Spintronics meets quantum spin liquids: a novel spectral …...2020/11/17 · Spintronics meets quantum spin liquids: a novel spectral probe of quantum magnets based on spin Hall

Spintronics meets quantum spin liquids: a novel spectral probe of quantum magnets based on

spin Hall phenomena

So Takei

SPICE Workshop, November 18, 2020

Queens College and The Graduate Center of the City University of New York

D. Joshi, A. P. Schnyder and S. Takei, Phys. Rev. B 98, 064401 (2018) J. Aftergood and S. Takei, Phys. Rev. Research 2, 033439 (2020)

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Joshua Aftergood (CUNY)

• collaborators:

• funding:

Andreas Schnyder (MPI Stuttgart)

Darshan Joshi (MPI Stuttgart Harvard)→

acknowledgments

2

• introduction to spin Hall noise spectroscopy

• summary & outlook

contents

• application to a quantum spin ladder: detection of topological phase transitions

• application to quantum spin liquids: detection of spin density of states- S=1/2 kagomé Heisenberg antiferromagnet, e.g., herbertsmithite ZnCu3(OH)6Cl2. - the antiferromagnetic Kitaev honeycomb model, e.g., α-RuCl3. - spinon Fermi surface coupled to gapless U(1) gauge fluctuations, e.g., organic salt com-

pounds, YbMgGaO4.

- dimerized quantum antiferromagnet + Dzyaloshinskii-Moriya interaction + external magnetic field, e.g., BiCu2PO6.

3

spin Hall magnetoresistance• spin Hall magnetoresistance (SMR): corrections to longitudinal and Hall resistivities of a

strongly spin-orbit coupled metal in contact with a magnetic material.

H. Nakayama et al., Phys. Rev. Lett. 110, 206601 (2013) Y.-T. Chen et al., Phys. Rev. B 87, 144411 (2013) 4



H. Nakayama et al., Phys. Rev. Lett. 110, 206601 (2013) Y.-T. Chen et al., Phys. Rev. B 87, 144411 (2013)

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• longitudinal resistance for in-plane magnetization: R = R0 + ΔR0 + R1 cos2 α

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4

spin Hall noise• Fluctuation-dissipation theorem: SMR gives additional contribution to the thermal voltage

noise across the metal.

A. Kamra et al., Phys. Rev. B 90, 214419 (2014)

V (t)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

SPIN HALL NOISE PHYSICAL REVIEW B 90, 214419 (2014)

FI

N

FIG. 3. (Color online) Schematic of the normal metal (N) andmagnetic insulator (FI) bilayers analyzed in the text. The bluedashed arrow indicates the equilibrium magnetization direction. Thecoordinate system is depicted in red. The black arrows define our signconvention for spin currents across the interface.

where mmm ≡ mmm(rrr,t) is the unit vector along the magnetizationdirection at position rrr , γ (>0) denotes the gyromagnetic ratio,andhhh0 and α0 are the internal Langevin stochastic field [26] andGilbert damping constant, respectively. The effective magneticfield, written in terms of the magnetic free-energy density Fm,

µ0HHH eff = −∂Fm

∂MMM= µ0HHH 0 +

2AMs

∇2mmm, (2)

includes Zeeman and anisotropy contributions in HHH 0, whilethe second term represents the exchange field in terms of theexchange constant A [27] and the saturation magnetizationMs . The N layer is incorporated by imposing continuityof spin-current density across the FI|N interface [28]. Onthe FI side, the spin current density, carried by collectivemagnetization dynamics, is given by −A(mmm × ∂ymmm). On theN side, the spin-current density consists of (i) spin pumping(JJJ sp) by the thermal fluctuations of the magnetization in theferromagnet [29], and (ii) spin-transfer torque (STT) (JJJ stt)generated by absorption of the thermal electronic spin-currentincident on the FI. The conserved net spin-current densityJJJ s(=JJJ sp − JJJ stt) from the FI to the N is then given by

JJJ s($$$,t) = −A(mmm × ∂ymmm), (3)

= !g̃r4π

(mmm × ṁmm) − Ms(mmm ×µ0hhh′), (4)

where $$$ ≡ (x,z) is the in-plane position vector, g̃r is the realpart of the spin-mixing conductance per unit area corrected forthe finite thickness and/or spin relaxation length in N leading toa backflow spin current into FI [30]. We disregard the typicallysmall [31] imaginary part of the mixing conductance forsimplicity. hhh′ represents the random STT with the correlationfunction [8,11,28],

〈µ0h′i($$$,t)µ0h′j ($$$′,t ′)〉 = &′δijδ(t − t ′)δ($$$ − $$$′), (5)

where 〈〉 denotes statistical averaging, &′ = !g̃rkBT /2πM2s ,(i,j ) = (x,y), kB is the Boltzmann constant, and T is thetemperature of the system.

Since the spin current flows across the interface alongthe out-of-plane (y) direction (see Fig. 3), its y-polarizedcomponent does not contribute to the ISHE signal [32], whilethe z-polarized component vanishes. Hence, we focus on the

x component [Eq. (4)]:

J sx = −!g̃r4π

ṁy + Msµ0h′y, (6)

with correlation function〈J sx ($$$,t)J

sx ($$$

′,t ′)〉= M2s 〈µ0h′y($$$,t)µ0h′y($$$′,t ′)〉

+(

!g̃r4π

)2〈ṁy($$$,t)ṁy($$$′,t ′)〉

− !g̃rMsµ04π

[〈ṁy($$$,t)h′y($$$′,t ′)〉

+ 〈ṁy($$$′,t ′)h′y($$$,t)〉]. (7)

Only the first term on the right-hand side of the equation aboveis appreciable [33] because the ac susceptibility, and thereforeṁy are negligibly small at frequencies under consideration(f ( f0). With Eq. (5),

〈J sx ($$$,t)J

sx ($$$

′,t ′)〉= !g̃r

2πkBT δ(t − t ′)δ($$$ − $$$′). (8)

In this low-frequency limit, all parameters of the ferromagnet,except for the interface spin-mixing conductance, convenientlydrop out.

For frequencies much smaller than the inverse spin-relaxation time in N, the spatially resolved spin-currentdensity is governed by the time-independent diffusion equation∂2µµµs/∂y

2 = µµµs/λ2sd for the spin-chemical potential µµµs withthe boundary conditions JJJ s(= −D∂µµµs/∂y) = J sx ($$$,t)x̂xx aty = 0 and JJJ s = 0 at y = tN [32]:

J sx (rrr,t) = J sx ($$$,t)sinh[(tN − y)/λsd]

sinh(tN/λsd). (9)

λsd is the spin-diffusion length, D is the diffusion constant in N,and the spin current flows along the y direction. This quasi-1Danalysis is rigorous because in-plane lateral spin diffusion doesnot contribute to the global electromotive force, as shownin Appendix. However, locally there might be significantcorrections to Eq. (9).

The ISHE converts the spin-current density to a charge-current density along the z direction:

J cz (rrr,t) = −θSH2e!

J sx (rrr,t), (10)

with θSH the spin Hall angle of N. We are interested here inthe global voltage noise over the sample edges as indicated inFig. 3 (see also Appendix), which amounts to

V (t) = −b∫

J sx ($$$,t)d2$, (11)

for frequencies far below the plasma frequency, whereb ≡ (ρθSHe/!w) (2λsd/tN ) tanh(tN/2λsd) and ρ = RwtN/l,with R the resistance of the N layer.

Employing Eqs. (8) and (11), we arrive at the autocorrela-tion

〈V (t)V (0)〉 = b2∫∫ 〈

J sx ($$$,t)Jsx ($$$

′,0)〉d2$ d2$′,

= b2 !g̃r2π

kBT wlδ(t). (12)

214419-3

5

spin Hall noise• Fluctuation-dissipation theorem: SMR gives additional contribution to the thermal voltage

noise across the metal.

A. Kamra et al., Phys. Rev. B 90, 214419 (2014)

V (t)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

SPIN HALL NOISE PHYSICAL REVIEW B 90, 214419 (2014)

FI

N

FIG. 3. (Color online) Schematic of the normal metal (N) andmagnetic insulator (FI) bilayers analyzed in the text. The bluedashed arrow indicates the equilibrium magnetization direction. Thecoordinate system is depicted in red. The black arrows define our signconvention for spin currents across the interface.

where mmm ≡ mmm(rrr,t) is the unit vector along the magnetizationdirection at position rrr , γ (>0) denotes the gyromagnetic ratio,andhhh0 and α0 are the internal Langevin stochastic field [26] andGilbert damping constant, respectively. The effective magneticfield, written in terms of the magnetic free-energy density Fm,

µ0HHH eff = −∂Fm

∂MMM= µ0HHH 0 +

2AMs

∇2mmm, (2)

includes Zeeman and anisotropy contributions in HHH 0, whilethe second term represents the exchange field in terms of theexchange constant A [27] and the saturation magnetizationMs . The N layer is incorporated by imposing continuityof spin-current density across the FI|N interface [28]. Onthe FI side, the spin current density, carried by collectivemagnetization dynamics, is given by −A(mmm × ∂ymmm). On theN side, the spin-current density consists of (i) spin pumping(JJJ sp) by the thermal fluctuations of the magnetization in theferromagnet [29], and (ii) spin-transfer torque (STT) (JJJ stt)generated by absorption of the thermal electronic spin-currentincident on the FI. The conserved net spin-current densityJJJ s(=JJJ sp − JJJ stt) from the FI to the N is then given by

JJJ s($$$,t) = −A(mmm × ∂ymmm), (3)

= !g̃r4π

(mmm × ṁmm) − Ms(mmm ×µ0hhh′), (4)

where $$$ ≡ (x,z) is the in-plane position vector, g̃r is the realpart of the spin-mixing conductance per unit area corrected forthe finite thickness and/or spin relaxation length in N leading toa backflow spin current into FI [30]. We disregard the typicallysmall [31] imaginary part of the mixing conductance forsimplicity. hhh′ represents the random STT with the correlationfunction [8,11,28],

〈µ0h′i($$$,t)µ0h′j ($$$′,t ′)〉 = &′δijδ(t − t ′)δ($$$ − $$$′), (5)

where 〈〉 denotes statistical averaging, &′ = !g̃rkBT /2πM2s ,(i,j ) = (x,y), kB is the Boltzmann constant, and T is thetemperature of the system.

Since the spin current flows across the interface alongthe out-of-plane (y) direction (see Fig. 3), its y-polarizedcomponent does not contribute to the ISHE signal [32], whilethe z-polarized component vanishes. Hence, we focus on the

x component [Eq. (4)]:

J sx = −!g̃r4π

ṁy + Msµ0h′y, (6)

with correlation function〈J sx ($$$,t)J

sx ($$$

′,t ′)〉= M2s 〈µ0h′y($$$,t)µ0h′y($$$′,t ′)〉

+(

!g̃r4π

)2〈ṁy($$$,t)ṁy($$$′,t ′)〉

− !g̃rMsµ04π

[〈ṁy($$$,t)h′y($$$′,t ′)〉

+ 〈ṁy($$$′,t ′)h′y($$$,t)〉]. (7)

Only the first term on the right-hand side of the equation aboveis appreciable [33] because the ac susceptibility, and thereforeṁy are negligibly small at frequencies under consideration(f ( f0). With Eq. (5),

〈J sx ($$$,t)J

sx ($$$

′,t ′)〉= !g̃r

2πkBT δ(t − t ′)δ($$$ − $$$′). (8)

In this low-frequency limit, all parameters of the ferromagnet,except for the interface spin-mixing conductance, convenientlydrop out.

For frequencies much smaller than the inverse spin-relaxation time in N, the spatially resolved spin-currentdensity is governed by the time-independent diffusion equation∂2µµµs/∂y

2 = µµµs/λ2sd for the spin-chemical potential µµµs withthe boundary conditions JJJ s(= −D∂µµµs/∂y) = J sx ($$$,t)x̂xx aty = 0 and JJJ s = 0 at y = tN [32]:

J sx (rrr,t) = J sx ($$$,t)sinh[(tN − y)/λsd]

sinh(tN/λsd). (9)

λsd is the spin-diffusion length, D is the diffusion constant in N,and the spin current flows along the y direction. This quasi-1Danalysis is rigorous because in-plane lateral spin diffusion doesnot contribute to the global electromotive force, as shownin Appendix. However, locally there might be significantcorrections to Eq. (9).

The ISHE converts the spin-current density to a charge-current density along the z direction:

J cz (rrr,t) = −θSH2e!

J sx (rrr,t), (10)

with θSH the spin Hall angle of N. We are interested here inthe global voltage noise over the sample edges as indicated inFig. 3 (see also Appendix), which amounts to

V (t) = −b∫

J sx ($$$,t)d2$, (11)

for frequencies far below the plasma frequency, whereb ≡ (ρθSHe/!w) (2λsd/tN ) tanh(tN/2λsd) and ρ = RwtN/l,with R the resistance of the N layer.

Employing Eqs. (8) and (11), we arrive at the autocorrela-tion

〈V (t)V (0)〉 = b2∫∫ 〈

J sx ($$$,t)Jsx ($$$

′,0)〉d2$ d2$′,

= b2 !g̃r2π

kBT wlδ(t). (12)

214419-3

• DC thermal voltage noise (Johnson-Nyquist noise):

SV (⌦) =

Z 1

�1hV (t)V (0)i e�i⌦tdt

⌦!0) 2kBT (R0 +�R0 +R1 cos2 ↵)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

AKASHDEEP KAMRA et al. PHYSICAL REVIEW B 90, 214419 (2014)

(a)

(b)

FIG. 1. (Color online) (a) Schematic of the voltage power spec-tral density measurements. The sample (gray) is connected to apreamplifier and a FFT spectrum analyzer. The symbols + and −define the sign convention for the voltage measurements. The setupand the amplification stage are shielded by a metal box (red thicklines). The applied magnetic field (blue arrow) makes an angle αwith the voltage measurement direction. (b) A typical noise spectrumcaptured using the setup described in (a). The individual data pointsshown in Fig. 2(c) are averaged over the frequency window between20 and 45 kHz. The dashed line depicts the white noise level expectedfrom the fluctuation-dissipation theorem.

sweep” [as in Fig. 1(b)] is obtained by averaging 15 000 suchspectra. A single average value of the white noise level is thenobtained by averaging the PSD sweep data in the frequencyrange 20–45 kHz. The frequency window is so chosen inorder to minimize the effects of the 1/f noise and externalelectromagnetic disturbances. The average of 19 such datapoints lead to the precision of 0.01% sufficient to resolve thespin Hall noise [Eq. (15)].2

The measurement configuration is depicted in Fig. 2(a). A60 mT magnetic field applied in the xz plane at an angle αwith the +z direction saturates the YIG magnetization along its

2The noise floor of our setup (output with zero voltage input, i.e.,short circuited amplifier input) 1.52 × 10−17 V2/Hz is subtracted fromall data points.

FIG. 2. (Color online) Sample and wire bonding assembly formeasuring (a) voltage power spectral density SV,long and (b) resistanceRlong. The applied magnetic field (blue arrow) makes an angle α withthe voltage measurement direction (zzz). (c) SV,long (squares) and Rlong(triangles) measured as a function of α. Both SV,long and Rlong exhibita cos2 α dependence and are related by SV,long = 4kBT Rlong withT = 291.5 K, consistent with the fluctuation-dissipation theorem.The α-dependent contributions to Rlong and SV,long are attributedto spin Hall effect-generated spin currents and spin Hall noise,respectively.

direction. The voltage noise PSD SV,long of the “longitudinal”voltage Vlong [Fig. 2(a)] averaged over 19 α sweeps isshown as white open squares in Fig. 2(c). We also carriedout conventional SMR measurement [15] of the longitudinalresistance Rlong along the Hall bar (zzz) direction [Fig. 2(b)] asa function of α for a charge current Iq = 40.5 µA along theHall bar. Rlong, shown as red triangles in Fig. 2(c), exhibits thecos2 α-dependence characteristic of the SMR effect [16]. Wefind that SV,long and Rlong are related by SV,long = 4kBT Rlong,with T = 291.5 K (room temperature), as expected from thefluctuation-dissipation theorem. Since the α-dependence ofRlong is attributed to SHE-generated spin currents [16], theanisotropic PSD must be caused by the spin Hall noise.

III. THEORY

To substantiate this claim, in the following we present astatistical linear response theory for the α-dependent noise thatelucidates the role of the spin currents. We restrict the analysisto frequencies far below the ferromagnetic resonance (FMR)frequency f0. We consider a bilayer of a normal metal (N) withspin Hall angle θSH deposited on a ferromagnetic insulator (FI)with its equilibrium magnetization pointing along ẑzz as shownin Fig. 3. The magnetization dynamics in the FI is describedby the Landau-Lifshitz-Gilbert (LLG) equation:

ṁmm = −γ [mmm × µ0(HHH eff + hhh0)] + α0(mmm × ṁmm), (1)

214419-2

5

spin Hall noise spectroscopy (SHNS)• heavy metal | quantum paramagnet (no long-range order) bilayer at temperature .T

D. Joshi, A. P. Schnyder and ST, Phys. Rev. B 98, 064401 (2018) J. Aftergood and S. Takei, Phys. Rev. Research 2, 033439 (2020)

metal

quantummagnet

noisy spincurrent

noisy voltageinverse spinHall effect

temperature 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V

Figure 1: A bilayer setup involving a metal with strong spin orbit coupling in contact with an unconventionalquantum magnet and in which two subsystems are held in thermal equilibrium. Spin fluctuations in the quantum

magnet generates noisy spin current fluctuations at the interface, which in turn converts into charge current fluctua-

tions inside the metal via inverse spin Hall e↵ect. The charge current fluctuations can be quantified via conductance

measurements across the metal owing to fluctuation-dissipation theorem.

1.1 Spin Hall noise spectroscopy

Our first system of interest will be a bilayer system in which a metal with strong spin orbit coupling(e.g., 5d transition metals such as Pt, Ta, and W) is placed in contact with a QSL, and the twosubsystems are held in thermal equilibrium (see Fig. 1). In the presence of exchange coupling atthe interface, spin injection (ejection) into (out of) the QSL occurs via the annihilation (creation)of S = 1 particle-hole excitations in the metal. Spin fluctuations in the QSL (both classical andquantum) will therefore generate equilibrium spin current fluctuations across the interface [49, 50],which, as a result of the inverse SHE, are converted into voltage fluctuations inside the metal— voltage fluctuations that may encode information about the microscopic structure of the spinsystem. The voltage noise can be quantified by measuring the resistance across the normal metallayer owing to the fluctuation-dissipation theorem. This bilayer system therefore o↵ers a table-top experimental technique — we hereafter refer to this technique as spin Hall noise spectroscopy(SHNS) — that directly probes the density of states for S = 1 excitations in the quantum magnet.

Bilayer devices involving 5d transition metals and conventional magnetic insulators are nowroutinely produced in spintronics laboratories for spin-transfer torque and spin pumping studies [38],and spin-Hall generated charge noise in a setup similar to Fig. 1, specifically, has garnered somestudy with respect to conventional quantum magnets hosting magnons [51]. Recently, Ref. [41]even succeeded in building an interface between Pt and a quantum spin chain material Sr2CuO3for the study of spin Seebeck e↵ect. We therefore anticipate that bilayers involving 5d transitionmetals and unconventional quantum magnets will become available in the near future by extendingcurrent spintronics technology.

1.2 Nonequilibrium quantum magnetism

Another attractive aspect of the bilayer setup in Fig. 1 is that it can be straightforwardly extendedfor the study of nonequilibrium QSLs. The augmented setup of interest is illustrated in Fig. 2, inwhich one a�xes two additional metals with strong spin orbit coupling atop the QSL. Inside the“spin injector” metal, a charge current Ic is applied parallel to the interface so that it generates

2

metallicbath

nonequilibriumquantum magnet

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 Is

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

spininjector

spinsink

spin current

spin/energy dissipation

Figure 2: An experimental setup that can be used to study two-terminal spin transport through a quantum magnet.A charge current Ic applied parallel to the interface generates a spin current Is oriented normal to the interface via

the direct SHE. This spin current enters the quantum magnet, is transmitted through the spin system by mobile spin

excitations, and is ejected into the spin sink. The dissipative metallic bath can act to stabilize the quantum magnet

toward an energy and spin flow equilibrium state.

a spin current Is oriented normal to the interface via the direct SHE. This spin current entersthe quantum magnet owing to the interfacial exchange coupling, is transmitted through the spinsystem by mobile spin-carrying excitations, and is ejected into the spin sink. While the spin injectorcontinuously pumps spin and energy into the QSL, the dissipative metallic bath can act to stabilizethe QSL toward an energy and spin flow equilibrium state. The setup in Fig. 2 thus realizestwo-terminal spin transport through insulating spin systems, and, in principle, allows the study ofunconventional quantum magnets driven into nonequilibrium steady states by spin currents.

Understanding unconventional quantum magnets driven out of equilibrium by spin currentsstill remains a largely unexplored problem. However, we have witnessed a flurry of spintron-ics experiments in recent years investigating two-terminal spin transport through both ferromag-netic [42, 43, 45] and antiferromagnetic [52, 53, 54, 71] insulators using setups very similar to the oneillustrated in Fig. 2. Two-terminal spin transport

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Spintronics meets quantum spin liquids: a novel spectral …...2020/11/17 · Spintronics meets quantum spin liquids: a novel spectral probe of quantum magnets based on spin Hall