8) ¯ ωop ) → π (Ef i − h ¯ ωop )2 + (Γ /2)2
=
3
2π op | ψi | HEPI | ψf |2 ×, h ¯ f
J. Phys.: Condens. Matter 26 (2014) 395302
and =
Z-W Wang et al
3. Results and discussion
Pav | i (Qiq ) | f (Qf q ) |2 δ(Ef i − h ¯ ωop ),
We choose the relaxation process from the first excited Landau level (n = 1) to zero Landau level (n = 0+ ) as an example to study the LO and SO phonon resonance scattering with spinconserving process (|1, ↓ −→ |0+ , ↓) and spin-flip process (|1, ↑ −→ |0+ , ↓), respectively. In a numerical simulation process, we apply the effective single-mode of the LA phonon to calculate the line-width of the relaxation rates [19, 20]. Throughout the paper, we fix the band gap 2G=40 meV and the deformation potential constant D=30 eV for the electron-LA phonon interaction and assume that the Fermi level is below the band gap. The Fermi level can be controlled by the electrical gate or doping. In fact, the position of the Fermi level and the modulation of the band gap have an essential influence on the transitions between the LLs [1], which is not the subject of the present paper.
f nq ,niq
(12)
e−β(nq +1/2)¯hωq i
Pav = ∞
niq =0
e−β(nq +1/2)¯hωq i
(13)
.
Here is the overlap integral of lattice vibrational wave function and is derived in detail in [17], Eif = Ei − Ef is the energy difference of the whole electron-lattice system before and after the transition including the energy of lattice relaxation, i.e. Ei = Ei − 1/2 q ωq2 Δ2iq . The average over the initial phonon states (Pav ) and summation over the final phonon states have been explicitly shown in the equation. The parameters β = 1/KB T . In the Condon approximation, op op the electronic matrix Mf i = ψf | HEPI | ψi can be regarded as independent of the lattice coordinate. Following the standard lattice relaxation procedure to carry out the average (summation) over initial (final) phonon states (the details of the procedure are given in [17]), we get
op | Mf i |2 +∞ dμeF (μ,Eif ) (14) W = h ¯ −∞ where ¯ ωop ) + F (μ, Eif ) = −iμ(Ef i − h
ωq q
2¯h
Δ2f iq
3.1. The longitudinal optical (LO) phonon scattering with spin-conserving and spin-flip process
Optical phonon in graphene is represented by the relative displacement of two sublattice carbon atoms in a unit cell. Phonon modes couple two neighboring carbon atoms through bond stretching and bending, so that the electron–phonon interaction becomes an off-diagonal matrix in the pseudospin space, which was originally derived in [21] and [22]. The Hamiltonian of the electron-optical phonon interaction can be written as:
(15)
β¯hωq × coth (cos μ¯hωq − 1) + i sin μ¯hωq . 2 The Δf iq = Δf q − Δiq depends on the LL states and strength of electron-acoustic phonon interaction. For multimode phonons, the integral of equation (14) can be obtained analytically. In such a case, the strongcoupling approximation is a popular and widely accepted approximation, i.e. by expanding the function F (μ, Eif ) to the second-order of μ. By using the steepest decent method [18, 19], equation (14) becomes 21 op | Mf i |2 2π F (16) W = h ¯ ST (¯hωq )2
op
HEPI
where the matrix N is N(q) =
F = exp −
¯ ωop − S¯hωq )2 (Ef i − h
2ST (¯hωq )2T ωq S¯hωq = Δ2f iq h ¯ ωq , 2¯ h q
and ST (¯hωq )2T
=
ωq q
2¯h
Δ2f iq (¯hωq )2
(17)
,
β¯hωq coth 2
(18) .
h ¯ † N(q)(aq + a−q )eiq·r , ρω0
0 NAB eiφq
NBA e−iφq 0
(20)
,
with NAB = −1 or i and NBA = 1 or i for longitudinal optical (LO) or transverse optical (TO) phonon, respectively, and φq = tan−1 (qy /qx ) is the azithmuthal angle of the wave vector q , ξ = dlnγ0 /dlnb ∼ 2 is a dimensionless parameter that denotes the change of the nearest-neighbor tight-binding matrix element with respect to the bond length b = 0.136 × 10−9 m, the parameter γ = 0.62×10−9 eV· m, ρ = 7.5×10−7 kg· m−2 is the mass density, ω0 is the optical phonon frequency for graphene. We only consider the electron-LO phonon interaction as an example in our analysis and h ¯ ωLO =196 meV is the LO phonon energy. All of the parameters mentioned above are from [21]. According to equations (16) and (20), we obtain the relaxation rate of the LO phonon resonance scattering 21 2 | MLO 2π f i |SC WSC = (21) h ¯ ST (¯hωq )2T (Ef i − h ¯ ωLO − S¯hωq )2 × exp − 2ST (¯hωq )2T
T
where
ξγ = 2 b q
(19)
The form factor F is a Gaussian distribution with a line-width 2ST (¯hωq )2T . Considering the temperature effect, the relaxation rate W should be multiplied by the factor Nop +1 for the optical phonon emission process. Nop = 1/(exp(¯hωop /KB T ) − 1) is the Bose distribution of optical phonon. 4
J. Phys.: Condens. Matter 26 (2014) 395302
Z-W Wang et al
for both the spin-conserving and spin-flip process. With increasing temperature, more LA phonons are thermally excited resulting in the increase of line-width. The detailed magnetic field and temperature dependence of the line-width were discussed in [20]. In general, the reciprocal of the relaxation rate (1/W) represents the relaxation time. From the comparison between the processes in figures 1(a) and (b), it can be deduced that the relaxation time of the spin-conserving process is three orders of magnitude shorter than that of the spin-flip process. This means that the spin-flip process needs more time than the spin-conserving process, which is consistent with the theoretical prediction as we expected and needs to be proved in further experiments. In the present model, the relaxation time of the spin-conserving process is in the hundreds of nanoseconds scale, which is longer than the processes of electron-hole recombination and generation via inter-and intra-valley optical phonon scattering in the absence of magnetic field [23–27]. Obviously, the energy levels were quantized into the LLs resulting in the inhibition of the optical phonon scattering. But the timescale is very close to the slow relaxation process mediated by electron-electron Auger scattering under strong magnetic field [28], thus to distinguish between them is very difficult. However, the Auger scattering sensitively depends on the carrier density but not for the optical phonon scattering. The Auger and LO phonon scattering belong to the non-radiation processes and were considered as the main obstacle for the fabrication of tunable far-infrared and terahertz laser based on inter-Landau-level emission [29]. Recently, Wendler and co-worker [30] presented a theoretical approach to study the optical phonon-induced carrier relaxation between the LLs in intrinsic graphene. They found that the optical phonon resonance scattering is enhanced at a specific field and the obtained relaxation time can access to the picosecond scale. When the energy of the optical phonon mismatches the LLs spacing, the two-phonon LO+LA relaxation process has also been proposed [31]. The study of these non-radiation processes has aroused enormous interest and is crucial for many potential applications in graphene. As for the LO phonon resonance scattering with spin-flip process shown in figure 1(b), the very long relaxation time is obtained in the microsecond scale. More important is that the relaxation time can be controlled by the modulation of the strength of the Rashba SOC as shown in figure 5. In the past years, the spin-flip relaxation assisted by the acoustic phonon scattering based on the SOC mixed mechanism in the graphene quantum dot and nanoribbon has been studied widely [12, 32–34]. But the investigations about the spin-flip via optical phonon resonance scattering between LLs in graphene are very few until now and need to be identified by the corresponding experimental measurement. The very interesting characteristic in figure 2 is that the opposite temperature dependences of the relaxation processes are presented in the resonance energy regime. From equations (16) and (21), we conclude that the factors NLO + 1, 1/ST (¯hωq )2T and F determine the temperature dependence of the relaxation rate. The contribution of the factor NLO + 1 is very weak due to the large LO phonon energy compared with the temperature range we considered. The latter two factors play a dominant role, which reduces and enhances the relaxation rate
Figure 2. The LO phonon relaxation rates with spin-conserving (a) and spin-flip (b) as a function of the energy separation between the LL n = 1 and n = 0+ at different fixed temperatures for the Rashba SOC constant ΔR = 0.05 meV.
for the spin-conserving process and 21 2 | A |2 | MLO 2π f i |SF WSF = h ¯ ST (¯hωq )2T (Ef i − h ¯ ωLO − S¯hωq )2 × exp − 2ST (¯hωq )2T
(22)
for the spin-flip process with A=
iΔR . E1 − E2 − 21 gμB B
Here, the parameter A denotes the magnitude of the admixture between different LLs due to the Rashba-type SOC. The 2 detailed expressions for the matrix elements | MLO f i |SC and LO 2 | Mf i |SF are given in appendix B. Figure 2 shows the relations of the LO phonon resonance relaxation rate (W) with the energy separation between the LL n = 1 and n = 0+ at different fixed temperature. Figures 2(a) and (b) show the spin-conserving and spin-flip process, respectively. It can be seen that the relaxation rates follow a Gaussian distribution with the small line-width of several meV due to the effect of LA phonon lattice relaxation 5
J. Phys.: Condens. Matter 26 (2014) 395302
Z-W Wang et al
respectively with increasing temperature. At the peak position regime, that is the condition of Ef i − h ¯ ωLO − S¯hωq ≈ 0 sat-
Table 1. Surface optical phonon modes for different polar substrates. Parameters have been taken from [36] and [37].
isfied, only the factor 1/ST (¯hωq )2T predominates the process. It can be clearly seen that the relaxation rate is inverse to the temperature, which means the effect of lattice relaxation of the LA phonon delays the emission of LO phonon with increasing temperature. Except for this peak position, the form factor F predominates the process, which results in the relaxation rate increasing with temperature. So the effect of lattice relaxation assists the emission of LO phonon in this regime. These two types of temperature dependence have been observed in experiments for many low-dimensional systems and further tests are needed to confirm the results for graphene. In fact, the temperature dependence of the surface optical phonon scattering in the next section is the same as that of the LO phonon scattering. The influences of the strength of Rashba-type SOC on the relaxation processes are discussed in detailed in figure 5.
Quantity(units)
SiC
SiO2
Al2 O3
HfO2
κ0 (ε0 ) κ∞ (ε0 ) h ¯ ωSO,1 (meV) h ¯ ωSO,2 (meV)
9.7 6.5 167 116
3.9 2.5 146 60
12.53 3.2 94 55
22.0 5.03 53 19
3.2. The surface optical (SO) phonon scattering with spin-conserving and spin-flip process
For the structure that the graphene monolayer is sandwiched between, the substrate and the air as plotted in figure 1(a), there exists the surface optical phonon modes, which correspond to a certain bulk transverse optical phonon mode of the substrate [35–37]. These modes interact with the electrons in graphene and have a potential influence on the basic properties of graphene [38–42]. The Hamiltonian of the electron-SO phonon interaction is given by SO HEPI =
e2 η¯hωSO,ν 2ε0
21 −qz e † iq·r (23) + a +)e (a √ q,ν −q,ν q q
with η=
(κ0 − κ∞ ) , (κ∞ + 1)(κ0 + 1)
where ωSO,ν is the frequency of SO phonon, ν = 1, 2 denotes the two branch, ε0 is the permittivity of vacuum and κ∞ (κ0 ) is the high (low) frequency dielectric constant. The parameter η is a combination of the known dielectric constants of the substrates and denotes the strength of the polarizability of the dielectric interface. Similar to the LO phonon resonance scattering, the relaxation rate of SO phonon resonance scattering is then given by 21 | MSO,ν |2SC 2π fi WSC = (24) h ¯ ST (¯hωq )2T (Ef i − h ¯ ωSO,ν − S¯hωq )2 × exp − 2ST (¯hωq )2T for the spin-conserving process and 21 | A |2 | MSO,ν |2SF 2π fi WSF = (25) h ¯ ST (¯hωq )2T (Ef i − h ¯ ωSO,ν − S¯hωq )2 × exp − 2ST (¯hωq )2T
Figure 3. The relaxation rates of the SO phonon resonance scattering with spin-conserving as a function of the energy separation between the LL n = 1 and n = 0+ for different polar substrates at fixed temperature T = 80 K, Rashba SOC constant ΔR = 0.05 meV and the distance d = 0.3 nm.
for the spin-flip process. The parameter A is same as the LO phonon scattering and the detailed expressions for the scattering matrix element | MSO,ν |2SC and | MSO,ν |2SF are fi fi given in appendix B. The adopted parameters in calculations are listed in table 1. Figure 3 shows the relaxation rates of the SO phonon resonance scattering with spin-conserving process as a function of the energy separation between the LL n = 1 and n = 0+ for different polar substrates. One can clearly see that the relaxation rate of SO phonon resonance scattering is two orders of magnitude larger than that of the LO phonon plotted 6
J. Phys.: Condens. Matter 26 (2014) 395302
Z-W Wang et al
in figure 2(a). The SO phonon scattering is enhanced: which can be attributed to the strong interaction between the electrons in graphene and the SO phonons of the polar substrates. In the past years, the influences of the SO phonon scattering on the low- and high-field carrier mobility, magneto-optical conductivity, hot electron relaxation and heat dissipation, plasmon emission and absorption and so on, were studied extensively [7, 38–42]. The LO phonon resonance scattering between the LLs in graphene has been proved in earlier experiments [1, 3, 8]. However, the SO phonon resonance scattering was mostly studied as a theoretical aspect and the related experimental identifications have been hard to make, until now, which may be because of its enhanced scattering. Equations (23) and (24) show that the rates of the SO phonon scattering are mainly determined by the energy (or frequency) of the phonon h ¯ ωSO and the dielectric constant η for the fixed temperature and distance parameter d. From figure 3, we can see that the relaxation rates of the high energy phonon branch are larger than that of the low branch in the same polar substrate. Through the comparisons between the high (or low) energy phonon branches in the different polar substrates, we notice that the relaxation rates increase from the SiC to Al2 O3 polar substrate due to the increase of their corresponding dielectric constant η as listed in table I. Although the dielectric constant of the polar substrate HfO2 is very high, the relaxation rates are smaller than other polar substrates, which arises from the energy of the phonon branch in HfO2 being very low. Therefore, the interplays between the energy of the phonon and the dielectric constant determine the relaxation rates of the SO phonon resonance scattering in the spin-conserving process and the former predominates the processes. In addition, the line-width of the high energy phonon is broader than that of the low energy one, which attributes to the high energy phonon resonance matching the large energy separation corresponding to the strong magnetic field, and thus results in the effect of lattice relaxation for LA phonon enhanced [20]. The relaxation rates of the SO phonon resonance scattering with spin-flip process as a function of the energy separation for different polar substrates are plotted in figure 4. The discrepancies are very obvious by comparing this process with the spin-conserving process as shown in figure 3. Firstly, the relaxation rate of the spin-flip is three orders of magnitude smaller than that of the spin-conserving process. Secondly, the relaxation rate of the low energy phonon branch is larger than that of the high energy one in the same polar substrates. The reason is that the relaxation rate is determined not only by the energy of phonon branch and the dielectric constant but also the parameter A. The parameter represents the magnitude of the admixture between different LLs due to the Rashba-type SOC and is in inverse proportion to the energy separation in its expression. So the large energy separation, that is the high energy phonon resonance, corresponds to the small value of A, which leads to the results that the relaxation rates decrease with increasing the phonon energy. The parameter A plays a dominate role in determining the relaxation rates in the spinflip processes. The relaxation rates increase with increasing the dielectric constants for different substrates, which is similar to the spin-conserving processes.
Figure 4. The relaxation rates of the SO phonon resonance scattering with spin-flip as a function of the energy separation between the LL n = 1 and n = 0+ for different polar substrates at fixed temperature T = 80 K, Rashba SOC constant ΔR = 0.05 meV and the distance d = 0.3 nm.
In figure 5, we choose the SiC polar substrate as an example to discuss quantitatively the influence of the Rashba SOC constant ΔR and the distance d on the relaxation rates for the fixed energy separation and temperature. Through the numerical calculation, we find that the influence of the Rashba SOC on the spin-conserving process can be neglected; we only illustrate its influence on the spin-flip process in figure 5. We can see that the relaxation rate increases quadratically with the Rashba SOC constant ΔR , which can be derived from the expression of the parameter A and equation (25). The conclusion is also fit for the LO phonon resonance scattering with spin-flip process. In fact, the modulation of the polar substrates on the Rashba-type SOC and the spin relaxation processes have been studied both theoretically and experimentally and have many potential applications on spintronics [15, 43, 44]. This topic is a hot research point up until now, but not the object of the present paper. In figure 5, the relaxation rates for spin-flip processes decreases with increasing the distance d, which arises from the interaction between the electrons in the graphene and the SO phonon in the polar substrate becoming weak due to the 7
J. Phys.: Condens. Matter 26 (2014) 395302
Z-W Wang et al
Acknowledgment
The authors are grateful to P-J Lin for the language support and helpful discussion. This work was supported by the National Natural Science Foundation of China (No.11304355). Appendix A. The mixed states for several lowest LLs due to the Rashba-type SOC
According to the selection rules of the Rashba-type SOC, the mixed states can be written R 1, ↓ |Hsoc |0+ , ↑(0) |1, ↓(0) , E0+ − E1 − 21 gμB B (A.1a) (0) (A.1b) |0+ , ↓ = |0+ , ↓ ,
|0+ , ↑ = |0+ , ↑(0) +
R 2, ↓ |Hsoc |1, ↑(0) |2, ↓(0) , 1 E1 − E2 − 2 gμB B
(0)
|1, ↑ = |1, ↑(0) + Figure 5. The SO phonon relaxation rate with the spin-flip process as functions of the Rashba SOC constant ΔR and the distance d at fixed energy separation E10+ = 167 meV and temperature T = 80 K for the SiC polar substrate.
|1, ↓ = |1, ↓(0) +
(0)
(A.1c)
R 0+ , ↑ |Hsoc |1, ↓(0) |0+ , ↑(0) , (A.1d) 1 E1 − E0+ − 2 gμB B
(0)
After the arithmetical operation for the matrix elements of R the Hsoc , the equations simplify to
increase of the distance between the graphene and polar substrate. The influence of the distance on the relaxation rate of the spin-conserving process is very similar to that of the spinflip process. These interesting properties need to be confirmed by related experiments in the future.
|0+ , ↑ = |0+ , ↑
(0)
√ i 2ΔR + |1, ↓, E0+ − E1 − 21 gμB B
|0+ , ↓ = |0+ , ↓(0) ,
(A.2a) (A.2b)
iΔR |2, ↓(0) , (A.2c) E1 − E2 − 21 gμB B √ i 2ΔR (0) |1, ↓ = |1, ↓ − |0+ , ↑(0) . (A.2d) E1 − E0+ − 21 gμB B
4. Conclusion
|1, ↑ = |1, ↑(0) +
In conclusion, we theoretically investigate the LO and SO phonon resonance scattering with spin-conserving and spinflip processes between LLs in graphene in the Huang–Rhys lattice relaxation model. We find that (i) the relaxation rate follows a Gaussian form with the line-width in several meV due to the effect of the lattice relaxation arising from the electrondeformation potential acoustic phonon interaction; (ii) the relaxation rates of the spin-conserving process are three orders of magnitude larger than the spin-flip process. Moreover, the SO phonon scattering is two orders of magnitude larger than the LO phonon scattering in the same process; (iii) the opposite temperature dependence of the relaxation rates are obtained in the resonance energy regime in the present model; (vi) the relaxation rate of the spin-conserving process for the SO phonon scattering increases with increasing the phonon energy, which is contrary to that of the spin-flip process; (v) the relaxation rates increase with increasing the dielectric constant for the different polar substrates in all processes; (vi) the influence of the strength of Rashba SOC on the relaxation rate follows a quadratical increasing trend in the spin-flip process and can be neglected in the spin-conserving process; (vii) the relaxation rates decrease dramatically with increasing the distance between the graphene and substrate, which arises from the interaction between the electrons in the graphene and the SO phonon in the polar substrate becoming weak.
Appendix B. The LO and SO phonon scattering matrix elements
In the Condon approximation, the LO and SO phonon scattering matrix elements for the spin-flip and spin-conserving process are given by
|2SF = | MSO,ν fi
8
2 | MLO f i |SF =
ξ 2γ 2 , 16π b4 ρωLO lB2
(B.1)
2 | MLO f i |SC =
ξ 2γ 2 , 32π b4 ρωLO lB2
(B.2)
e2 ηωSO,ν 256π ε0 2 √ 3 12d 2 8d 4 2d2 × 2π + 3 + 5 e lB 2lB lB lB √ 2d 10d 8d 3 × 1 − Erf − 2 − 4 , lB lB lB
(B.3)
J. Phys.: Condens. Matter 26 (2014) 395302
| MSO,ν |2SC = fi
2 e2 ηωSO,ν √ 1 2d 2 2d2 2π + 3 e lB 32πε0 2lB lB √2d 2d − 2 . × 1 − Erf lB lB
Z-W Wang et al
[19] [20] [21] [22] [23]
(B.4)
Here the subscript ν = 1, 2 denotes the different branch of the SO phonon modes.
[24] [25] [26]
References [27] [1] Goerbig M O 2011 Rev. Mod. Phys. 83 1193 [2] Li G H and Andrei E Y 2007 Nat. Phys. 3 623 [3] Orlita M and Potemski M 2010 Semicond. Sci. Technol. 25 063001 [4] Yao X H and Belyanin A 2012 Phys. Rev. Lett. 108 255503 [5] Tokman M, Yao X H and Belyanin A 2013 Phys. Rev. Lett. 110 077404 [6] Glazov M M and Ganichev S D 2014 Phys. Rep. 535 101 [7] Low T and Avouris P 2014 ACS Nano 8 1086 [8] Yan J, Goler S, Rhone T D, Han M, He R, Kim P, Pellegrini V and Pinczuk A 2010 Phys. Rev. Lett. 105 227401 [9] Kim Y, Ma Y, Imambekov A, Kalugin N G, Lombardo A, Ferrari A C, Kono J and Smirnov D 2012 Phys. Rev. B 85 121403 [10] Orlita M, Tan L Z, Potemski M, Sprinkle M, Berger C, de Heer W A, Louie S G and Martinez G 2012 Phys. Rev. Lett. 108 247401 [11] Wang C M and Luan X Y 2013 Europhys. Lett. 103 37012 [12] Struck P R and Burkard G 2010 Phys. Rev. B 82 125401 [13] Zhou S Y, Gweon G H, Fedorov A V, First P N, DeHeer W A, Lee D H, Guinea F, CastroNeto A H and Lanzara A 2007 Nat. Matter. 6 770 [14] Varykhalov A, Scholz M R, Kim T K and Rader O 2010 Phys. Rev. B 82 121101 [15] Ertler C, Konschuh S, Gmitra M and Fabian J 2009 Phys. Rev. B 80 041405 [16] Viljas J K and Heikkil¨o T T 2010 Phys. Rev. B 81 245404 [17] Huang K and Rhys A 1950 Proc. R. Soc. Ser. A 204 406 [18] Li X Q and Arakawa Y 1997 Phys. Rev. B 56 10423
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Li X Q and Arakawa Y 1999 Phys. Rev. B 60 1915 Wang Z W, Liu L, Li W P and Xu K 2014 Phys. Lett. A 378 65 Ando T 2006 J. Phys. Soc. Japan 75 124701 Ando T 2007 J. Phys. Soc. Japan 76 104711 Rana F, George P A, Strait J H, Dawlaty J, Shivaraman S, Chandrashekhar M and Spencer M G 2009 Phys. Rev. B 79 115447 Suzuura H and Ando T 2009 J. Phys: Conf. Ser. 150 022080 Xu W, Dong H M, Li L L, Yao J Q, Vasilopoulos P and Peeters F M 2010 Phys. Rev. B 82 125304 Kim R, Perebeinos V and Avouris P 2011 Phys. Rev. B 84 075449 Winnerl S, Orlita M, Plochocka P, Kossacki P, Schneider H and Helm M 2011 Phys. Rev. Lett 107 237401 Plochocka P, Kossacki P, Golnik A, Kazimierczuk T, Berger C, de Heer W A and Potemski M 2009 Phys. Rev. B 80 245415 Morimoto T, Hatsugai Y and Aoki H 2008 Phys. Rev. B 78 073406 Wendler F, Knorr A and Malic E 2013 Appl. Phys. Lett. 103 253117 Li W P, Yin J W, Yu Y F and Wang Z W 2013 Solid State Commun. 163 19 Droth M and Burkard G 2011 Phys. Rev. B 84 155404 Volk C, Neumann C, Kazarski S, Fringes S, Engels S, Haupt F, M¨uller A and Stampfer C 2013 Nat. Commun. 4 1753 Droth M and Burkard G 2013 Phys. Rev. B 87 205432 Fratini S and Guinea F 2008 Phys. Rev. B 77 195415 Konar A, Fang T and Jena D 2010 Phys. Rev. B 82 115452 Lin I T and Liu J M 2013 Appl. Phys. Lett. 103 081606 DaSilva A M, Zou K, Jain J K and Zhu J 2010 Phys. Rev. Lett. 104 236601 Zou K, Hong X, Keefer D and Zhu J 2010 Phys. Rev. Lett. 105 126601 Scharf B, Perebeinos V, Fabian J and Avouris P 2013 Phys. Rev. B 87 035414 Hwang E H and Das Sarma S 2013 Phys. Rev. B 87 115432 ˇ c I 2013 Phys. Rev. B Scharf B, Perebeinos V, Fabian J and Zuti´ 88 125429 Han W, Pi K, McCreary K M, Li Y, Wong J J I, Swartz A G and Kawakami R K 2010 Phys. Rev. Lett. 105 167202 Shakouri K H, Masir M R, Jellal A, Choubabi E B and Peeters F M 2013 Phys. Rev. B 88 115408
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The optical phonon resonance scattering with spin-conserving and spin-flip processes between Landau levels in graphene
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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 395302 (9pp)
doi:10.1088/0953-8984/26/39/395302
The optical phonon resonance scattering with spin-conserving and spin-flip processes between Landau levels in graphene Zi-Wu Wang1 , Zhi-Qing Li1 and Shu-Shen Li2 1
Tianjin Key Laboratory of Low Dimensional Materials Physics and Preparing Technology, Department of Physics, Tianjin University, Tianjin 300072, People’s Republic of China 2 Institute of Semiconductor, CAS, Beijing 100083, People’s Republic of China E-mail: [email protected] Received 8 May 2014, revised 22 July 2014 Accepted for publication 24 July 2014 Published 5 September 2014 Abstract
In the frame of Huang–Rhys’s lattice relaxation model, we theoretically investigate the electron relaxation assisted by optical phonon resonance scattering among Landau levels with spin-conserving and spin-flip processes in graphene. We not only consider the longitudinal optical (LO) phonon scattering, but also the surface optical (SO) phonon scattering induced by the polar substrate under the graphene. The relaxation rate displays a Gaussian distribution by considering the effect of lattice relaxation that arises from the electron-deformation potential acoustic phonon interaction. We find that the relaxation rate of the spin-conserving process is three orders of magnitude larger than that of the spin-flip process for the same phonon mode. Moreover, the discrepancy of relaxation rates between the SO and LO phonon scattering is at two orders of magnitude for the same process. The opposite temperature dependence of the relaxation rates are also obtained in the resonance energy regime in the present model. In addition, the influences of the strength of Rashba spin–orbital coupling, the dielectric constant of different polar substrates and the distance between the graphene and substrate on the relaxation rates are also discussed quantitatively for the SO phonon scattering. The obtained results could be useful for the graphene-based applications on the mid-infrared and terahertz modulation and spintronic devices. Keywords: graphene, optical phonon scattering, spin-flip process (Some figures may appear in colour only in the online journal)
properties such as the giant optical nonlinearity and very high rate of photon production in the mid-infrared and THz range [4, 5]. These properties may be used for the development of novel electronic devices for the mid-infrared, THz and optoelectronic applications proposed very recently [6, 7]. Therefore, the detailed studies of relaxation processes between LLs in graphene are crucial for the performance of these devices. Optical phonon scattering between the LLs is a very important relaxation process, which belongs to the non-radiation channels and hinders seriously the radiation of the infrared and THz photons. It is also known as magnetophonon resonance
1. Introduction
The basic properties of graphene in the strong magnetic field have been extensively discussed in literatures [1–3]. The unique feature is that the Landau levels (LLs) in graphene are unequal between them due to its linear band structure and this has been observed in early experiments [2, 3], which differs from the conventional two-dimensional materials and structures. Transitions between LLs in graphene fall into the mid-infrared to terahertz (THz) range for a magnetic field in the range 0.01–10 Tesla, which results in a series of novel 0953-8984/14/395302+09$33.00
1
© 2014 IOP Publishing Ltd
Printed in the UK
J. Phys.: Condens. Matter 26 (2014) 395302
Z-W Wang et al
(MPR) [3] where the optical phonon energy matches the energy separation between LLs. The MPR has been widely used to explore the effective mass of electron, the frequency of optical phonon, the strength of electron-optical phonon coupling in conventional semiconductor and low-dimensional structures. In fact, the MPR for the longitudinal optical (LO) phonon in monolayer and mutilayer graphene has been observed from the magneto-Raman scattering and magneto-optical conductivity experiments [8–10]. Several theoretical models have also been presented to explain these experimental results [1, 3]. In recent years, the SO phonon scattering, which is induced by the polar substrate of the graphene as shown in figure 1(a), aroused immense attention as it gives rise to a series of special properties, such as the saturation of the current, the change of the resistivity at room temperature, and the polaronic shifts of the absorption peaks in magneto-optical conductivity. Recently, the MPR for surface optical (SO) phonon modes were also studied theoretically in n-doped graphene using the balance-equation scheme for nonlinear magnetotransport [11]. Although the LO and SO phonon scattering have been extensively studied in various experiments, the detailed studies for the relaxation rates of the LO and SO phonon scattering and the comparisons between them have not yet been mentioned until now. The knowledge of this aspect is significant not only for the potential applications for mid-infrared and THz radiation, but also the electrical and thermal transport studies in graphene. In addition, most of the studies for the optical phonon scattering between LLs in graphene considered the spin-conserving processes and the spin-flip processes are neglected. In the present paper, we calculate numerically the relaxation rates of the LO and SO phonon resonance scattering with spin-conserving and spin-flip processes between LLs of the graphene based on the Huang–Rhys lattice relaxation model. The relaxation rates display a Gaussian distribution with the line-width of several milli-electron volts (meV) by taking into account the effect of lattice relaxation that arises from the electron-deformation potential acoustic phonon interaction. Through the numerical calculation, we find that the relaxation rate of the spin-conserving process is three orders of magnitude larger than that of the spin-flip process for the same phonon mode. Moreover, the SO phonon scattering is two orders of magnitude larger than the LO phonon scattering in the same process. We also discuss the temperature dependence of the relaxation rates, the opposite temperature dependence are obtained in the resonance energy regime. In addition, the dependencies of relaxation rates on the strength of Rashba-type spin–orbital coupling (SOC), the dielectric constant of different polar substrates and the distance between the graphene and substrate are also analyzed quantitatively for the SO phonon scattering. The obtained theoretical results related to the possible experimental phenomenon are also discussed. We hope that our results can stimulate and highlight related experimental work to identify these conclusions.
Figure 1. (a) The schematic diagram of the sandwich structure (air/graphene/substrate), d denotes the distance between the graphene and substrate, the magnetic field B exerts in the perpendicular direction. (b) The schematic diagram for the distribution of the Landau levels with Zeeman spin splitting and the optical phonon scattering from the LL n = 1 to n = 0+ with the spin-conserving and spin-flip processes, 2G is the opened gap of graphene, ↑ (↓) stands for the spin up (down) state.
in figure 1(a)). d denotes the distance of the graphene above the substrate, the air has the permittivity of free space ε0 . The existing surface optical phonon modes are described in detail in section 4. In the presence of a uniform static perpendicular magnetic field B the total Hamiltonian is given by H = He + Hz + Hph + Hep
(1)
with the first Hamiltonian He = H0 + Hso . H0 is the single free carrier Hamiltonian without the SOC [12]: H0 = vF (p + eA) · σ + τ Gσz ,
(2)
where the first term is the well-known Dirac Hamiltonian for graphene. The second term represents the opened band gap 2G which can be induced, e.g. by the substrate or a monolayer of intercalated structure [13, 14]. τ = ±1 denote the K and K’ valleys. We can restrict the scattering in a single valley in the absence of valley scattering, e.g. τ = +1. Hsoc is the term of SOC, which is the key element to spin-flip relaxation. Hz = 21 gμB B·s describes the Zeeman coupling of the electron † spin to the magnetic field. Hph = ¯ ωqλ aqλ aqλ is the qλ h phonon Hamiltonian with ωqλ standing for the phonon energy spectrum of branch λ and momentum q. Hep is the Hamiltonian of electron–phonon interaction. The single free carrier Hamiltonian in equation (2) can be solved analytically. In the frame of symmetric gauge for A = (B/2)(−y, x, 0) , the eigenstate and eigenenergy are given by [1, 12] T C m−1 φ (r)ei(m−1)θ χs , (3) |n, s(0) = ψn (r, θ )χs = n An m A iTn CBn φB (r)eimθ En = Kn (|n|¯hωB )2 + G2 , (4) with
2. Theoretical model
m−1 = CAn
The graphene monolayer we considered is sandwiched between the substrate and the air (see the schematic diagram
n!/2π lB2 (n + |m| − 1)!,
φA (r) = ξ |m|−1 e−ξ 2
2
/2
|m|−1
Ln
,
J. Phys.: Condens. Matter 26 (2014) 395302
and m CBn =
Z-W Wang et al
where Γ is the line-width that arises from the mixing of many mechanisms, e.g. the impurity, disorder, electron–phonon interaction. There were many researches for the line-width Γ in bulk and low-dimensional systems. In general, the line-width Γ is set to a fixed value to discuss the theoretical and experimental results quantitatively. However, in the present study, the delta function is replaced by the Fourier expansion and we consider the contribution of the electron-acoustic phonon interaction to the line-width due to the effect of lattice relaxation, which is convenient to investigate the temperature dependence of the line-width in the relaxation and transition processes. For the coupled electron lattice system, the LL state in graphene can be decomposed into the Born–Oppenheimer approximation as Φi (, Qiq ) = ψi (R, Qiq ) i (Qiq ), where R(r, θ ) is the electron position coordinate, Qiq models the normal coordinate to describe the lattice vibration. ψi and
i are the electron and lattice wave functions, respectively. In the electron wave function, Qiq is only a parameter. The lattice wave function i is a direct product of states of harmonic oscillators whose oscillating equilibrium origins are influenced by the electron state, especially in the presence of a strong magnetic field, which can be expressed as:
i (Qiq ) =
nq (Qiq ) =
nq (Qq + Δiq ) (9)
n!/2πlB2 (n + |m|)!,
φB (r) = ξ |m| e−ξ
2
/2
|m|
Ln .
√ Here Tn = 1 or (1/ 2) are for the cases of n = 0 or (n = 0); |m| n = n + (|m| + m)/2, (n = 0, ±1, ..., m = 0, ±1, √...). Ln is the √ Laguerre polynomial. The parameter ξ = r/√2lB with ¯ /eB denotes the magnetic length; h ¯ ωB = 2¯hvF / lB lB = h is the effective magnetic energy; vF = 106 m/s is the Fermi velocity near the Dirac point; χs is the spin state with s = ↑, ↓, ±1. Furthermore, Kn = 1 (−1) is for the electrons (holes) when n > 0 (n < 0), and Kn = 0 for n = 0± . The schematic diagram of the distributions for the zero and first excited LLs with spin states are shown in figure 1(b). In general, the SOC is composed of the Dresselhaus-type and Rashba-type D R Hsoc = Hsoc + Hsoc = ΔD τ σz νz + ΔR (τ σx νy − σy νx ),
(5)
where ΔD denotes the Dresselhaus coupling strength, which is very weak due to the symmetrical lattice structure of graphene and is disregarded in the present paper. ΔR is the Rashba coupling strength that we mainly considered, which can be modulated by an external factor, such as the electrical field or the substrate [15]. The SOC can be treated as a small perturbation to the energy of LLs in equation (4). However, the effect of the SOC is to weakly mix the eigenstates equation (3), e.g. the state |n, ↑(0) mixes with the other states |n , ↓(0) (n = n ). In the first-order perturbation, the SOC corrects the electron spin product states |n, ↑ = |n, ↑(0) and |n, ↑ = |n, ↑(0) to |n, ↑ = |n, ↑(0) + |n, ↓ = |n, ↓(0) +
n , ↓ |Hsoc |n, ↑(0) |n , ↓, En − En − 21 gμB B
(6a)
n , ↑ |Hsoc |n, ↓(0) |n , ↑, En − En − 21 gμB B
(6b)
q
q
ωq Qiq h ¯
LA where Δiq = ψi |HEPI |ψi /ωq2 describes the shift of the lattice normal oscillator originated before and after the electron transiLA denotes the electron-longitudinal tion. The Hamiltonian HEPI acoustic (LA) phonon interaction and can be written as [16]: Dq i 0 † LA HEPI = )eiq·r , (10) (aq + a−q 0 i ρω /¯ h q q
(0)
(0)
2π op | Φi | HEPI | Φf |2 δ(Ef i − h ¯ ωop ), h ¯ f
21
ωq = Hiq √ n q 2 nq ! π h ¯ ωq Q2iq × exp − , 2¯h
respectively. We illustrate the zero (n = 0+ ) and first excited (n = 1) LLs mixing with the adjacent LLs for the Rashba-type SOC in appendix A. According to the selection rule L|n|+1 → L|n| in graphene for the transitions between LLs [4], based on the Fermi golden rule, the relaxation rate assisted by the optical phonon scattering can be written as: W =
√
† ) is the creation (annihilation) operator for the where aq (a−q LA phonon with the wave vector q, D denotes the deformation potential constant, whose value is in 10–50 eV intervals [16]. ωq = υq is the frequency of the LA phonon and satisfies the linear dispersion relation and υ = 2 × 104 m s−1 is the acoustic velocity. We emphasize that different electron-acoustic phonon interactions correspond to different changes in the lattice induced by phonons. Dominant electron–phonon interaction for long wavelength acoustic mode is LA phonon through the deformation potential mechanism, which is caused by an area change in the unit cell and gives the main contributions to the effect of lattice relaxation and the line-width. Hiq is the Hermitian polynomial, nq is the LA phonon number. Considering the effect of lattice relaxation, equation (7) can be rewritten as [17]: 2π op W = | Φi | HEPI | Φf |2 δ(Ef i − h ¯ ωop ) (11) h ¯ f
(7)
where Φi (Φf ) is the initial (final) electron state of the LLs in graphene as described in equation (1). Ef i is the energy separation between the initial and final LLs with spin states, h ¯ ωop is the optical phonon energy with dispersionless relation. For usual cases, in order to evaluate the relaxation rate of equation (7), the delta function is replaced by a Lorentzian line-shape function: Γ /2 1 δ(Ef i − h (8) ¯ ωop ) → π (Ef i − h ¯ ωop )2 + (Γ /2)2
=
3
2π op | ψi | HEPI | ψf |2 ×, h ¯ f
J. Phys.: Condens. Matter 26 (2014) 395302
and =
Z-W Wang et al
3. Results and discussion
Pav | i (Qiq ) | f (Qf q ) |2 δ(Ef i − h ¯ ωop ),
We choose the relaxation process from the first excited Landau level (n = 1) to zero Landau level (n = 0+ ) as an example to study the LO and SO phonon resonance scattering with spinconserving process (|1, ↓ −→ |0+ , ↓) and spin-flip process (|1, ↑ −→ |0+ , ↓), respectively. In a numerical simulation process, we apply the effective single-mode of the LA phonon to calculate the line-width of the relaxation rates [19, 20]. Throughout the paper, we fix the band gap 2G=40 meV and the deformation potential constant D=30 eV for the electron-LA phonon interaction and assume that the Fermi level is below the band gap. The Fermi level can be controlled by the electrical gate or doping. In fact, the position of the Fermi level and the modulation of the band gap have an essential influence on the transitions between the LLs [1], which is not the subject of the present paper.
f nq ,niq
(12)
e−β(nq +1/2)¯hωq i
Pav = ∞
niq =0
e−β(nq +1/2)¯hωq i
(13)
.
Here is the overlap integral of lattice vibrational wave function and is derived in detail in [17], Eif = Ei − Ef is the energy difference of the whole electron-lattice system before and after the transition including the energy of lattice relaxation, i.e. Ei = Ei − 1/2 q ωq2 Δ2iq . The average over the initial phonon states (Pav ) and summation over the final phonon states have been explicitly shown in the equation. The parameters β = 1/KB T . In the Condon approximation, op op the electronic matrix Mf i = ψf | HEPI | ψi can be regarded as independent of the lattice coordinate. Following the standard lattice relaxation procedure to carry out the average (summation) over initial (final) phonon states (the details of the procedure are given in [17]), we get
op | Mf i |2 +∞ dμeF (μ,Eif ) (14) W = h ¯ −∞ where ¯ ωop ) + F (μ, Eif ) = −iμ(Ef i − h
ωq q
2¯h
Δ2f iq
3.1. The longitudinal optical (LO) phonon scattering with spin-conserving and spin-flip process
Optical phonon in graphene is represented by the relative displacement of two sublattice carbon atoms in a unit cell. Phonon modes couple two neighboring carbon atoms through bond stretching and bending, so that the electron–phonon interaction becomes an off-diagonal matrix in the pseudospin space, which was originally derived in [21] and [22]. The Hamiltonian of the electron-optical phonon interaction can be written as:
(15)
β¯hωq × coth (cos μ¯hωq − 1) + i sin μ¯hωq . 2 The Δf iq = Δf q − Δiq depends on the LL states and strength of electron-acoustic phonon interaction. For multimode phonons, the integral of equation (14) can be obtained analytically. In such a case, the strongcoupling approximation is a popular and widely accepted approximation, i.e. by expanding the function F (μ, Eif ) to the second-order of μ. By using the steepest decent method [18, 19], equation (14) becomes 21 op | Mf i |2 2π F (16) W = h ¯ ST (¯hωq )2
op
HEPI
where the matrix N is N(q) =
F = exp −
¯ ωop − S¯hωq )2 (Ef i − h
2ST (¯hωq )2T ωq S¯hωq = Δ2f iq h ¯ ωq , 2¯ h q
and ST (¯hωq )2T
=
ωq q
2¯h
Δ2f iq (¯hωq )2
(17)
,
β¯hωq coth 2
(18) .
h ¯ † N(q)(aq + a−q )eiq·r , ρω0
0 NAB eiφq
NBA e−iφq 0
(20)
,
with NAB = −1 or i and NBA = 1 or i for longitudinal optical (LO) or transverse optical (TO) phonon, respectively, and φq = tan−1 (qy /qx ) is the azithmuthal angle of the wave vector q , ξ = dlnγ0 /dlnb ∼ 2 is a dimensionless parameter that denotes the change of the nearest-neighbor tight-binding matrix element with respect to the bond length b = 0.136 × 10−9 m, the parameter γ = 0.62×10−9 eV· m, ρ = 7.5×10−7 kg· m−2 is the mass density, ω0 is the optical phonon frequency for graphene. We only consider the electron-LO phonon interaction as an example in our analysis and h ¯ ωLO =196 meV is the LO phonon energy. All of the parameters mentioned above are from [21]. According to equations (16) and (20), we obtain the relaxation rate of the LO phonon resonance scattering 21 2 | MLO 2π f i |SC WSC = (21) h ¯ ST (¯hωq )2T (Ef i − h ¯ ωLO − S¯hωq )2 × exp − 2ST (¯hωq )2T
T
where
ξγ = 2 b q
(19)
The form factor F is a Gaussian distribution with a line-width 2ST (¯hωq )2T . Considering the temperature effect, the relaxation rate W should be multiplied by the factor Nop +1 for the optical phonon emission process. Nop = 1/(exp(¯hωop /KB T ) − 1) is the Bose distribution of optical phonon. 4
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for both the spin-conserving and spin-flip process. With increasing temperature, more LA phonons are thermally excited resulting in the increase of line-width. The detailed magnetic field and temperature dependence of the line-width were discussed in [20]. In general, the reciprocal of the relaxation rate (1/W) represents the relaxation time. From the comparison between the processes in figures 1(a) and (b), it can be deduced that the relaxation time of the spin-conserving process is three orders of magnitude shorter than that of the spin-flip process. This means that the spin-flip process needs more time than the spin-conserving process, which is consistent with the theoretical prediction as we expected and needs to be proved in further experiments. In the present model, the relaxation time of the spin-conserving process is in the hundreds of nanoseconds scale, which is longer than the processes of electron-hole recombination and generation via inter-and intra-valley optical phonon scattering in the absence of magnetic field [23–27]. Obviously, the energy levels were quantized into the LLs resulting in the inhibition of the optical phonon scattering. But the timescale is very close to the slow relaxation process mediated by electron-electron Auger scattering under strong magnetic field [28], thus to distinguish between them is very difficult. However, the Auger scattering sensitively depends on the carrier density but not for the optical phonon scattering. The Auger and LO phonon scattering belong to the non-radiation processes and were considered as the main obstacle for the fabrication of tunable far-infrared and terahertz laser based on inter-Landau-level emission [29]. Recently, Wendler and co-worker [30] presented a theoretical approach to study the optical phonon-induced carrier relaxation between the LLs in intrinsic graphene. They found that the optical phonon resonance scattering is enhanced at a specific field and the obtained relaxation time can access to the picosecond scale. When the energy of the optical phonon mismatches the LLs spacing, the two-phonon LO+LA relaxation process has also been proposed [31]. The study of these non-radiation processes has aroused enormous interest and is crucial for many potential applications in graphene. As for the LO phonon resonance scattering with spin-flip process shown in figure 1(b), the very long relaxation time is obtained in the microsecond scale. More important is that the relaxation time can be controlled by the modulation of the strength of the Rashba SOC as shown in figure 5. In the past years, the spin-flip relaxation assisted by the acoustic phonon scattering based on the SOC mixed mechanism in the graphene quantum dot and nanoribbon has been studied widely [12, 32–34]. But the investigations about the spin-flip via optical phonon resonance scattering between LLs in graphene are very few until now and need to be identified by the corresponding experimental measurement. The very interesting characteristic in figure 2 is that the opposite temperature dependences of the relaxation processes are presented in the resonance energy regime. From equations (16) and (21), we conclude that the factors NLO + 1, 1/ST (¯hωq )2T and F determine the temperature dependence of the relaxation rate. The contribution of the factor NLO + 1 is very weak due to the large LO phonon energy compared with the temperature range we considered. The latter two factors play a dominant role, which reduces and enhances the relaxation rate
Figure 2. The LO phonon relaxation rates with spin-conserving (a) and spin-flip (b) as a function of the energy separation between the LL n = 1 and n = 0+ at different fixed temperatures for the Rashba SOC constant ΔR = 0.05 meV.
for the spin-conserving process and 21 2 | A |2 | MLO 2π f i |SF WSF = h ¯ ST (¯hωq )2T (Ef i − h ¯ ωLO − S¯hωq )2 × exp − 2ST (¯hωq )2T
(22)
for the spin-flip process with A=
iΔR . E1 − E2 − 21 gμB B
Here, the parameter A denotes the magnitude of the admixture between different LLs due to the Rashba-type SOC. The 2 detailed expressions for the matrix elements | MLO f i |SC and LO 2 | Mf i |SF are given in appendix B. Figure 2 shows the relations of the LO phonon resonance relaxation rate (W) with the energy separation between the LL n = 1 and n = 0+ at different fixed temperature. Figures 2(a) and (b) show the spin-conserving and spin-flip process, respectively. It can be seen that the relaxation rates follow a Gaussian distribution with the small line-width of several meV due to the effect of LA phonon lattice relaxation 5
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respectively with increasing temperature. At the peak position regime, that is the condition of Ef i − h ¯ ωLO − S¯hωq ≈ 0 sat-
Table 1. Surface optical phonon modes for different polar substrates. Parameters have been taken from [36] and [37].
isfied, only the factor 1/ST (¯hωq )2T predominates the process. It can be clearly seen that the relaxation rate is inverse to the temperature, which means the effect of lattice relaxation of the LA phonon delays the emission of LO phonon with increasing temperature. Except for this peak position, the form factor F predominates the process, which results in the relaxation rate increasing with temperature. So the effect of lattice relaxation assists the emission of LO phonon in this regime. These two types of temperature dependence have been observed in experiments for many low-dimensional systems and further tests are needed to confirm the results for graphene. In fact, the temperature dependence of the surface optical phonon scattering in the next section is the same as that of the LO phonon scattering. The influences of the strength of Rashba-type SOC on the relaxation processes are discussed in detailed in figure 5.
Quantity(units)
SiC
SiO2
Al2 O3
HfO2
κ0 (ε0 ) κ∞ (ε0 ) h ¯ ωSO,1 (meV) h ¯ ωSO,2 (meV)
9.7 6.5 167 116
3.9 2.5 146 60
12.53 3.2 94 55
22.0 5.03 53 19
3.2. The surface optical (SO) phonon scattering with spin-conserving and spin-flip process
For the structure that the graphene monolayer is sandwiched between, the substrate and the air as plotted in figure 1(a), there exists the surface optical phonon modes, which correspond to a certain bulk transverse optical phonon mode of the substrate [35–37]. These modes interact with the electrons in graphene and have a potential influence on the basic properties of graphene [38–42]. The Hamiltonian of the electron-SO phonon interaction is given by SO HEPI =
e2 η¯hωSO,ν 2ε0
21 −qz e † iq·r (23) + a +)e (a √ q,ν −q,ν q q
with η=
(κ0 − κ∞ ) , (κ∞ + 1)(κ0 + 1)
where ωSO,ν is the frequency of SO phonon, ν = 1, 2 denotes the two branch, ε0 is the permittivity of vacuum and κ∞ (κ0 ) is the high (low) frequency dielectric constant. The parameter η is a combination of the known dielectric constants of the substrates and denotes the strength of the polarizability of the dielectric interface. Similar to the LO phonon resonance scattering, the relaxation rate of SO phonon resonance scattering is then given by 21 | MSO,ν |2SC 2π fi WSC = (24) h ¯ ST (¯hωq )2T (Ef i − h ¯ ωSO,ν − S¯hωq )2 × exp − 2ST (¯hωq )2T for the spin-conserving process and 21 | A |2 | MSO,ν |2SF 2π fi WSF = (25) h ¯ ST (¯hωq )2T (Ef i − h ¯ ωSO,ν − S¯hωq )2 × exp − 2ST (¯hωq )2T
Figure 3. The relaxation rates of the SO phonon resonance scattering with spin-conserving as a function of the energy separation between the LL n = 1 and n = 0+ for different polar substrates at fixed temperature T = 80 K, Rashba SOC constant ΔR = 0.05 meV and the distance d = 0.3 nm.
for the spin-flip process. The parameter A is same as the LO phonon scattering and the detailed expressions for the scattering matrix element | MSO,ν |2SC and | MSO,ν |2SF are fi fi given in appendix B. The adopted parameters in calculations are listed in table 1. Figure 3 shows the relaxation rates of the SO phonon resonance scattering with spin-conserving process as a function of the energy separation between the LL n = 1 and n = 0+ for different polar substrates. One can clearly see that the relaxation rate of SO phonon resonance scattering is two orders of magnitude larger than that of the LO phonon plotted 6
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in figure 2(a). The SO phonon scattering is enhanced: which can be attributed to the strong interaction between the electrons in graphene and the SO phonons of the polar substrates. In the past years, the influences of the SO phonon scattering on the low- and high-field carrier mobility, magneto-optical conductivity, hot electron relaxation and heat dissipation, plasmon emission and absorption and so on, were studied extensively [7, 38–42]. The LO phonon resonance scattering between the LLs in graphene has been proved in earlier experiments [1, 3, 8]. However, the SO phonon resonance scattering was mostly studied as a theoretical aspect and the related experimental identifications have been hard to make, until now, which may be because of its enhanced scattering. Equations (23) and (24) show that the rates of the SO phonon scattering are mainly determined by the energy (or frequency) of the phonon h ¯ ωSO and the dielectric constant η for the fixed temperature and distance parameter d. From figure 3, we can see that the relaxation rates of the high energy phonon branch are larger than that of the low branch in the same polar substrate. Through the comparisons between the high (or low) energy phonon branches in the different polar substrates, we notice that the relaxation rates increase from the SiC to Al2 O3 polar substrate due to the increase of their corresponding dielectric constant η as listed in table I. Although the dielectric constant of the polar substrate HfO2 is very high, the relaxation rates are smaller than other polar substrates, which arises from the energy of the phonon branch in HfO2 being very low. Therefore, the interplays between the energy of the phonon and the dielectric constant determine the relaxation rates of the SO phonon resonance scattering in the spin-conserving process and the former predominates the processes. In addition, the line-width of the high energy phonon is broader than that of the low energy one, which attributes to the high energy phonon resonance matching the large energy separation corresponding to the strong magnetic field, and thus results in the effect of lattice relaxation for LA phonon enhanced [20]. The relaxation rates of the SO phonon resonance scattering with spin-flip process as a function of the energy separation for different polar substrates are plotted in figure 4. The discrepancies are very obvious by comparing this process with the spin-conserving process as shown in figure 3. Firstly, the relaxation rate of the spin-flip is three orders of magnitude smaller than that of the spin-conserving process. Secondly, the relaxation rate of the low energy phonon branch is larger than that of the high energy one in the same polar substrates. The reason is that the relaxation rate is determined not only by the energy of phonon branch and the dielectric constant but also the parameter A. The parameter represents the magnitude of the admixture between different LLs due to the Rashba-type SOC and is in inverse proportion to the energy separation in its expression. So the large energy separation, that is the high energy phonon resonance, corresponds to the small value of A, which leads to the results that the relaxation rates decrease with increasing the phonon energy. The parameter A plays a dominate role in determining the relaxation rates in the spinflip processes. The relaxation rates increase with increasing the dielectric constants for different substrates, which is similar to the spin-conserving processes.
Figure 4. The relaxation rates of the SO phonon resonance scattering with spin-flip as a function of the energy separation between the LL n = 1 and n = 0+ for different polar substrates at fixed temperature T = 80 K, Rashba SOC constant ΔR = 0.05 meV and the distance d = 0.3 nm.
In figure 5, we choose the SiC polar substrate as an example to discuss quantitatively the influence of the Rashba SOC constant ΔR and the distance d on the relaxation rates for the fixed energy separation and temperature. Through the numerical calculation, we find that the influence of the Rashba SOC on the spin-conserving process can be neglected; we only illustrate its influence on the spin-flip process in figure 5. We can see that the relaxation rate increases quadratically with the Rashba SOC constant ΔR , which can be derived from the expression of the parameter A and equation (25). The conclusion is also fit for the LO phonon resonance scattering with spin-flip process. In fact, the modulation of the polar substrates on the Rashba-type SOC and the spin relaxation processes have been studied both theoretically and experimentally and have many potential applications on spintronics [15, 43, 44]. This topic is a hot research point up until now, but not the object of the present paper. In figure 5, the relaxation rates for spin-flip processes decreases with increasing the distance d, which arises from the interaction between the electrons in the graphene and the SO phonon in the polar substrate becoming weak due to the 7
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Acknowledgment
The authors are grateful to P-J Lin for the language support and helpful discussion. This work was supported by the National Natural Science Foundation of China (No.11304355). Appendix A. The mixed states for several lowest LLs due to the Rashba-type SOC
According to the selection rules of the Rashba-type SOC, the mixed states can be written R 1, ↓ |Hsoc |0+ , ↑(0) |1, ↓(0) , E0+ − E1 − 21 gμB B (A.1a) (0) (A.1b) |0+ , ↓ = |0+ , ↓ ,
|0+ , ↑ = |0+ , ↑(0) +
R 2, ↓ |Hsoc |1, ↑(0) |2, ↓(0) , 1 E1 − E2 − 2 gμB B
(0)
|1, ↑ = |1, ↑(0) + Figure 5. The SO phonon relaxation rate with the spin-flip process as functions of the Rashba SOC constant ΔR and the distance d at fixed energy separation E10+ = 167 meV and temperature T = 80 K for the SiC polar substrate.
|1, ↓ = |1, ↓(0) +
(0)
(A.1c)
R 0+ , ↑ |Hsoc |1, ↓(0) |0+ , ↑(0) , (A.1d) 1 E1 − E0+ − 2 gμB B
(0)
After the arithmetical operation for the matrix elements of R the Hsoc , the equations simplify to
increase of the distance between the graphene and polar substrate. The influence of the distance on the relaxation rate of the spin-conserving process is very similar to that of the spinflip process. These interesting properties need to be confirmed by related experiments in the future.
|0+ , ↑ = |0+ , ↑
(0)
√ i 2ΔR + |1, ↓, E0+ − E1 − 21 gμB B
|0+ , ↓ = |0+ , ↓(0) ,
(A.2a) (A.2b)
iΔR |2, ↓(0) , (A.2c) E1 − E2 − 21 gμB B √ i 2ΔR (0) |1, ↓ = |1, ↓ − |0+ , ↑(0) . (A.2d) E1 − E0+ − 21 gμB B
4. Conclusion
|1, ↑ = |1, ↑(0) +
In conclusion, we theoretically investigate the LO and SO phonon resonance scattering with spin-conserving and spinflip processes between LLs in graphene in the Huang–Rhys lattice relaxation model. We find that (i) the relaxation rate follows a Gaussian form with the line-width in several meV due to the effect of the lattice relaxation arising from the electrondeformation potential acoustic phonon interaction; (ii) the relaxation rates of the spin-conserving process are three orders of magnitude larger than the spin-flip process. Moreover, the SO phonon scattering is two orders of magnitude larger than the LO phonon scattering in the same process; (iii) the opposite temperature dependence of the relaxation rates are obtained in the resonance energy regime in the present model; (vi) the relaxation rate of the spin-conserving process for the SO phonon scattering increases with increasing the phonon energy, which is contrary to that of the spin-flip process; (v) the relaxation rates increase with increasing the dielectric constant for the different polar substrates in all processes; (vi) the influence of the strength of Rashba SOC on the relaxation rate follows a quadratical increasing trend in the spin-flip process and can be neglected in the spin-conserving process; (vii) the relaxation rates decrease dramatically with increasing the distance between the graphene and substrate, which arises from the interaction between the electrons in the graphene and the SO phonon in the polar substrate becoming weak.
Appendix B. The LO and SO phonon scattering matrix elements
In the Condon approximation, the LO and SO phonon scattering matrix elements for the spin-flip and spin-conserving process are given by
|2SF = | MSO,ν fi
8
2 | MLO f i |SF =
ξ 2γ 2 , 16π b4 ρωLO lB2
(B.1)
2 | MLO f i |SC =
ξ 2γ 2 , 32π b4 ρωLO lB2
(B.2)
e2 ηωSO,ν 256π ε0 2 √ 3 12d 2 8d 4 2d2 × 2π + 3 + 5 e lB 2lB lB lB √ 2d 10d 8d 3 × 1 − Erf − 2 − 4 , lB lB lB
(B.3)
J. Phys.: Condens. Matter 26 (2014) 395302
| MSO,ν |2SC = fi
2 e2 ηωSO,ν √ 1 2d 2 2d2 2π + 3 e lB 32πε0 2lB lB √2d 2d − 2 . × 1 − Erf lB lB
Z-W Wang et al
[19] [20] [21] [22] [23]
(B.4)
Here the subscript ν = 1, 2 denotes the different branch of the SO phonon modes.
[24] [25] [26]
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