ivchenko_bookreg

where Iss = ps ω/(η0τs).

The measurements illustrated in Fig. 8.8 indicate that the photocurrent jx at a low power level depends linearly on the light intensity and gradually saturates with increasing intensity, jx I/(1 + I/IsL,C), where IsL,C is the saturation parameter for linearly and circularly polarized radiation. This corresponds to a constant absorbance at low values of I and decreasing absorption with rising I. From (8.86, 8.87) we obtain

One can see from Fig. 8.8 that the measured saturation intensities IsL,C are di erent, namely IsC < IsL. This is in agreement with theory. The saturation of the absorption of linearly polarized radiation is governed by the energy relaxation time τε whereas in case of the circular polarization it is governed by both τε and τs. If τs is of the order of τε or larger, the saturation becomes spin sensitive and the saturation intensity of circularly-polarized radiation drops below that for the linear polarization.

Taking into account that Iss = IsLIsC/(IsL − IsC) and using the measured values of IsL, IsC one can estimate the parameter Iss = ps ω/(η0τs) and even the time τs. The latter is possible if the absorbance η0 is known from an independent experiment or theoretical calculation, see details in [8.40, 8.41].

8.6 Chirality E ects in Carbon Nanotubes

The point-group symmetry of a chiral medium makes no di erence between polar and axial vectors. Thus, by definition, the chirality e ects are those where a polar vector and an axial vector (or a pseudovector) are interconnected by a phenomenological equation. Here we consider chirality related e ects in photogalvanic, electron-transport and optical properties of carbon nanotubes. For this purpose we apply the e ective mass theory used previously in Chaps. 3, 4 for the analysis of absorption spectra of one-dimensional structures. It will be shown that the chiral properties are determined by terms in the electron e ective Hamiltonian describing the coupling between the electron wave vector along the tube principal axis and the orbital momentum around the tube circumference (Chap. 2). We derive equations for the circular photocurrent and magneto-chiral dc electric currents, which are linear in the external magnetic field and quadratic in the bias voltage or amplitude of the electro-magnetic field. Moreover, we perform analytic estimations for the natural circular dichroism and magneto-spatial e ect in the light absorption.

392 8 Photogalvanic E ects

8.6.1 Circular Photocurrent in Nanotubes

In chiral nanotubes, the circular PGE is an electron analog of a screw tread or a plane with a propeller. In this case the tensor γ in the general phenomenological equation (8.1) has only one nonzero component γzz and the dc circular photocurrent is described by

where γ ≡ γzz is a real coe cient. For a wave propagating along the nanotube principal axis z, one has instead of the above equation

We remind that the circular photocurrent reverses its direction under inversion of the light circular polarization and vanishes for linearly-polarized excitation.

The consideration of the circular PGE and spin-galvanic e ect in QWs in Sects. 8.1 and 8.2 is based on allowance of spin-dependent linear-in-k terms in the electron e ective Hamiltonian. In carbon nanotubes the spin-orbit interaction is negligible and the similar role is played by the coupling between n and kz described by (2.82). Here we take a chiral carbon nanotubes with ν = 1, assume that the tube is n-doped and consider the direct intersubband optical transitions of electrons from the lowest to the first excited conduction subbands. Taking into account the conservation of the angular momentum z-component the allowed transitions are as follows: (c, n = 0, kz , K) → (c, n = 1, kz , K) under σ+ excitation and (c, 0, kz , K) → (c, −1, kz , K) under σ− excitation. In accordance with (8.90) they lead to circular photocurrents of opposite polarities.

In a 1D system with parabolic subbands, only two values of kz satisfy the energy conservation law

¯ + −

Their average value kz = (kz + kz )/2 is nonzero because of the linear-kz terms. The electron average velocities in the excited and initial subbands coincide and are equal to

m1β1 − m0β0 v¯z = (m1 − m0) .

It follows then that the photoexcited carriers contribute to an electric current before they loose their average velocity because of the momentum relaxation. Under the steady-state optical excitation this mechanism leads to the photocurrent

where W10 is the probability rate for the intersubband transitions and τp(n) is the momentum scattering time in the n-th subband. We ignore in (8.91) the small frequency region where the terms linear-in-kz exceed or are comparable with the terms quadratic in kz . Then one can use the expression [8.42]

derived for βn = 0. Here I is the light intensity in units energy · length−1 · time−1, k ,0 is the size-quantized wave vector (2.78) at n = 0, the 1D reduced density of states equals to

the factor of two makes allowance for the spin degeneracy, θ(x) is the step function, −µ−101 is the inverse reduced e ective mass m−1 1 − m−0 1 [since m1 > m0, a value of µ10 is positive and g10(E) is defined for negative values of E], and fc0(E0) is a value of the equilibrium distribution function at the energy E0 = ( 2kz2/2m0) = (µ10/m0)(∆10 − ω) = 2(∆10 − ω).

Equation (8.92) gives one of four possible contributions to the circular PGE. Three others are also related to the photoexcitation asymmetry in the kz space. They arise due to the kz dependence of the optical matrix element, momentum relaxation time and electron equilibrium distribution function. A sum of all four contributions is given by

where the lengths lv , lm, lτ , lf are related to the photoexcitation asymmetry arising due to the kz -dependence of the velocity and density of states (lv ), of the squared matrix element (lm), of the momentum relaxation time (lτ ) and of the equilibrium distribution function (lf ). For the optical transitions under consideration the straightforward derivation results in [8.42]

Here f 0 is the equilibrium distribution function, we assume that in equilibrium the upper subband (c, 1, K) is unoccupied, and E0 is the initial energy of an electron photoexcited from the subband n = 0. Obviously, the momentum relaxation time τp(1) is shorter than τp(0) because a photoelectron excited

394 8 Photogalvanic E ects

to the subband (c, n = 1) can be readily scattered to the subband (c, n = 0). For crude estimations one can use the equation

which is just another form of the estimation (8.12). For the band parameters

˚

of carbon nanotubes, L < 100 A and under the saturation of the optical transitions (high-intensity pulsed laser radiation), the photocurrent jCP GE can amount a value of 10−9 A. It can be significantly enhanced in recently discovered carbon-nanotube crystals [8.43]. Note that in undoped nanotubes the circular PGE disappears because the contribution to the circular photocurrent due to interband optical transitions from (v, n, K) to (c, n + 1, K) is completely compensated by that due to transitions from (v, −n − 1, K ) to (c, −n, K ) as a result of a particular charge conjugation symmetry of carbon nanotubes.

We conclude this subsection by comparing the circular photocurrent (8.96) with the photon drag current which is independent of the sign of the circular polarization. In accordance with (8.61) the photon drag e ect is

where q is the photon wave vector (in vacuum q = ω/c). Thus, for ω = 0.1 eV we have jCPGE/jPD β0m0/( 2q) ≈ 3.

It is worth-while to remind that, in addition to the circular PGE, in noncentrosymmetric media a photocurrent of another kind can be induced by the electro-magnetic wave. It is the linear PGE described in (8.1) by a third-rank tensor χλµν symmetrical with respect to interchange of the indices µ and ν. Ideal carbon nanotubes are unpolar with their principal axis, z, being two-sided which forbids nonzero components χzµν . In BxCy Nz nanotubes the symmetry is reduced, the principal axis is polar and the linear PGE becomes allowed as predicted in [8.44]. In the next subsection we show that, in the presence of an external magnetic field B z, the photocurrent can be induced by linearly-polarized or unpolarized photoexcitation even in an ideal carbon nanotube.

8.6.2 Magneto-Chiral Currents and Optical Absorption

Other chirality e ects allowed in chiral nanotubes are as follows.

(a) Magneto-chiral photogalvanic e ect described by

It was predicted in [8.1] for bulk gyrotropic crystals and observed first in QW structures [8.45,8.46] (ez , e are the longitudinal and transverse components of the polarization unit vector e, Bz is the external longitudinal magnetic

field). For the interband transitions (v, 0) → (c, 0) excited in an undoped carbon nanotube by the linearly polarized light E z, one has the following estimation for the magneto-chiral photocurrent

where Φ is the magnetic flux, Bz L2/4, passing through the cross section of a tube and Φ0 is the magnetic flux quantum, c /e. For Bz = 10 T and L = 100

˚ ≈ · −2

A, the ratio Φ/Φ0 2 10 . While deriving the above equation we took into account that, in an external magnetic field, in the expression for the quantized value of kn one should change n by n + (Φ/Φ0). Then, for ν = 0, the band gaps ∆(K) and ∆(K ) di er and the contributions from the K and K valleys to the magneto-chiral photocurrent do not compensate each other.

(b) Magneto-chiral conductivity described by the second term in the

predicted for gyrotropic crystals [8.47] and simple spiral nanotubes [8.48] and observed recently by Rikken et al. in chiral carbon nanotubes [8.49] (Fz is the dc electric field, σ is the conventional conductivity). The magnetochiral conductivity becomes nonzero if one takes into account that in the case of a chiral carbon nanotube the electron-phonon deformation potential has contributions both even and odd in kz . For a doped nanotube with the Fermi energy EF one can obtain [8.42]

where the parameter γ is the tight-binding parameter defined in (2.72). For

(c) Natural optical activity and circular dichroism, the latter described by

with qz being the z-component of the light wave vector and W (σ+), W (σ−) being the absorption rates for the σ+ and σ− circularly-polarized radiation. For carbon nanotubes, the e ects (c) were considered theoretically in the framework of the microscopic tight-binding model by Tasaki et al. [8.50]. Below we present an estimation for the natural dichroism derived for the interband optical transitions taking place in the vicinity of the gap, ∆cv10, between the (v, 1, K) valence and (c, 1, K) conduction bands

The product βqz can be estimated as 0.1 meV. It is instructive to present here as well an estimation for the magneto-induced dichroism

396 8 Photogalvanic E ects

(d) Magneto-chiral absorption (or emission) described by the second term in the equation for the light absorption (emission) probability rate

and observed in few chiral media, see [8.51,8.52] and references therein. Note that the first magneto-spatial dispersion e ect was reported by Gross et al. [8.53]). The magneto-chiral correction is expected to be more remarkable for optical transitions in the vicinity of the gap ∆cv00. In this case it is given by

It is worth to mention e ects which are inverse to the circular and magneto-chiral PGEs. In the absence of magnetic field the dc current in a carbon nanotube should induce the circular dichroism of the optical absorption or the circular polarization of the photoluminescence. Up to now the e ect of the electric current on the optical activity has been observed only in bulk gyrotropic tellurium [8.54]. In the presence of an external magnetic field parallel to a nanotube the current jz should induce a change in the photoluminescence intensity proportional to jz Bz .

9 Conclusion

“It is not the end yet!”

Hexagram N 64 of Chinese “Book of Changes”

At the opening and concluding sessions of International Conferences on the Physics of Semiconductors (ICPS) the speakers are traditionally encouraged to describe in broad terms the state and future of semiconductor physics. In his introduction to the 6th ICPS (Exeter, 1962) Sir Nevill Mott predicted the death of semiconductor physics on about a ten year time scale. In the closing address at the 11th ICPS (Warsaw, 1972) J.J. Hopfield reminded those words of Mott. Nevertheless, he also expressed his own doubts in the future of semiconductor physics as a part of the fundamental physics and, among clear promises for continued science, mentioned “systems of lower dimensionality” and “opto-electronics”. In fact, the expectations that in the seventies-eighties all fundamental problems would have been solved were rather strong. Surprisingly, the approaching crisis had been resolved in a positive way. It is true, nowadays the interest in investigation of bulk semiconductor crystals is limited. However, they represent only a small area of a big field occupied mainly by semiconductor nanostructures. L.J. Sham openly expressed his optimism in the program summary at the 17th ICPS (San Francisco, 1984) and predicted the continued flourishing of semiconductor physics. More recently, in his welcome address to participants of the 25th ICPS (Osaka, 2000), H. Yoshikawa emphasized that “semiconductor physics is one of the most important research areas of basic physics, and many important discoveries or the creation of new concepts in this area have greatly influenced the other research fields of basic science”.

At present the optimism in liveliness of semiconductor physics is founded on the exciting progress in nanotechnology. The latter is aimed at the development and use of devices that have a size of only a few nanometers. The top-down and bottom-up approaches to produce nanostructures on a molecular level by a suitable sequence of chemical reactions or lithographic techniques may result in a close and e cient link between physics, chemistry and biology as it took place in physics and chemistry during 30 years after

398 9 Conclusion

appearance of the first semiconductor rectifiers and photocells at the end of 1920’s.

Now it is widely accepted that basic studies are crucial to expose new ideas and opportunities to fabricate by design the three-dimensional complicated nanoobjects by using atoms, molecules, clusters, nanoparticles, superlattices, quantum wells, wires, dots, rings, nanotubes, nanomagnets etc. as building blocks. Optics can play an important role for the characterization and optimization of such objects provided it retains its freedom and status of fundamental science. For example, the extension of quantum electrodynamics to describe the interaction between N -dimensional (N = 0, 1, 2, 3) photons and N -dimensional (N = 0, 1, 2, 3) excitons has direct implication for the development of lasers based on quantum microcavities. The nanotechnology related to making artificial heterostructures of di erent dimensionalities needs the basic knowledge on interfaces and their properties. The interface physics seems to be growing out of its infancy. The better understanding in this area can be also achieved in the basic research of optical and transport interface-dependent phenomena, i.e., the phenomena dependent on the character and orientation of chemical bonds at the interface. And one more example: Semiconductor spintronics has reached a stage of understanding various mechanisms of spin injection, accumulation, detection and manipulation, both optical and nonoptical. But the common feeling is that novel conceptual ideas are needed to trigger the realization of e ective spin devices.

Appendix. Character Tables of Point Groups

Below we present character tables of some point groups and examples of bases of irreducible representations.

Table A.1. Group Cs. The mirror plane σh is perpendicular to the axis x and contains the axes y, z. Ji are components of an axial vector.

e σh

A+ 1 1 y; z; yz; x2; y2; z2; Jx

A− 1 −1 x; xy; xz; Jy ; Jz

Table A.2. Group C2v . The mirror planes σv , σv are perpendicular to the axes x and y respectively, the two-fold rotation axis is parallel to z.

Table A.3. Group D2d. The mirror-rotation axis S4 is parallel to z, the mirror planes σd, σd are perpendicular to the axes x and y respectively, the two-fold rotation axes u2, u2 are obtained from x, y by the rotation around z by 45◦.

400 9 Conclusion

Table A.4. Group C3v . The three-fold rotation axis is parallel to z, one of the mirror planes σv contains the axis x.

Table A.5. Group Td. The symmetry operation σd is mirror reflection with

¯ ¯ ¯

respect to one of these six planes: (110), (110), (101), (101), (011), (011); x, y, z are the principal axes [100], [010] and [001], respectively.