EM wave generated by oscillating line charge current in waveguide

In the same way, we solve the electric field induced by ac line charge current in waveguide with the method we consider, for checking if our calculation method is practical, again.

The calculation also gives us a simpler exercise for solving electric field induced by ac in-plane polarized spin current.

When the wave length becomes comparable with the thickness of the waveguide, we should consider the near-field light instead of far-field light. The waveguide structure confines the field and leads far-field into near-field wave. The near-field wave is more complicated than far-field wave. Fortunately, plane wave expansion method is useful method for us to deal with our problem. Therefore, the quantization field in

two-parallel-CONSIDER

Figure 3.2: An infinite long wire is located at the middle of the waveguide and towards the x-direction.

Figure 3.3: The side view of the waveguide structure and the wire. The waveguide is the same as one we discussed in 2.2 which has permeability µ and primitivity ε. The up and down electrode slabs are made of perfect conductor.

planes waveguide we derived in Chapter 2 can be a complete set. The structure of the waveguide we consider about is the same as we discussed in Chapter 2. An infinite long wire parallel with x-axis is located in the middle of two parallel metal gates as shown the following picture.

Assume that the semiconductor between two metal gates has the permittivity ε and permeability µrespectively. The wire is very thin, comparing with the distance d, and this wire carries flow of electron charge I = eλ_e^~k_me^ex where λˆ _e is the particle density per unit length, me is the mass of a free electron, and ke is the electron wave number, where

~ is the Plank constant. The wave function of the electrons is

Ψ (r) = hr|ψi =p

λ_eexp [ik_ex] ϕ (y, z) . (3.16)

At the beginning, we do not consider about ac current. We will add the oscillating information in the effective Hamiltonian later. The perturbation operator of oscillating line current in the waveguide is given by:

H⁰ = e

2m_e (p · A + A · p) = e

m_ep · A = e

m_eA · p, (3.17)

where e > 0. For the transverse gauge p · A − A · p = −i~∇ · A = 0. We care about only the transverse field without the longitudinal field. It means that we consider about the field induced by ac charge current without the Coulomb field. Therefore, the perturbing Hamiltonian does not include the Coulomb potential.

Effective Hamiltonian

We sandwich Eq. (3.17) with electron state to get the effective Hamiltonian of the photons in the waveguide which is given by the following:

H⁰ef f = hψ| H⁰|ψi = hψ| e

m_eA · p |ψi

= ke~λe

e m_e

Z dx

Z dy

Z

dzϕ (y, z) Axϕ (y, z) − i~λ e m_e

½Z dx

Z dy

Z

dzϕ (y, z)

× Ay

∂

∂yϕ (y, z) + Z

dx Z

dy Z

dzϕ (y, z) Az

∂

∂zϕ (y, z)

¾ ,

(3.18)

CONSIDER

where A_x, A_y, A_zis the x, y, z-component of vector potential respectively. Vector potential near the position¡

x, y = 0, z = ^d₂¢

changes violently, but the wave function of the electron ϕ (y, z) changes smoothly, so we may take some approximation. Then we have:

H⁰_{ef f} = ~λ_e e

The detailed derivation from Eq. (3.18) to Eq. (3.19) is left in the Appendix C. Putting Eq. (2.16) and Eq. (2.17) into Eq. (3.19), we obtain:

H⁰_{ef f} = ~λ_e e

According to Fourier analyze, R_∞

−∞e^ikxxdx = 2πδ (kx). The terms with kzδ (kz) in the equation above will be vanished after integration overkx. We can drop it. Then we have

H⁰_{ef f} = −eλ_e~k_e shown in Fig. 3.4

”δ (k_x) cos¡_π

2 + φ_k¢

” in equation above will lead to the form ”−Sgn(k_y)” after

inte-Figure 3.4: The illustration shows the relation between φ_k, ˆk, and ˆz × ˆk.

gration with respect to kx, so we substitute δ (kx) cos¡_π

2 + φk

¢to −δ (kx) Sgn(ky) to avoid complicated calculation. Actually, the magnitude of electron current is equal to eλe~ke

me

by definition, so that we let I₀ = eλ_e^~k_m^e

e.Then, we have H⁰_{ef f} = 2πI₀X

m,κ

1

iω_mk2 sin³mπ 2

´ ©b_mk2e^−iωmk2t− b_mk2⁺e^iωmk2tª

δ (k_x) Sgn(k_y) (3.22)

If we allow the current to oscillate harmonically in time, we can substitute I₀cos (Ωt) for I₀ where Ω is the oscillation angular frequency. The perturbation operator becomes:

H⁰_{ef f} = 2πI₀cos (Ωt)X

m,κ

1

iω_mk2 sin³mπ 2

´ ©b_mk2e^−iωmk2t− b_mk2⁺e^iωmk2tª

× δ (k_x) Sgn(k_y)

(3.23)

The first order perturbation coefficient

Again, we apply time-dependent perturbation theory to solve the new eigenstate of pho-ton, and the first order perturbation coefficient in H⁰_{ef f} is given by:

f_kλ⁽¹⁾ = −i

~ Z _t

−∞

h{0, 0, ..., 0, 1_kλ, 0, ..., 0}| H⁰_{ef f} |{0}idt. (3.24)

CONSIDER

We impose the mechanism of adiabatic turn-on to simulate the more realistic system and simplify calculation. Therefore, we put the term e^ηt in the integration form. We consider about one photon emission. It means the photon state from |{0}i to |{0, 0, ..., 0, 1_kλ, 0, ..., 0}i

We add a factor of to describe adiabatic turned on . η is a constant smaller than 1 far, switched on very gradually in the past, and we are looking at times much smaller than_η¹. We can then take the initial time to be −∞. For TE modes We have the first order perturbation coefficient by:

f_m⁽¹⁾0κ⁰2 = −i

~ Z _t

−∞

h{0, 0, ..., 0, 1_mk2, 0, ..., 0}| e

mA · pe^ηt|{0}idt

= i

~ Z _t

−∞

e^ηth{0, 0, ..., 0, 1_mk2, 0, ..., 0}| I₀cos (Ωt) 2πi

×X

m,k

1

ω_mk2sin³mπ 2

´ ©b_mke^−iωmk2t− b_mk⁺e^iωmk2tª

δ (k_x) Sgn(k_y) |{0}i dt

(3.25)

b_mk and b_mk2⁺ in Eq. (3.25) is corresponded to the annihilation operator and the creation operator respectively. We use Eq. (2.21) as well as Eq. (2.22) and Simplify Eq. (3.25), so we have:

f_mk2⁽¹⁾ = I₀

~2π 1 ωmk2

r~ω_mk2

εV sin³mπ 2

´

× 1 2i

½exp [i (ω_mk2− iη + Ω) t]

ω_mk2− iη + Ω + exp [i (ω_mk2− iη − Ω) t]

ω_mk2− iη − Ω

¾

δ (k_x) Sgn(k_y)

(3.26)

For TM modes

f_n⁽¹⁾0k⁰1 = −i

~ Z _t

−∞

h{0, 0, ..., 0, 1_n⁰_k⁰₁, 0, ..., 0}| H⁰_{ef f}|{0}idt = 0.

The expectation value of vector potential

We are interesting the expectation value of vector potential, because the electric field can be obtained easily after knowing it. The time-dependent expectation value of vector

potential in the photon state |Ψi is given by:

A (r, t) = hΨ| A^(op)|Ψi

= {h{0, 0, ..., 0, ...}| +X

mk

f⁽¹⁾_mk2^∗h{0, ..., 0, 1mk2, 0, ...}|}A^(op){|{0, 0, ..., 0, ...}i

+X

mk

f_mk2⁽¹⁾ |{0, ..., 0, 1mk2, 0, ...}i}.

(3.27)

Let’s remember that f_nk1⁽¹⁾ is equal to zero, and we already derived f_mk2⁽¹⁾ in Eq. (3.26). We combine Eq. (2.16), Eq. (3.26) and Eq. (3.27), and we get:

A (r, t)

=X

m,κ

I₀

~π 1 ω²_mk2

~ω_mk2

εV sin³mπ 2

´

sin³mπ d z

´ ½

e^{i(k·ρ−ωmk2t)}· exp [i (ω_mk2+ Ω) t]

ω_mκ2− iη + Ω + e^{i(k·ρ−ωmk2t)}exp [i (ω_mk2− Ω) t]

ωmk2− iη − Ω + e^{−i(k·ρ−ωmk2t)}exp [−i (ω_mk2+ Ω) t]

ωmk2+ iη + Ω +e^{−i(k·ρ−ωmk2t)}exp [−i (ω_mk2− Ω) t]

ω_mk2+ iη − Ω

¾

× δ (k_x) Sgn(k_y)

³ ˆ z × ˆk

´

(3.28)

We can find the first two terms in the curve bracket in above equation is the complex conjugate of the last two terms. It is just like that we add the complex conjugate of electric field in equation, and it keeps the physical quantity be real number. The summation over k takes arbitrary directions and arbitrary magnitudes wave number. For arbitrary orientations and magnitudes of k, the summation over k can be generalized to integration over k, just like what we do in Chapter 3.

X

k

→X

k

∆k_x

∆k_x

∆k_y

∆k_y → 1

(∆k_x) (∆k_y) Z

dkxdky = V d(2π)²

Z

dkxdky

CONSIDER

Because the z direction is quantized and described by summation over m, it cannot change to representation of integration. The integral only respects to x and y.

A (r, t) = I₀ π

Substituting the dispersion relation kz2+ k² = ωnk12µε into above equation, we have:

A (r, t) = − I₀

Solving these integrals is not easy. Even though we can apply complex integral meth-ods solve these integrals, the branch cuts make the complex integrals complicated. The

detailed derivation is saved for Appendix D. Considering y > 0, we have

By the same way, we can derive the vector potential for y < 0 which is given by:

A (r, t) = − I₀

From Eq. (3.31) and Eq. (3.32) , we can rewrite the vector potential as following:

A (r, t) = A^>(r, t) + A^<(r, t) (3.33)

CONSIDER

µεΩ² > kz2 means the frequency of the current oscillation beyond the cutoff frequency of the parallel-plates capacitor. The wave is in propagating modes. µεΩ² < k_z² indicates the frequency of the current oscillation above the cutoff frequency of the parallel-plates capacitor. And the corresponding electric field is given by:

E (r, t) = E^>(r, t) + E^<(r, t) (3.36)

The electric field only couples to TM wave and it only has the x-component field, because the line current oscillates in the x-direction. The direction of the electric field satisfies the expectations of classical electrodynamics. When µεΩ² > k_z² = ¡_mπ

d

¢₂ , the situation will insure wave propagation. When µεΩ² < k_z² = ¡_mπ

d

¢₂

, the wave becomes evanescent mode. An evanescent wave is a nearfield standing wave with an intensity that exhibits exponential decay with distance and it does not propagate. We can see this in Eq. (3.38) which has a term exponentially decaying from y = 0.

The electric field is the same as the classical expectation. Again, we prove that our

calculation method is practical and the quantization wave in the waveguide we deduce is correct. The detailed calculation process of this problem in classical method is left for Appendix E.

3.3 Brief summary

In previous two sections, we drove the vector potential and electric field induced by oscillating charge in free space and ac line current in waveguide. The electromagnetic wave induced ac line current in waveguide only couples to TM modes. The results of the two different systems solved by our calculation method are identified with the calculation method of classical electrodynamics. We did show that our method is practical.

Chapter 4

The electric field induced by ac spin current

We already know that our calculation method is practical, and we will solve the electric field induced by ac in-plane polarized spin current in the waveguide in this chapter.

4.1 Effective Hamiltonian for photon

We had discussed the structure of the waveguide of our system previously. 2DEG (two dimensional electron gas) in our system is at the middle of the two parallel metal gates as Fig. 2.2. Ac in-plane polarized spin current will generate out-off-plane electric field and we can use a voltmeter to measure the electrical potential difference between the two metal gates.

We will calculate line spin current instead of surface spin current, because the field induced by line spin current is easy to analyze its physical meaning. Moreover, if we directly calculate the field induced by surface current, the field has singularity owing to the waveguide structure. Actually we may decompose the surface spin current into countless line spin currents. We can integrate over the electric field distribution of line spin current into one of surface spin current. Assume that the line spin current flow is

parallel to x-axis and this ”line” is located at (x,0,d/2). The spin polarization direction is towards the negative y-axis. We do not let this spin current oscillate at first, and the wave function of the spins is given by:

ψ (r) = hr|ψi = p

where λ_sis the line density of the spins and k_sis the wave number of the spins.

·

√1

2 −^√i₂

¸_T is the spin state, whose spin direction always point towards the negative y-axis. In semicon-ductor, the spin orbit coupling term is given by H⁰ = e^Λ_~σ ¦ (p × E) , where we discussed in Chapter 1. The effective perturbing Hamiltonian for photons in the system can be obtain by sandwiching H⁰ with electron state as given by:

H⁰ef f = hψ| H⁰|ψi

where E_x and E_z is the x and z component of the electric field respectively. p is operator for electron, it operate on the electron state. E is classical physical quantity for electron.

The Eq. (2.12) and Eq. (2.15) lead us to obtain E_x and E_z by dot product. Hence, the

Figure 4.1: The figure shows the relationship between φ_k, k, and ˆx.

perturbing Hamiltonian becomes:

H⁰_{ef f} = −eλΛ Z ©

e^−iksxϕ (y, z)ª( 1 µε

X

n,k

1 ωnk1

³nπ d

´₂

cos³nπ d z´ £

−c_nke^{i(kxx+kyy−ωnk1t)}

+ cnk+e^{−i(kxx+kyy−ωnk1t)}¤

cos φk− iX

m,k

³mπ d

´

cos³mπ d z´ £

bmke^{i(kxx+kyy−ωmk2t)}

+ b_mk⁺e^{−i(kxx+kyy−ωmk2t)}¤

sin φ_k+ 1 µε

X

n,k

k

ω_nk1cos³nπ d z´ £

−k_xc_nke^{i(kxx+kyy−ωnk1t)}

+k_xc_nk⁺e^{−i(kxx+kyy−ωnk1t)}¤

− iE_x ∂

∂z + iE_z ∂

∂x

¾©

e^iksxϕ (y, z)ª dr,

(4.3)

where φ_k is the angle between the k and ˆx. The relationship between φ_k, k, and ˆx is shown in Fig. 4.1.

Because the cross section of the y-z plane of the line spin current is far smaller than the thickness of the waveguide d, and the electric field near the line current¡

x, y = 0, z = ^d₂¢ change smoothly in the space and the electric field near the line current changes violently

in the space, we can write the integrationR

For the same reason, the integrationR dxR

We substitute Eq. (4.4) into Eq. (4.5) and express E_x , and E_z in waveguide modes. After simple integral process, the effective perturbing Hamiltonian becomes:

H⁰_{ef f} = −eλΛ after integrating over k_x. We drop it to avoid unnecessary calculation. The Hamiltonian

of the effective perturbation becomes: cnk and cnk+couples to the TM wave. We only care about the z-component of the electric field, because only this component of the electric field distributes the electrical potential difference between the two metal slabs of the waveguide. The z-component electric field is corresponding to the TM wave. (c_nκ1 or c_nκ1⁺ couples to photons of TM wave.) Actually, if we keep calculating the vector potential with the perturbing Hamiltonian H⁰_{T E}, the vector potential of TE modes will be equal to zero. It means that only photons corre-sponding to the transverse magnetic (TM) waveguide modes are exited. If we allow the spin current to become oscillatory, with an ac frequency Ω, we can incorporate this into our present framework by changing: k_s → k_scos (Ωt). Hence, the Hamiltonian of the effective perturbation becomes as the following:

H⁰_{T M} = 2πeλ_sk_scos (Ωt) Λ

With the perturbing Hamiltonian for photons, we can obtain the wave function by

using perturbation theorem. We will solve the photon state in our system in next section.

4.2 The new photon state in the waveguide

We have obtained the effective Hamiltonian of the system of ac in-plane polarized spin current for TM wave in parallel-planes waveguide. We desire to know the photon state of TM wave in our system. The photon state gives us all information of the wave in the waveguide including the z-component electric field. The photon state of TM wave in our system becomes:

|Ψi = |Ψ₀i +X

nk1

f_nk1⁽¹⁾ |{0, 0, ..., 0, 1_nk1, 0, ..., 0}i

= |{0, 0, ..., 0, ..., 0}i +X

nk1

f_nk1⁽¹⁾ |{0, 0, ..., 0, 1_nk1, 0, ..., 0}i

(4.9)

where |Ψ0i is the original wave function. f_nk1⁽¹⁾ is the first order perturbation coefficient of H⁰T M. f_nk1⁽¹⁾ can be solved by:

f_n⁽¹⁾0k⁰1 = −i

~ Z _t

−∞

h{0, 0, ..., 0, 1n⁰k⁰1, 0, ..., 0}| H⁰ef fe^ηt|{0}idt

= −i

~ Z _t

−∞

e^ηth{0, 0, ..., 0, 1_n⁰_k⁰₁, 0, ..., 0}| 2πeλ_sk_scos (Ωt) Λ µε

×X

n,κ

k

ω_nk1cos³nπ 2

´ £cnk1e^−iωnk1t+ cnk1+e^iωnk1t¤

δ (kx) |{0}i dt

(4.10)

After integration, the first order perturbation coefficient becomes:

f_n⁽¹⁾0k⁰1 = −πeλsks

~ Λ µε

k

ω_nk1 cos³nπ 2

´ r

µ~ωnk1

V

½exp [i (ωnk1+ Ω) t]

ω_nk1− iη + Ω +exp [i (ωnk1− Ω) t]

ω_nk1− iη − Ω

¾ δ (kx)

(4.11)

Since H⁰_{T M} does not include the b_mk or b_mk⁺, H⁰_{T M} would not couple to the TE wave but TM wave which we care about. From above equation, we know that f_n⁽¹⁾0k⁰1 vanishes for odd ”n” which imply the electric field for TM wave will only couple to even n mode.

Also, we know the photon state of TM wave in our system.

4.3 The expectation value of the vector potential in our system

The expectation value can give us the predicted mean value of the result. In the previous section, we obtained the photon state of TM wave |Ψi. The expectation value of the vector potential of TM wave in the photon state |Ψi is given by:

A_{T M}(r, t) = hΨ| A^(op)|Ψi

= (

h{0, 0, ...., 0}| + X

m⁰kλ

f⁽¹⁾_m0kλ

∗h{0, .., 0, 1_m⁰_kλ, 0, ..., 0}|

)

A^(op){|{0, 0, ..., 0}i

+X

m⁰kλ

f_m⁽¹⁾0kλ|{0, ..., 0, 1m⁰kλ, 0, ..., 0}i )

.

(4.12)

Substituting Eq. (2.17) into Eq. (4.12), we have:

A_{T M}(r, t)

= − 1 µε

X

nk

¡_nπ

d

¢

ωnk12 sin³nπ d z

´r

µ~ω_nk1 V

n

f_nk1⁽¹⁾e^{i(k·ρ−ωnk1t)}+ f⁽¹⁾_nk1^∗e^{−i(k·ρ−ωnk1t)} oˆk

+ 1 µε

X

nk

k

ω_nk1² cos³nπ d z

´ r

µ~ω_nk1 V

n

−if_nk1⁽¹⁾e^{i(k·ρ−ωnk1t)}+ if⁽¹⁾_nk1^∗e^{−i(k·ρ−ωnk1t)} o

ˆ z (4.13)

From above equation, we notice that the vector potential has two components that are k and z directions. The two directions are the same as the directions of electric field.

Combining above equation Eq. (4.11) and Eq. (4.13), we obtain:

For arbitrary orientation and arbitrary magnitude of wave number k, We can generalize the summation to representation of integration for continuous k. It meansP

k

R dk_xdk_y. Because the z direction is quantized and described by summation m, it may not change to representation of integration. The integral only respects to x and y. Then we get

AT M(r, t) = Λ

We use the dispersion relation k_z² + k² = ω_nk²µε and replace ω_nk1 by q

1 µε

¡k_y²+ k_z²¢ .

”δ (k_x)” in above equation make the integration with respect to k_x easy. Then the vector potential becomes:

We find that F₁ , F₂ , G₁ , G₂ is the complex conjugate of F₃ , F₄ , G₃ , G₄. When we deal with vector potential induced by charge oscillation or ac line current problem in Chapter 3, we ever both experience this condition. The integrals in Eq. (4.16) seem not easy. We have to consider about the situation for y > 0 and y < 0. Complex integral methods can simplify this problem. Using Complex integral method, we must choose different contours for the corresponded integral. The detailed derivation is saved for Appendix F. We instead λ^~k_m∗^s to I_sc. I_sc is the magnitude of spin current in unit of Ampere.

After the complex contour integration, the vector potential can be divided by two part. the transverse gauge in which the scalar potential vanishes. The electric field is given by

E = −^∂A(r,t)_∂t . Therefore, the electric field for TM modes can be shown as: electric field in the situation µεΩ² <¡_nπ

d

After complicated calculations we finally get the electric field induced by ac spin-polarized current in the waveguide. ”P

n=0

cos¡_nπ

2

¢” in Eq. (4.21) and Eq. (4.22) ask that only the even modes of electric field exist. It is different from the electric field that is induced by charge current in the waveguide we discussed in Chapter 3 only and couples to odd modes.

The electric field has the term exp¡

−^nπ_d |y|¢

which decays in the positive y-axis and negative y-axis from the y = 0, even though we do not understand that it’s physical meaning, we find that the result is extremely different with the ac charge current. From mathematics view point, the ac in-plane polarized spin current couples to TM wave which the electric field is proportional to _ω¹

nk1 . But charge current oscillation which couples to the TE wave is not. This difference results that the final residue integral methods generate

difference number of poles, so that the electric field induced by the ac in-plane polarized spin current exists the decay terms. From Eq. (4.22), we notice that the electric field vanishes if the oscillation frequency is equal to zero. This checks with the ac nature of our results in this work. We will explain in next chapter that only the n=0 waveguide mode will contribute to the potential difference between the two metal gates of the waveguide.

4.4 Brief summary

We drove electric field induced by ac spin current in waveguide and coupling to TM modes.

The electric field only couple to even mode and has exponential decaying terms. With the electric field, we can calculate the ac electrical potential difference induced by ac spin current.

Chapter 5

Result and discussion

In the last chapter, we will discuss the final result of the ac electrical potential difference induced by as in-plane polarized spin current and explain the signal is measurable if we add a film bulk acoustic-wave resonator in the measuring circuit.

5.1 Discussion

In this section, we deduce the magnitude of the electrical potential difference induced by ac in-plane polarized spin current and discuss the result. The potential difference induced by the ac spin current can be deduced from equation by integrating the z-component of the electric field over the thickness of the waveguide. Then the ac electrical potential difference is given by

Vz = Z _d

0

Ezdz = Λ εem^∗

~ Isc

npµεΩ²sin³p

µεΩ²y − Ωt

´o

(5.1)

Only the mode n = 0 wave distributes the potential difference, because the electric field of every mode of the wave oscillates in the z direction and is canceled by integrating over the thickness of the waveguide except the ground mode. We notice that the electrical potential difference is independent from the thickness of the waveguide d. It is good news for us, because we neglect the skin effect in our calculation.

Skin depth

Skin depth, also known as classical skin depth, is the depth to which electromagnetic radiation can penetrate the surface of a conductor. It will cause energy loss or change of the field. Under the condition that the electronic mean free path becomes comparable with or greater than the classically calculated skin depth, we should consider about anomalous skin depth instead of classical skin depth [15]. The anomalous skin depth is given by:

δ =

µ l

2πf⁰aµσ

¶_1/3

√2 3

where ”a” is a real coefficient of the order of unity, l is the mean free path in material, µ is permeability in material, f⁰ is the frequency of incident electromagnetic wave, σ is the conductivity of material. The higher frequency gives the less anomalous skin depth.

If the thickness of slabs is longer than the skin depth, we do not worry about this problem. Essentially, for any material the ratio of electronic mean free path l to absolute conductivity σ is a constant independent of temperature. For aluminum and gigahertz of incident electromagnetic wave, the skin depth is about 0.5µm. If the thickness of slabs is more than 0.5µm, we do not worry about the impact of skin effect.

Surface spin current

The electric field in Eq. (4.20) is induced by the ac line spin current in the waveguide. Ac-tually spin current should flow though the 2DEG. We can calculate the total distribution of surface spin current on the 2DEG by integration over the y-direction. That is

E_z(r, t) = Λ εdem^∗

~ I_sccos³nπ d z

´

×p µεΩ²

Z

dy⁰sin³p

µεΩ²(y − y⁰) − Ωt

´

(5.2)

where we only consider about the ground mode. If the oscillation frequency is about giga-hertz, when we integrate the electric field with respect to y⁰, the wave length 2π/p

where we only consider about the ground mode. If the oscillation frequency is about giga-hertz, when we integrate the electric field with respect to y⁰, the wave length 2π/p

在文檔中 全電性自旋流量測的研究 (頁 37-96)