Proposal for the momentum-resolved and time-resolved optical measurement of the current distribution in semiconductors

Jiang-Tao Liu; Fu-Hai Su; Xin-Hua Deng; Hai Wang

doi:10.1364/OE.20.011694

1. Introduction

Directly mapping the amplitude and the angular distribution of current in real time plays important roles both in basic physics and applied engineering. For example, the graphene transistor is promising to extend to THz frequency [1, 2], which require the ability to detect the ultrafast current with sub-picosecond time resolution. Real-time current detection in momentum space is also important in the studies of Bloch oscillator [3], quantum transport in nano-electronics [4], spin Hall effect [5–8], and scattering processes in semiconductors such as the scattering between carriers and impurity [9–11], lattice vibration, and crystal defects etc.

Some experimental techniques such as angle-resolved photoemission spectroscopy (ARPES) [12, 13] have been used to observe the distribution of electrons in the momentum space. Especially, the time-resolved two-photon photoemission experiment (2PPE) was used to access the electron current dynamics at a metal surface on the femtosecond time scale [14]. However, the photoemission spectroscopy normally is only appropriate for the mapping of the surface current. Khurgin proposed that the current induced second harmonic generation (SHG) can be used to map the current in semiconductor [15] and have been successfully verified by experiments very recently [16]. However, this effect is quite small. For instance, a laser pulse with the intensity of 10¹⁰ W/cm² only can generate 10³ harmonic photons with the presence of 1 A/cm current density. Therefore, the current-induced SHG may be suppressed by the surface SHG and bulk SHG with small current density.

To achieve high time resolution, ultrafast optical measurement techniques must be used. But the traditional absorption spectrum can not be used to detect the charge current with uniform spatial charge distributions because the optical transitions are the same at k and −k. Therefore, an asymmetric optical transition process in k space is essential. The optical quantum-interference process QUIC between one- and two-photon absorption is such a process, which has been used to inject charge current and spin current [17–23]. And as a detection technique, the Faraday rotation in QUIC has also been proposed to detect the amplitude and direction of the spin current [24]. Based on these studies, in this paper we suggest a different scheme by utilizing a momentum-resolved two-color optical coherence absorption spectrum (QUIC-AB) for the direct detection of charge current. Our proposed current-detection method is based on the principle of two-color quantum interference that create an asymmetric optical transition process in k space, which is able to offer the information of amplitude and distribution of charge current in real time.

2. Theory

In the two-color quantum interference control (QUIC), the optical transition rate depends strongly on the direction of the wave vector k; e.g., the transition rate of QUIC can be enhanced at +k but disappears at −k. Thus, if the band states near +k are already filled with electrons [see Fig. 1(a)], the optical absorption of QUIC is reduced due to the state filling effects. However, if the electrons are located at −k, these electrons have no effect on the QUIC-AB because the optical transition at −k is forbidden.

Fig. 1 (a) Schematic illustration of the transition of two parallel-linearly polarized beams in semiconductor materials. (b) Illustration of the detection of the angular distribution of pure charge current by using two-color optical coherence absorption spectrum.

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To reveal the relation between the electron distribution function and the QUIC-AB, we calculate the transition rate of QUIC using the Volkow-type wave function [22–24]. In optical QUIC, the total vector potential of ω and 2ω laser pulse is given by A = a₁A₁ cos(ωt +φ₁) + a₂A₂ cos(2ωt + φ₂). The transition rate of QUIC can be written as

W (k) = 2 π {(\frac{e}{2 \bar{h} m c})}^{2} δ [ω_{c v} (k) - 2 ω] \times (W_{k}^{n} + W_{k}^{i}) [f_{v} (k) - f_{c} (k)],

where

\begin{array}{l} W_{k}^{n} = {(k \cdot a_{1})}^{2} {| p_{v c} \cdot a_{1} |}^{2} η_{1}^{2} A_{1}^{2} + {| p_{v c} \cdot a_{2} |}^{2} A_{2}^{2}, \\ W_{k}^{i} = k \cdot a_{1} η_{1} A_{1} A_{2} [{(p_{v c} \cdot a_{1})}^{*} (p_{v c} \cdot a_{2}) e^{i Δ φ} + c . c .], \end{array}

where m is the electron mass, e the electron charge magnitude, c the speed of light in vacuum, ω_cv the optical transition frequency, f_v(k) (f_c(k)) the distribution function of the valence (conduction) electrons,

η_{1} = \frac{e A_{1}}{2 ω c m_{c v}}

, 1/m_cv = 1/m_c − 1/m_v, m_c (m_v) the effective mass of electrons (holes), p_vc = 〈c|p|v〉 the dipole transition matrix element, and Δφ = 2φ₁ − φ₂ the relative phase of the ω and 2ω laser beams.

W_{k}^{n}

and

W_{k}^{i}

describe the single-photon and two-photon transition and the quantum interference between the two-color photon excitations. Thus, the optical absorption coefficient is given by

α (ω) = \frac{\bar{h} ω W (ω)}{u (c / n_{0})}

, where

u = \frac{n_{0}^{2} A_{2}^{2} ω^{2}}{2 π c^{2}}

is the electromagnetic energy density, and n₀ the refractive index of the semiconductors.

From Eq. (1), only the interference term $W_{k}^{i}$ depends on both the direction and the magnitude of electron vector k. In the current paper, we focus on the detection of pure electron current. Thus, two parallel linearly polarized probe laser beams (i.e., a₁||a₂) are used [see Fig. 1(b)]. In this case, the interference term $W_{k}^{i}$ reaches the maximum for Δφ = 0° but disappears for Δφ = 90°. The normal single-photon and two-photon transition term $W_{k}^{n}$ is invariant for different Δφ. If we measure the difference in absorption between Δφ = 0° and Δφ = 90°, the normal single-photon and two-photon transition term $W_{k}^{n}$ can be eliminated. Thus we can define the absorption coefficient of QUIC-AB as

α_{QUIC} (ω) = α_{Δ φ = 0} (ω) - α_{Δ φ = 90 °} (ω) .

Compared with the absorption of the fundamental radiation of the ω and 2ω light beams, using this definition, the background induced by other absorption process (e.g., the intraband absorption, the heating effect, and the electro-optic effect) can also be eliminated, and therefore the precision of measurement can be improved. It worth noted that a constant electric field may change the QUIC absorption in a semiconductor due to the nonlinear optical Franz-Keldysh effect [25]. But this effect is quit small, e.g., the amplitude of the differential transmission is smaller than 10⁻⁶ even with a 100 V/cm electric field.

Using Eq. (1), the absorption coefficient of QUIC-AB in the presence of a charge current can be written as

α_{QUIC} (ω) = \frac{π^{2} e^{2}}{\bar{h} c n_{0} m^{2} ω A_{2}} η_{1} A_{1}^{2} ζ_{h, l} P^{2} \times \sum_{k} [f_{v} (k) - f_{c} (k)] δ [ω_{c v} (k) - 2 ω] k cos θ,

where P² = |〈S|P_x|X|〉|², |S〉 and |X〉 are the Kohn-Luttinger amplitudes, and θ denotes the angle between the electron wave vector k and the polarization direction of laser beam a_l, ζ_h,l is a coefficient dependent on the mixing of the light- and heavy- hole states. For bulk GaAs, the hole states can be obtained from the multiband Luttinger-Kohn (LK) effective-mass Hamiltonian [24, 26, 27]. Thus we can find that

ζ_{h} = \frac{1}{c_{h}^{2}} [{(R_{h} + E_{+}^{'})}^{2} - ({| M |}^{2} + {| L |}^{2}) / 3]

for the transition from the heavy valence band to the conduction band, and

ζ_{l} = \frac{1}{c_{h}^{2}} [{| M |}^{2} + {| L |}^{2} + {(R_{h} - E_{-}^{'})}^{2} / 3]

for the transition from the light valence band to the conduction band [24], where

E_{\pm}^{'} = \pm \sqrt{{| M |}^{2} + {| L |}^{2} + R_{h}^{2}}

is the energy difference between the hh and lh bands, R_h = −γ₂(sin² θ_e − 2 cos² θ_e),

M = - \sqrt{3} γ_{2} {sin}^{2} θ_{e} e^{- 2 i φ_{e}}

,

L = i 2 \sqrt{3} γ_{3} sin θ_{e} cos θ_{e} e^{- i φ_{e}}

, γ₂ and γ₃ are the Luttinger parameters, θ_e is the polar angle of wave vector k,

Thus, the difference in absorption between that with and that without an electron current can be written as $ℜ (ω, Δ φ) = \frac{e^{- α (ω, Δ φ) L} - e^{- α^{0} (ω, Δ φ) L}}{e^{- α^{0} (ω, Δ φ) L}} \approx [α^{0} (ω, Δ φ) - α (ω, Δ φ)] L$ , where L is the thickness of the sample, and α(ω, Δφ) [α⁰ (ω, Δφ)] is the corresponding absorption coefficient with [without] a electron current. Thus, the differential absorptivity of QUIC-AB is

Δ 𝒜_{QUIC} (ω) = ℜ_{Δ φ = 0} (ω) - ℜ_{Δ φ = 90 °} (ω) \approx [α_{QUIC}^{0} (ω) - α_{QUIC} (ω)] L,

where

α_{QUIC} (α_{QUIC}^{0})

is the corresponding QUIC-AB absorption coefficient with (without) a electron current.

When valence states are fully occupied or the holes in the valence-band follow a uniform angular distribution in k-space (i.e., f_v(k) ≡ f_v(k)), from Eq. (3) we can get $α_{QUIC}^{0} - α_{QUIC} \sim \frac{1}{2} \sum_{k} [f_{c} (- k) - f_{c} (k)] δ_{ω_{c v}, 2 ω} k cos θ$ , which is linearly proportional to the current density along the optical pulse polarization direction $J_{e}^{a_{l}} = - e \sum n_{e} (k) v_{e} (k) • a_{l} \sim - \frac{\bar{h} e}{m_{e}} \sum_{k} [f_{c} (k) - f_{c} (- k)] k cos θ$ . Therefore, using the QUIC-AB, the direction and amplitude of the electron current can be detected directly.

3. Numerical results

A numerical simulation is shown in Fig. 2. In this example, an electric field is applied in the plane of the n-doped GaAs sheet and along the −x direction. Thus, the distribution of the electrons can be described by f(k) = f_e(k − k_e) [15,16,28], where f_e(k) = {exp[(E_k − E_F)/k_bT]+ 1}⁻¹ is the equilibrium Fermi function, E_F the Fermi energy, k_e = −eτE_D/h̄ is the electric field induced shifts in k space in the relaxation approximation, E_D and τ is the applied electric field strength and the electron relaxation time, respectively. Thus the whole Fermi surface has been shifted in the +x direction, the occupation number of electrons at +k is larger than that at −k [see Fig. 2(a)], and a DC current is flowing in the −x direction. The corresponding QUIC-AB is shown in Fig. 2(b) for a 1 μm GaAs sheet with E_F = 50 meV, τ_e = 0.1 ps, E_D = 5 V/cm, and T = 77 K. Since the electron occupation number at +k_x is larger than that at −k_x, the absorption of probe laser beam decreases when the polarization of ω and 2ω optical pulse is in the x direction with φ₁ = φ₂ = 0 and Δφ = 0 (i.e., θ_al = 0°), a negative QUIC-AB is achieved. But when the polarization of ω and 2ω optical pulse is in the x direction with φ₁ = φ₂ = π and Δφ = π (i.e., θ_{a_l} = 180°), the absorption of probe laser beam increased because of the low electron occupation number at −k_x, and the QUIC-AB is positive. When the sheet current density is about 1 A/cm, the correspond differential absorption is $Δ 𝒜_{QUIC} / J_{e}^{x} \approx 1.4 \times 10^{- 3}$ . Since the resolution of the differential transmission ΔT/T in the experiment can be high as 10⁻⁶ by the phase control technique [21], the QUIC-AB can resolve the current difference smaller than 0.001 A/cm. From Fig. 2(b) we can also find that absorption peak appears at E_2ω = 1562 meV, which is slight lager than the Femi energy E_F at 77 K, and the full width at half maximum (FWHM) of the absorption peak is about 35 meV. When the tuning E_2ω − E_F is about 30 meV, the correspond differential absorption $Δ 𝒜_{QUIC} / J_{e}^{x} \approx 2 \times 10^{- 4}$ .

Fig. 2 (a) Polar contour plot of the electron distribution function f_e(k,θ) − f_e(k,θ + π) under an electric field. (b) Polar contour plot of QUIC-AB $Δ 𝒜_{QUIC} / J_{e}^{x} \times 10^{6}$ as a function of the energy E₂_ω and the angle θ_{a_l} between the electron wave vector and the polarization direction of laser beam.

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We now turn to the discussions for the influence on the QUIC-AB from the temperature, applied electric field strength, and relaxation time τ_e as shown in Fig. 3. In k-space, the electrons forming the current are located at the Fermi energy surface. Therefore, the energy spreading of the inject current is about 2k_BT, where k_B is the Boltzmann constant. Thus, the energy dispersion of the inject current in k-space will increase linearly with the temperature. But the optical absorption depends both on the optical transition line width and the energy of the electrons. If the optical transition energy of the corresponding electron current is far from the center energy of laser beam, these electrons have little contribution to the QUIC-AB. The current resolution of QUIC-AB will decrease with temperature increase [see Fig. 1(a)]. The applied electric field and relaxation time τ_e have small effect on the current resolution of QUIC-AB because electric field induced shifts k_e is usually quite small. Only with an strong applied electric field and slow relaxation, the the current resolution of QUIC-AB decreases rapidly [see Fig. 3(b)].

Fig. 3 (a) The maximum QUIC-AB $Δ 𝒜_{QUIC} / J_{e}^{x} \times 10^{6}$ as a function of temperature. (b) The maximum QUIC-AB $Δ 𝒜_{QUIC} / J_{e}^{x} \times 10^{6}$ as a function of electron relaxation time τ_e with different applied electric field, E = 5 V/cm (black solid line), E = 10 V/cm (red dashed line), E = 30 V/cm, (blue dotted line).

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Finally, we discuss the experimental realization of our theory predication. Benefitted from ultrafast laser technology, the time resolution of the pump-probe detection can be down to the femto-second. However, the pump and probe pulse length should not be too short. A short pulse with a too large energy range will lower the energy resolution and reduce the interference term $W_{k}^{i}$ . As the probe pulse also injects a pure charge current, a relatively low energy density 2ω probe pulse must be chosen. In the detection of QUIC-AB, there is no spin or charge accumulation in real space. Thus, laser beams need not be focused, and the disturbance of edge states can be avoided.

4. Conclusion

In conclusion, the momentum-resolved and time-resolved QUIC-AB is investigated. The QUIC-AB can be used to detect both the amplitude and the direction of the charge current in real time. The QUIC-AB may play an important role in investigating the ultrafast transport process or ultrafast electronics and spintronics.

Acknowledgment

We would like to thank Hui Zhao for fruitful discussion. This work was supported by the NSFC Grant Nos. 10904059, 10904097, and 11004199.

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Proposal for the momentum-resolved and time-resolved optical measurement of the current distribution in semiconductors

Abstract

1. Introduction

2. Theory

3. Numerical results

4. Conclusion

Acknowledgment

References and links

Cited By

Figures (3)

Equations (5)

Optics Express