Optical properties of Li 1+x Ti 2 O 4 (x = 0.10 and 0.15)

Chapter 7

Optical properties of Li _1+x Ti ₂ O ₄ (x = 0.10 and 0.15)

Optical reflectivity and Raman-scattering measurements of Li 1+x Ti 2 O 4 (x =0.10 and 0.15) were investigated in this work. These polycrystalline samples were growth by Y. C. Liao et al. ^[34]

7-1 Optical reflectance

The optical reflectivity spectra of both samples are shown in Fig. 7 - 1 (a). The optical conductivity spectra are shown in Fig. 7 - 1 (b). An expanded plot of the optical conductivity in the infrared regime of Li 1.1 Ti 2 O 4 is shown in Fig. 7 - 2, where the fitting curves based on Drude-Lorentz model are also presented. There are two notable features seen in the optical conductivity spectra: first, an inter-band transition at around 38000 cm ^-1 which can be associated with charge-transfer transitions between O 2p to Ti 3d states. A first-principle theoretical band structure calculation of LiTi 2 O 4

shows a charge-transfer transition energy of about 3.2 eV between O 2p and Ti 3d states, which is compatible with our observation; ^[120] second, in the infrared regime, the optical conductivity can be described as in terms as a Drude behavior, a broad peak at about 380 cm ^-1 , and an over-damped mid-infrared component. The mid-infrared band centered around 2200 cm ^-1 , which is usually observed in high-T C

cuprate oxides. This feature might be due to the electron-electron correlation. ^[112] The

broad peak around 380 cm ^-1 , which is present in spectrum of Li 1.1 Ti 2 O 4 , might be

related to the disorder band. We can extrapole the frequency-dependent optical conductivities to the zero frequency for these samples. The values of the dc conductivity have the same order of magnitude with those obtained from dc resistivity measurements, which are labled on the vertical axis in Fig. 7 - 1 (b) with red (x = 0.15) and black (x = 0.10) square. Besides, the value of the x = 0.15 sample is larger than that of x = 0.1 sample in dc and optical conductivity cases, consistening with Li doping effect. The absoult value of the room-temperature Drude plasma frequency (~

1570 cm ^-1 for the Li 1.1 Ti 2 O 4 sample) implies a carrier density as n = 2.76 10 × ¹⁹ cm ^{− 3} under the assumption of m ^* = m _e (free electron mass). Notably, this value is much smaller than that obtained from first-principle theoretical band structure calculation ( n = 2.7 10 × ²² cm ^{− 3} ), which might be due to the large value of effective mass of carriers.

To get more understanding about the carrier density from the optical spectra, a useful method, called partial sum rule, is usually be applied. In a solid, the integral weight of the whole optical conductivity spectrum should be proportional to the number of valence electrons. The relation is called sum rule which can be written as ^[62]

2

1 *

0

( ') '

2 _cell ^cell

d e N

m V σ ω ω π

∞

∫ = . (7.1.1)

Then, the partial sum rule can be written as following integral formula,

* 2 1

0

( ) 2 ( ') '

c

cell

eff c

cell

N m V d

e N

ω ω σ ω ω

= π ∫ , (7.1.2)

where m and ^* e are effective mass and charge of electron, respectively. V and _cell

N are the volume and valence electron number of unit cell. N represents the

number of electrons participating in the optical response under the incident photon ω c

= in single unit cell. Therefore, ω _c is usually called cut-off frequency. Using the partial sum rule, we obtain the effective numbers of electrons, shown in Fig. 7 - 3.

Immediately, we can see the effective electron number of Li 1.15 Ti 2 O 4 is larger than Li 1.1 Ti 2 O 4 , suggesting that the Li doping indeed increases the carrier density of Li 1+x Ti 2 O 4 . Another feature is that both curves show fast increasing as the incident photon energy larger than 30000 and 28000 cm ^-1 for x = 0.1 and 0.15, respectively.

This frequency is the boundary between intraband and interband transition, or in other words, the charge transfer frequency. The effective electron number below charge transfer frequency is related to the number of electrons in the valence band, which interact with all the low energy excitations. These characteristic values of effective carrier density, n tot , of Li 1.1 Ti 2 O 4 and Li 1.15 Ti 2 O 4 are 0.195 × 10 cm ^{21 − 3} and 0.274 × 10 cm ^{21 − 3} , respectively. In Fig. 7 - 4, for comparison, we show the relation of T C and n tot of Li 1+x Ti 2 O 4 and other superconductors, which are in the under- or optimal doping regions. ^[112] Although the carrier density and superconducting critical temperature of Li 1+x Ti 2 O 4 compounds is much smaller than high-T c superconductors, they obey the universal linear correlation.

7-2 Raman scattering

Fig. 7 - 5 shows the room-temperature Raman scattering spectra of Li 1.1 Ti 2 O 4 and

Li 1.15 Ti 2 O 4 . Four observed phonon peaks at around 340, 432, 498, and 627 cm ^-1 are

assigned to be F 2g (3), E g , F 2g (2), and A 1g modes, respectively. ^[113,114] The sketch of

vibration of these phonon modes is plotted in Fig. 7 - 6. Temperature-dependent

Raman spectra are shown in Fig. 7 - 7. There are several important features: (i) The

shape of spectrum doesn’t change dramatically as temperature was varied, indicating

the crystal structure remain unchanged; (ii) As temperature is lowered, all phonon

modes are hardening and narrowing, and (iii) The background intensity is increasing

as temperature is decreasing. To quantitatively analysis the changes of the phonon

frequencies with decreasing temperature, a standard Lorentzian profile, Eq (3.2.22),

was used to fit the phonon modes. We show the fitting result in Fig. 7 - 8, where the

individual phonon modes are plotted with different colors. All fitting parameters were

listed in the following Table 7.1.

Table 7.1. Parameters of the Lorentzian fit for the temperature-dependent Raman spectra data of two samples.

x = 0.1 (cm ^-1 ) 10 K 50 K 100 K 150 K 170 K 200 K 250 K 300 K ω 1 347.6 347.3 345.0 342.8 342.4 343.3 339.9 339.8 Γ 1 12.9 16.2 17.2 20.1 22.7 19.9 28.6 25.3 F 2g (3)

A 1 0.8 1.0 0.9 1.0 1.0 0.8 1.3 0.8

ω 2 439.9 441.2 439.3 436.9 435.0 434.6 430.3 430.9 Γ 2 11.5 13.3 14.8 16.0 18.2 20.1 26.6 25.2 E g

A 2 0.9 1.0 1.0 0.9 0.9 0.9 1.0 0.8

ω 3 503.3 503.9 502.1 500.3 500.3 499.6 498.1 498.5 Γ 3 15.6 16.3 16.8 19.9 19.8 22.0 18.8 20.0 F 2g (2)

A 3 1.0 1.0 0.9 0.8 0.8 0.7 0.7 0.6

ω 4 629.7 630.0 629.2 629.0 628.2 629.8 626.4 626.7 Γ 4 15.8 16.2 18.6 23.5 27.7 30.7 22.6 47.2 A 1g

A 4 0.9 0.9 0.9 1.0 1.0 0.9 0.6

BG A 1.0 0.8 0.9 0.8 0.7 0.6 0.4 0.2

x = 0.15 (cm ^-1 ) 10 K 50 K 100 K 150 K 200 K 250 K 300 K ω 1 347.6 347.7 345.8 342.8 342.4 342.0 339.9 Γ 1 15.5 17.3 19.1 26.4 26.1 29.1 30.9 F 2g (3)

A 1 0.6 0.6 0.7 0.7 0.7 0.5

ω 2 439.9 440.7 438.6 436.4 434.7 433.6 431.7 Γ 2 13.7 11.9 13.6 16.2 16.6 20.7 21.4 E g

A 2 0.8 0.7 0.7 0.6 0.7 0.4

ω 3 504.5 504.7 503.3 501.2 500.7 499.0 497.5 Γ 3 14.4 14.6 15.2 16.6 19.5 18.9 24.4 F 2g (2)

A 3 0.9 0.8 0.8 0.7 0.8 0.4

ω 4 633.2 632.0 630.3 630.9 628.8 629.6 626.8 Γ 4 40.7 31.1 25.8 34.9 30.7 34.6 36.3 A 1g

A 4 1.0 0.9 0.7 0.6 0.7 0.4

BG A 1.0 0.6 0.7 0.4 0.4 0.3

Here, the ω _i and Γ denote the frequency and linewidth of phonon peaks. _i A , _i

related to the area of phonon peak, is shown by the normalized value. Besides, the

normalized area of background (BG) is also listed in Table 7.1. To trace the

temperature dependence of the phonon frequency and linewidth more clearly, ω _i and Γ vs. temperature plots are presented in Fig. 7 - 9 and Fig. 7 - 10. Obviously, all the i

phonons show hardening and narrowing with decreasing temperature. However, we did not see any anomalous around the Curie temperature, which are 180 and 170 K for the x = 0.1 and 0.15 samples, respectively. The phonon frequencies harden only in the order of several wavenumber. This temperature dependence of phonon can be fitted with the anharmonic effect model. ^[115,116]

Anharmonic effect is involved with the higher-order terms of the atomic vibrations beyond traditional harmonic term. In the words of many-body physics, anharmonic effect includes the process which an optical phonon decays into two or three phonons. Fig. 7 - 11 shows the sketch of anharmonic processes of Raman-active phonons. It describes the scattering process involving the absorption of incident photon = ω _I , the emission of a photon = ω _S , and the creation of an optical phonon

o K j

which then decays via anharmonicity into two phonons , three phonons , and etc.

Fig. 7 - 11 (a) and (c) show the processes of which one phonon decays into two and three phonons. At non-zero temperature, the anharmonic processes can occur in which the decay of o K _j

is accompanied by the absorption of another phonon, and then emission one or two phonons, which are the case of Fig. 7 - 11 (b) and (d). The phonon frequency and linewidth as function of temperature can be written as following equations.

0 2

2 3 3

( ) (1 ) (1 )

1 1 ( 1)

x y y

T A B

e e e

Γ = Γ + + + + +

− − − , (7.1.1)

0 2

2 3 3

( ) (1 ) (1 )

1 1 ( 1)

x y y

T C D

e e e

ω = ω + + + + +

− − − , (7.1.2)

0 2 _B ; 0 3 _B

x = = ω k T y = = ω k T , (7.1.3)

where Γ , ₀ ω ₀ , A, B, C and D are constants. The second terms of Eqs. (7.1.1) and (7.1.2) are related to the processes involved with decay into two phonons while the third terms are related to the processes involved with decay into three phonons. In our experimental conditions, sample temperature is not very high, so the influence of third terms can be neglected. The fitting results of temperature-dependent phonon parameters of F 2g (2) mode of Li 1.15 Ti 2 O 4 are shown in Fig. 7 - 12 and Fig. 7 - 13. The agreement between the experimental points and simulation curves is reasonable good.

We also obtained the value of Γ , ₀ ω ₀ , A, and C from the fitting results. They are 12.3, 505.7, 1.1 and -0.8 cm ^-1 , respectively. The “+” sign of parameter A indicates the linewidths widen with increasing temperature. In contrast, the “-“ sign of parameter C indicates the phonon frequencies soften with increasing temperature. In addition, if the temperature goes to zero, the value of x and y in Eq. (7.1.3) become infinity, and then only the constants A, B, C and D remain in Eqs. (7.1.1) and (7.1.2). Therefore, the phonon frequencies and linewidth at zero temperature could be obtained. All the fitting parameters are list in Table 7.2.

Table 7.2. The resulting parameters which are obtained by the anharmonic effect model.

Unit (cm ^-1 ) ω ₀ Γ ₀ A C ω ( T = 0) Γ ( T = 0) F 2g (3) 348.2 12.8 1.0 -0.6 347.6 13.7

E g 442.5 9.5 1.4 -1.0 441.4 10.9 F 2g (2) 504.2 15.8 0.5 -0.6 503.6 16.2 x = 0.1

A 1g 630.6 8.2 4.3 -0.5 630.1 12.5 F 2g (3) 348.4 14.6 1.2 -0.6 347.9 15.8 E g 441.6 11.0 0.9 -0.8 440.8 11.9 F 2g (2) 505.7 12.3 1.1 -0.8 505.0 13.4 x = 0.15

A 1g 633.3 27.2 1.0 -0.7 632.6 28.2

According to our analysis, the temperature dependence of the observed

Raman-active phonons in Li 1+x Ti 2 O 4 is mainly due to the anharmonic effect. In

another words, it is nothing but thermal expansion effects. The interesting phenomena of these compounds are the fact that there are no any excitations coupling with phonons, i.e. lattice degree of freedom. It is very different from the situation in high-T c cuprate oxides and manganite oxides. ^[111,92]

7-3 Summary

There is no structural phase transition, which could be evidenced by the temperature-dependent Raman-scattering spectra. Our result shows only typical thermal expansion effect. In the infrared conductivity spectrum, there is a mid-infrared band around 2000 cm ^-1 which is associated with electron-electron correlation. This mid-infrared band is usually seen in high-T c superconductors.

Besides, the relation between T c and n tot of these two samples follows the universal

correlation of high-T c cuprates.

Fig. 7 - 1. (a) The room-temperature optical reflectivity and (b) the optical conductivity spectra of Li 1+x Ti 2 O 4 ( x = 0.10 and 0.15).

0 1000 2000 3000 4000

0 50 100 150

Op tic al co n du cti vi ty ( Ω -1 cm -1 )

Frequency (cm-1)

Experimental data Fitting result Drude peak Lorentz peak1 Lorentz peak2

Fig. 7 - 2. Black line is the optical conductivity which was calculated from measured

reflectance of x = 0.10 sample at 300 K. Color lines are Drude-Lorentz fit to the data.

0 20000 40000 0.0

0.4 0.8 1.2

N

( ω )

Frequency (cm

) Li

Ti

O

Li

Ti

O

Fig. 7 - 3. The effective carrier number as a function of the photon energy for two samples.

0 1 2 3 4 5 6

0 10 20 30 40 50 60 70 80 90 100 110 120

T

(K )

n

(10

cm

) LiTiO

BKBO La214 Bi2212 Y123 Tl2212

Fig. 7 - 4. The superconducting transition temperature (T c ) as a function of n tot for Li 1+x Ti 2 O 4 (LiTiO, black square), Ba 1-x K x BiO 3 (BKBO, red circle), La 2 CuO 2.14

(La214, green triangle), Bi 2 Sr 2 CaCu 2 O 8+

(Bi2212, blue triangle), YBa 2 Cu 3 O 7-

(Y123, diamond), and Tl 2 Ba 2 CaCu 2 O 8 (Tl2212, magenta triangle). The datas of

100 200 300 400 500 600 700 800

E _g

A _1g F _2g (2)

In te ns ity (a rb . u ni ts)

Raman shift (cm ^-1 ) x = 0.15

x = 0.10

λ = 514.5 nm T = 300 K Li _1+x Ti ₂ O ₄ F _2g (3)

Fig. 7 - 5. The Raman scattering spectra of Li 1.1 Ti 2 O 4 and Li 1.15 Ti 2 O 4 at room temperature. Red arrows indicate the identified Raman-active phonons.

Fig. 7 - 6. Atomic displacements of the Raman-active phonon modes. ^[113,114]

Raman Shift (cm ^-1 )

200 400 600 800

200 400 600 800

Raman shift (cm ^-1 ) (b) x = 0.15

10 K

150 K 250 K 300 K

Intensity (arb. uni ts)

Raman shift (cm ^-1 ) (a) x = 0.10

10 K

100 K

200 K

300 K

Fig. 7 - 7. The temperature-dependent Raman-scattering spectra of (a) Li 1.1 Ti 2 O 4 and (b) Li 1.15 Ti 2 O 4 .

100 200 300 400 500 600 700 800

In te ns ity (a rb . u n its)

Raman shift (cm ^-1 )

Exp Fit

Background F _2g (3) E _g F _2g (2) A 1g

Li 1.15 Ti

2 O

4

300 K

Fig. 7 - 8. The 300-K Raman spectrum compared to the Lorentzian fitting results for the x = 0.15 compound.

Raman Shift (cm ^-1 ) Raman Shift (cm ^-1 )

Raman Shift (cm ^-1 )

0 100 200 300 12

18 24 30 36 42 48 340

342 344 346 348

0 100 200 300 12

16 20 24 28 32

432 436 440

0 100 200 300 8

12 16 20 24 28

32 498

499 500 501 502 503 504

0 100 200 300 14

16 18 20

22 626

627 628 629 630

Temperature (K)

F req uen cy (c m

) F

(3)

Li ne wid th (cm

)

Temperature (K)

E

Temperature (K)

F

(2)

Temperature (K)

A

Fig. 7 - 9. Frequencies and linewidths of phonons as a function of temperature for Li 1.1 Ti 2 O 4 .

0 100 200 300 24

30 36 42

0 100 200 300 16

20 24 28 32

432 436 440

0 100 200 300 12

16 20 24

496 498 500 502 504 506

0 100 200 300 16

20 24

626 628 630 632 634

340 342 344 346 348

Temperature (K) Temperature (K)

Line widt h ( cm

)

F

(3)

Temperature (K) E

Temperature (K) F

(2)

F req uen cy (c m

) A

Fig. 7 - 10. Frequencies and linewidths of phonons as a function of temperature for

Li 1.15 Ti 2 O 4 .

Fig. 7 - 11. Diagrams representing three- and four-phonon anharmonic processes contributing to the decay of the Raman-active phonon mode. ^[116] (a) One phonon decays into two phonons; (b) One phonon absorbs another phonon; (c) One phonon decays into three phonons; and (d) One phonon absorbs another phonon and then decays into two phonons.

0 50 100 150 200 250 300

497 498 499 500 501 502 503 504 505 506

Lorentian fitting parameters Anhamonic simulation Fre que ncy (cm -1 )

Temperature (K)

Li _1.15 Ti ₂ O ₄ F _2g (2)

ω ₀ = 505.7 cm ^-1 C = -0.8 cm ^-1

Fig. 7 - 12. Anharmonic effect modeling of the F 2g (2) phonon frequency for the x =

0.15 compound.

0 50 100 150 200 250 300 12

14 16 18 20 22 24

26 Li _1.15 Ti ₂ O ₄ F _2g (2)

Linewidt h ( cm -1 )

Temperature (K)

Lorentian fitting parameters Anhamonic simulation Γ ₀ = 12.50 cm ^-1

A = 0.959 cm ^-1

Fig. 7 - 13 Anharmonic effect modeling of F 2g (2) phonon linewidth for the x = 0.15

compound.