Boltzmann approach to the triplon SSE

in the CuGeO₃ sample. Therefore, even in the nominally pure sample, the effect of triplon scattering by impurities cannot be ignored. Considering the Cu-Cu inter-atomic distance is 0.2926 nm [96], the average distance between two free spins in the nominally pure sample is about 1.5 µm.

As shown in Figs. 3.18(a)-(c), significant suppression ofV˜_SSEis recognized in the 1% and 3 % Zn-doped CuGeO3. The linear fitted values ofV˜_SSEin the range of−1 T< µ₀H <1T are shown in Fig. 3.18(d). In all temperature range, the V˜_SSE/µ₀H of Cu0.99Zn0.01GeO3/Pt is lower than the CuGeO₃/Pt sample. TheV˜_SSEof the Cu_0.97Zn_0.03GeO₃/Pt sample, on the other hand, is always in the same order as the noise level. In these doped samples, triplon spin-excitations are still present in the SP phase, but their transport may be blocked by impurities. Furthermore, the similarT andHdependence ofV˜_SSEin pure and Zn-1% doped sample indicates the same origin ofV˜_SSE in both samples: the triplon spin current.

We did not consider the effect of the thermal conductivity across different samples in Fig. 3.18(d). Generally, doping of impurities reduces the κ of CuGeO3. Theκ of a 1.6 % Mg-doped sample is only ∼ 1/3 of the undoped sample [112]. According to discussion in Section3.3, we should considerV˜_SSE·κas the intrinsic value of the spin current. Sinceκ_Pure is larger than κ_1%Zn, the magnitude ofJ_s is still larger in the pure sample even if we take κ into consideration.

3.5 Boltzmann approach to the triplon SSE

In this section, we discuss the Boltzmann approach to the triplon SSE in CuGeO3. From the experimental results, we know the SSE signal is critically suppressed by impurity doping.

This implies the scatterings from non-magnetic impurities and free spins are likely playing a crucial role in the triplon transport. We also notice that the non-monotonicH-dependence of V_SSE may come from the competition of two effects. The Zeeman effect tends to enlarge

|V_SSE|because the unbalance ofJ_S^z₌₊₁andJ_S^z₌₋₁is monotonical increases withH. On the other hand, the scattering from free Cu²⁺ spins are more likely to occur at high H (where the free spins are fully polarized) and causes a decrease of pure spin current at high H.

Therefore we choose to formalize theH-dependence of V_SSE with the Boltzmann approach, which can naturally include the scattering effect as phenomenological relaxation term for triplons. The calculation was done in collaboration with associate professor Dr. Masahiro Sato.

The model Hamiltonian for the CuGeO3 spin chain is H_SP=^!

j

J₁(1 +δ(−1)^j)Sj·S_j+1+J₂S_j·S_j+2+B^!

j

S_j^z. (3.4)

3.5 Boltzmann approach to the triplon SSE 60

Here, B ≡gµ_Bµ₀H,g = 2.1 for CuGeO₃[71], andµ_B is the Bohr magneton. For CuGeO₃, the exchange constants are estimated to be J₁ ∼ 120.6 K[67], J₂ ∼ 0.36J1[68] and δ ∼ 0.0022[68]. In the case ofα ≡J₁/J₂ <0.241≡α^critical and δ = 0, the spin chain is in the same universality class as the Tomonaga-Luttinger liquid, the ground state is non-magnetic spin liquid and the excitation are gapless spinons. For a system with α >α_c, the ground state spontaneously dimerizes even atδ= 0and an excitation gap opens even aboveT_SP[59, 60,61]. In the general case ofαandδ, the exact solution of the ground state and excitations are unknown. However, the system can still be mapped to a sine-Gordon model with an additional cosine 4φterm with a coefficient g₂ ∝(α−α^critical)as[47,53]

H_eff =ˆ dx v

2π

$1

K(∂xφ)²+K(∂xθ)^2%+ 2g1

(2πa)² ˆ

dxsin(2φ) + 2g2

(2πa)² ˆ

dxcos(4φ). (3.5) Where φ and θ are dual boson fields and v and K are, respectively, the spinon velocity and the Luttinger parameter. The g₁ ∝ δ term comes from the bond alternation. For α = α^critical, the additional 4φ term disappears and the Hamiltonian becomes an exactly solved model[47]. In the case of CuGeO3, α is close to α^critical, so we use the parameters derived from the exact solution of the model in the case of α^critical.

The spin excitation of theJ₁-J₂^critical spin chain at zero magnetic fields is known to be the three-ford degenerated triplons. The dispersion relation aroundk∼0for three branches of triplon, anti-soliton (S^z =−1), breather (S^z = 0), and soliton (S^z = 1) are respectively (see Eq. 1.80)

,_AS=⁸∆²+v²k²+B, forS^z =−1; (3.6) ,_B=⁸∆²+v²k², forS^z = 0; (3.7) ,_S=⁸∆²+v²k²−B, forS^z = 1. (3.8) Here,Bis defined asB ≡gµ_Bµ₀H with thegfactor ofg= 2.1[71],µ_Bis the Bohr magneton and µ₀H is the external magnetic flux density. The exact solution[59] gives v = 1.174J1a, and the spin gap ∆ ∼2.36 meV (∼ 27.4 K) is estimated by a ESR experiment[71]. If we consider the triplon as a wave packet located atrand have a wave numberk, the distribution functionf(k,r, t) of it follows the Boltzmann equation as

∂

∂tf+v_k·∇rf+dk

dt ·∇kf = ∂

∂tf|scattering. (3.9)

The group velocity of triplon is v_k= ¹_!^∂+(k)_∂k .

In the SSE measurement, a constant temperature gradient is applied along the x-direction (spin chain) to the sample. For a system of length , the temperature gradient

3.5 Boltzmann approach to the triplon SSE 61

will be ∂T /∂x = −^Thigh_L⁻_x^Tlow ≡ ∆Tx. For a small ∆Tx, the system is close to equilibrium and the distribution function is approximated as

f =f₀(k) +g(k,r, t). (3.10) Here, f₀(k) is the equilibrium distribution function, and g(k,r, t) is the deviation from equilibrium. We take

f₀(k) = 1

e^β(+(k))±1. (3.11)

Here we consider both fermionic (+) and bosonic (−) distributions, as the excitations may follow anyonic statistics in a pseudo-2D system. In three-dimensional dimerized systems, triplon obeys bosonic statistics and in such systems, the Bose-Einstein condensation is observed[113]. Our numerical results, however, do not show a distinguishable difference between fermionic and bosonic distribution. This is because the large energy gap makes the Boltzmann factor exp(β,) much larger than 1 and the distribution function is reduced to Boltzmann distribution.

During the measurement, the system is in a nonequilibrium steady state (∂f /∂t = dk/dt= 0). For the scattering term, we adapt relaxation time (τk) approximation:

∂

∂tf|scattering=−f(k,r, t)−f₀(k)

τ_k =−g(k,r, t)

τ_k . (3.12)

Since we assume ∂T /∂x and g are small, we omit the∂g/∂T term and Eq. 3.9becomes g(k) =τ_kv^x_kk_B∆xT ∂

∂(k_BT)f₀(k) (3.13)

We consider three different distribution functions for three triplon branches and the total spin current along the x-direction is

J_s= ^!

S^z=1,0,−1

!

k

!S^zv^(x)_Sz,kf_S^z(k)

=^!

S^z

!

k

!S^zv^(x)_Sz,kg_S^z(k)

=^!

S^z

!

k

!S^z(v^(x)_Sz,k)²τ_kk_B∆xT ,_S^z(k) (k_BT)²

e^β+Sz(k) (e^β+Sz(k)±1)²

=^!

S^z

ˆ

B.Z.

dk

2π!S^z(v^(x)_Sz,k)²τ_kk_B∆T ,_S^z(k) (kBT)²

e^β+Sz(k)

(e^β+Sz(k)±1)². (3.14) To numerically calculate theH-dependence of Js, it is necessary to evaluate the relax-ation timeτ_k based on appropriate impurity scattering mechanisms. With impurity density n_imp,τ_k is given by[114]

τ_k(,)⁻¹ ∼n_impD(,)V². (3.15)

3.5 Boltzmann approach to the triplon SSE 62

Here, D(,) and V are the density of states of triplon and impurity potential, respectively.

In CuGeO3, the two-dimensionality is strong because of anisotropic exchange constants J_c ∼120K,J_b ∼0.1Jc andJ_a∼ −0.01Jc[67]. Therefore,D(,) at the bottom of the band is assumed to be a constant. Consider the scattering from both nonmagnetic impurities and unpaired Cu spins, the total relaxation time is

τ_k,total⁻¹ =τ_k,non−mag⁻¹ +τ_k,mag⁻¹ . (3.16) Where,

τ_k,non−mag⁻¹ =n_impC₀V₀²; (3.17)

τ_k,mag⁻¹ =^"n_impB_S(T, H)^#·C_mag·^"V_magB_S(T, H)^#2. (3.18) C₀ and C_mag are constants, and in the case of CuGeO3,B_S(T, H) is the Brillouin function with S= 1/2. The numerical results ofJ_s(H) atT = 2.4 K are shown in the Fig. 3.19for the case of C_magV_mag² - C₀V₀². The result dose not show a significant difference between the fermionic and bosonic distributions.

| JS | (a.u.)

μ₀H (T)

0 1 2 3 4

μ₀H (T)

0 1 2 3 4 5

0

| J

_S^z₌₊₁

|

| J

_S^z_{= 1-}

|

| J

_s

| ^{| J}

^Sz=+1

^|

| J

_S^z_{= 1-}

|

| J

_s

|

Fermi distribution Bose distribution

Figure 3.19: Calculation results of the H dependence of the triplon spin current atT = 2.4 K. |J_S^z_=±1|is the spin current carried by soliton/antisoliton. |J_S^z₌₊₁|−|J_S^z₌₋₁|gives the total spin current |J_s|.

In the low H regime, where the spin of magnetic impurities is not aligned and the scattering effect is weak, the Zeeman effect dominates the generation and transport of the spin current. With the increase ofH, the triplon mode ofS^z =−1(S^z= 1) shifts downward (upward) due to the Zeeman effect. The imbalance betweenS^z =±1triplon becomes larger,

3.5 Boltzmann approach to the triplon SSE 63

impurities are magnetized and magnetic scattering dominates the spin current transport, causing suppression of V_SSE. The competition between the Zeeman effect and scattering results in the appearance of a J_s peak at ∼ 2 T, which is very close to the experimental results.

T (K)

4 9 10 16

Γ (meV)

0.003 1

0.01 0.03 0.1 0.3

Γ

∝

T² Γ

∝

^T3

Γ

∝

^T4

Regnault 1996

Δ (meV)

0 2.5

0.5 1 1.5 2

T (K)

0 4 8 12 16

T_SP T_SP

(a) (b)

Lussier 1996 Δ =Δ(0)(1-T/T_SP)^0.12

... ...

5 6 7 8

Γ

∝

^T5

Γ

∝

^T6

Kikuchi 1994

Figure 3.20: (a) T-dependence of the SP gap ∆ at k = (0,1,1/2)[115]. Dashed-dotted line is the fitted result of ∆(T) up to T_SP. The fitting function is described in the main text. (b) T-dependence of the triplon line width of the neutron scattering peakΓ[64,116].

Dashed-dotted lines are the fitted results of several power laws.

Finally, to evaluate theT-dependence ofJ_s, we need information on theT-dependence of the triplon gap ∆(T) and the triplon lifetime 1/τk(T). By fitting neutron scattering results of spin-gap ∆(T) [115] with∆(T) =∆(0)⁽1−T /T_SP^)α, we obtain∆(0)∼2.05 meV and α)0.12, as shown in Fig. 3.20(a). In the numerical calculations, ∆in Eq. 3.6-3.8 is replaced by the fitted function of ∆(T).

On the other hand, theT-dependence of the lifetime of triplon is obtained as the peak width Γ(T) of triplon excitation from inelastic neutron scattering experiments[64, 115].

Here, the explicitT-dependence of1/τkis set as a new term1/τk(T), and the total relaxation time is

τ_k,total⁻¹ =τ_k,non⁻¹ _−mag+τ_k,mag⁻¹ +τ_k(T)⁻¹. (3.19) 1/τk,non−mag is independent of T and 1/τk,mag only weakly depends on T via the Brillouin function.

The experimental results of triplon peak width Γ(T) obtained from Ref. [64, 116] are shown in Fig. 3.20(b). We fitted the Γ(T) by seveal power law Γ = Γ(0)T^α, where Γ(0) are the fitting parameter and α is set to be 2,3,4,5, and 6. As shown in Fig. 3.20(b),

3.5 Boltzmann approach to the triplon SSE 64

T (K)

2 3 4 5 6 7 ... ... 16 (c)

V SSE(a.u.) 10¹

Γ

∝

^T5

10⁰ 10^-1 10^-2 10^-3 (b)

V SSE(a.u.) 10¹

Γ

∝

^T4

10⁰ 10^-1 10^-2 (a)

V SSE(a.u.) 10¹

Γ

∝

^T3

10⁰ 10^-1 10^-2

V

∝

^T2.7

V

∝

^T-1.9

V

∝

^{T1.7 V}

∝

^T-2.8

V

∝

^T0.9 _V

∝

^T-3.7

T_max

T_max

T_max

Figure 3.21: Calculation results of theT dependence of the triplon spin current atµ₀H = 2.4 T. τ_k(T)⁻¹ ∝T^3,4,5 are assumed for (a), (b), and (c).

the best power fit is Γ ∝ T⁴. The microscopic mechanism of this T-dependence is not clear. The scattering with thermally excited phonons and other triplons maybe two origins of the T-dependence of Γ. In this work, we treat these T-dependent scattering processes phenomenologically. Specifically, as

τ_k(T)⁻¹ =C_thV_th²Γ(0)^" T T_SP

#α

. (3.20)

Where C_th and V_th areT-independent constants.

As mentioned earlier in section 3.4, the T-dependence of thermal conductivity of the sample must also be considered: we should divide the calculated spin current byκto obtain V_SSE ∝ Js/κ. The numerical result of V_SST(T) with different α values at µ₀H = 3 T is

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