Carrier Temperature Effect

The effect of the carrier temperature on the PPC is studied. We show the AELR as a function of sheet densities on the two temperatures of 100K and 500K in Fig. 4.4.

1E10 1E11 1E12 10

100

Bare interaction With DS

With PPC With HP With HD With SS

Average energy-loss r a tes AELRs (meV/ps)

Carrier density n_2D (cm^-2)

Figure 4.3: Illustration of plasmon-phonon coupling on average energy-loss rate in a 10nm GaAs/Al_.24Ga_.76As quantum well. Bare: the AELR without any effects, DS: with the dynamical screening, HP: with the hot phonon effect, PPC: with the plasmon-phonon coupling (including HP and DS), HD: with HP and DS, SS: with the static screening.

1E10 1E11 1E12 10

100

Average energy-loss rat e s AELRs (m eV/ps) 500K (PPC)

500K (HD)

100K (PPC) 100K (HD)

Carrier density n_2D (cm^-2)

1

1E10 1E11 1E12

1

Deviation ra tio

500K 300K T

_C

=100K

Density (cm

^-2

)

Figure 4.4: Illustration of carrier temperature effect on plasmon-phonon coupling in a 10nm GaAs/Al.24Ga.76As quantum well. HD: with the hot phonon effect and the dynamical screening. PPC: with the plasmon-phonon coupling (including HD). Inset figure shows the deviation ratio caused by the PPC.

The PPC effect on hot carrier relaxations at the carrier temperature of 100K is found to enhance the AELR more strongly than that at 500K. In the inset of Fig. 4.4 we show the deviation ratio ((AELR_PPC-AELR_HD)/AELR_HD) as a function of sheet densities on the different carrier temperatures. It is found that the carrier temperature does not significantly influence the sheet densities where the maximum deviation ratio is shown. As the carrier temperature increases, the enhancement on the AELR by the PPC effect becomes smaller. This is the consequence of hot phonon effect. It more greatly drops the net plasmon-phonon generation rate at higher carrier temperature so that the enhancement of ALER by the PPC effect is shown to gradually disappear.

4.4 Summary

Using renormalized phonon propagators, six plasmon-phonon coupled branches in a quantum well are obtained and used to study the influence of the PPC on hot carrier relaxations. When hot carriers can relax their excess energies via the plasmon-like modes, the considerable enhancement of the AELR by the PPC is found. The effect is significant when the sheet carrier densities are around 10¹¹ cm^-2 and the carrier temperatures are low. At higher sheet densities, the effect gradually evanesces because hot carriers are more difficult to excite the plasmon-like modes due to the increased energies. The enhancement of the AELR also strongly depends on hot phonon effect. Thus, at the carrier temperature as low as possible but higher than³⁶ 40K, decreasing nonequilibrium phonons would lead more noticeable effect of the PPC on hot carrier relaxations around the sheet carrier densities of 10¹¹ cm^-2.

References

1. F. Stern, Phys. Rev. Lett. 18, 546 (1967).

2. S. D. Sarma and J. J. Quinn, Phys. Rev. B 25, 7603 (1982).

3. A. C. Tselis and J. J. Quinn, Phys. Rev. B 29, 3318 (1984).

4. S. D. Sarma and E. H. Hwang, Phys. Rev. Lett. 81, 4216 (1998).

5. E. H. Hwang and S. D. Sarma, Phys. Rev. B 64, 165409-1 (2001).

6. J. K. Jain and P. B. Allen, Phys. Rev. Lett. 54, 2437 (1985).

7. S. J. Cheng and R. R. Gerhardts, Phys. Rev. B 63, 035314-1 (2001).

8. W. H. Backes, F. M. Peeters, F. Brosens, and J. T. Devreese, Phys. Rev. B. 45, 8437 (1992).

9. B. Vinter. Phys. Rev. B 13, 4447 (1976); 15, 3947 (1977).

10. D. Dahl and L. J. Sham, Phys. Rev. B 16, 651 (1977).

11. W. L. Bloss, Solid. State Commun. 46, 779 (1983).

12. W. P. Chen, Y. J. Chen, and E. Burstein, Surf. Sci. 58, 263 (1976).

13. A. Tselis, G. Gonzales de la Cruz, and J. J. Quinn, Solid. State Commun. 46, 779 (1983).

14. D. Olego, A. Pinczuk, A. C. Gossard, and W. Wiegmann, Phys. Rev. B 26, 7867 (1982).

15. G. Abstreiter and K. Ploog, Phys. Rev. Lett. 42, 1308 (1979).

16. R. Sooryakumar, A. Pinczuk, A. Gossard, and W. Wiegmann, Phys. Rev. B 31, 2578 (1985).

17. G. Fasol, N. Mestres, H. P. Hughes, A. Fischer, and K. Ploog, Phys. Rev. Lett.

56, 2517 (1986).

18. G. Fasol, R. D. King-Smith, D. Richards, U. Ekenberg, N. Mestres, and K. Ploog,

19. A. Pinczuk, M. G. Lamount, and A. C. Gossard, Phys. Rev. Lett. 56, 2092 (1986).

20. R. Merlin, K. Bajema, R. Clarke, F.-Y Juang, and P. K. Bhattacharya, Phys. Rev.

Lett. 55, 1768 (1985).

21. S. Oelting, D. Heitmann, and J. P. Kotthaus, Phys. Rev. Lett. 56, 1846 (1986).

22. J. K. Jain, R. Jalabert, and S. D. Sarma, Phys. Rev. Lett. 60, 353 (1988).

23. S. D. Sarma, J. K. Jain, and R. Jalabert, Phys. Rev. B 37, 6290 (1988).

24. S. D. Sarma, J. K. Jain, and R. Jalabert, Phys. Rev. B 37, 1228 (1988), S. D.

Sarma, J. K. Jain, and R. Jalabert, Phys. Rev. B 37, 4560 (1988).

25. J. F. Young and P. J. Kelly, Phys. Rev. B 47, 6316 (1993).

26. L. Wendler and R. Pechstedt, Phys. Rev. B 35, 5887 (1987).

27. W. Xiaoguang, F. M. Peeters, and J. T. Devreese, Phys. Rev. B 32, 6982 (1985).

28. K. J. Yee, D. S. Yee, D. S. Kim, T. Dekorsy, G. C. Cho, and Y. S. Lim, Phys. Rev.

B 60, R8513 (1999).

29. J. J. Baumberg and D. A. Williams, Phys. Rev. B 53, R16140 (1996).

30. T. Dekorsy, A. M. T. Kim, G. C. Cho, H. Kurz, A. V. Kuznetsov, and A. Forster, Phys. Rev. B 53, 1531 (1996).

31. J. J. Licari and R. Evrard, Phys. Rev. B 15, 2254 (1977).

32. N. Mori and T. Ando, Phys. Rev. B 40, 6175 (1989).

33. A. K. Sood, J. Menendez, M. Cardona, and K. Ploog, Phys. Rev. Lett. 54, 2111 (1985); A. K. Sood, J. Menendez, M. Cardona, and K. Ploog, Phys. Rev. Lett. 54, 2115 (1985).

34. G. D. Mahan, “Many-particle physics” (Kluwer Academic, Plenum Publishers, New York, 2000), p.468-474.

35. J. T. Devreese and F. M. Peeters, “The Physics of the Two-Dimensional Electron Gas” (Plenum, New York, 1987), p.183-225.

36. J. Shah, “Ultrafast Spectroscopy of Semiconductors and Semiconductor Nanostructures” (Springer, Berlin, 1996), p.163.

Chapter 5

Structure Effect on Fröhlich Interaction

5.1 Introduction

Electron polar-optical-phonon interaction in III-V semiconductor quantum wells plays an important role for hot carrier relaxations, which influence the high-speed responses of many quantum devices. In the past, electron-phonon scattering rates in a quantum well were typically calculated using the bulk phonon model or the bulk-like phonon model^1-3. In the bulk-like phonon model, the optical phonon modes are assumed to be the same as those in the bulk material while the electron wave functions incorporate quantum confinement. More recently, the DCM^4-7 and Huang-Zhu model⁸ (HZM) were developed for dielectric slab problems and were more accurate than the bulk and the bulk-like phonon models. The fundamental types of phonon modes^4,7,8 and the electron-phonon Hamiltonian^5,6,8 in heterostructures have become an interesting subject. Experimentally, Plooget al.^9,10 discovered the evidence of the confined LO, TO phonons and interface phonons in GaAs/AlAs superlattices using Raman scattering. An order of magnitude reduction in the intersubband scattering rates in GaAs/AlxGa1-xAs quantum wells was reported by Schlapp et al. using an infrared bleaching technique¹¹. The reduced scattering rates were explained successfully by Sarma et al. using the DCM¹².

In the last decade, techniques involving ultrafast spectroscopy became very

powerful tools in studying carrier dynamics in semiconductors. Ploog et al.^13-15 used time-resolved photoluminescence to study the hot carrier relaxation in a quasi-two dimensional system. Their experimental results were analyzed with the AELR¹⁶ and indicated that the width of a quantum well had little effect on the hot carrier relaxation.

However, the AELR in their analysis was calculated using the bulk phonon model.

More recently, a better method using the hot-electron neutral-acceptor luminescence¹⁷, shown in Fig. 5.1, was developed to study the carrier relaxation mechanisms. It gives a better spectral resolution at lower carrier excitation densities than those of the ultrafast spectroscopy technique. This method has been used^18-20 to determine the effective phonon energy in GaAs/AlxGa1-xAs quantum wells with various structure parameters. The effective phonon energy can be estimated in our calculations and be compared with experimental measurements.

The purpose of the report is to calculate the electron-phonon scattering rates in GaAs/AlxGa1-xAs quantum wells with various structure parameters based on the DCM model. Especially, we focus on the dependence of the electron-optical phonon interaction on the Al composition in the barrier, which is the subject that is still lacking in earlier reports. The calculated results are compared with earlier experimental results²⁰.

5.2 Dielectric Continuum Model

5.2.1 Phonon energy in GaAs/Al x Ga 1-x As quantum wells

Based on DCM, there are six types of optical-phonon modes⁶ in a dielectric slab.

However, due to selection rules for the intra-subband scattering, only the confined LO mode, the half-space LO mode, and the symmetric interface modes were taken into consideration in our calculations. The confined phonons propagate in the well,and the

Figure 5.1: Schematic diagram of hot photoluminescence at energy

hν

_PL through electron-neutral acceptor (marked by A) recombination. Incident photons with hν_ex excite electrons, and subsequent scattering processes are illustrated including electron-electron, electron-phonon, electron-plasmon scatterings. (quoted from Ref.[21])

q_z is quantized. The half-space phonons, whose z component of the phonon wave vector is not restricted, propagate in the barrier. Symmetric interface phonons propagate along the interface, and the in-plane atomic displacement is symmetric with respect to the center of the well. Symmetric interface mode can be further divided into the S+ and the S- branches. These two phonon branches also have different dispersion characteristics, which is given by the solution of

where the lattice dielectric function is given by

where the optical phonon energy in the AlxGa1-xAs layer has two modes: the GaAs-like mode and the AlAs-like mode.

< ω

_LO1(TO1)

>

represents the LO (TO)

energy in the GaAs layer. <ω_LO2(TO2) > represents the LO (TO) energy in the Al_xGa_1-xAs layer, and is taken as the average of those of the AlAs-like mode

)

The material parameters used in the dissertation are listed in Table II²². Fig. 5.2 shows an example of the interface phonon dispersion in a 50nm-width GaAs/Al_0.3Ga_0.7As quantum well. At the long wave length limit, the S+ and the S- interface modes go to the LO phonon energy in AlGaAs barrier and TO energy in GaAs well, respectively, while the antisymmetric plus (A+) and the antisymmetric minus (A-) interface modes go to the LO phonon energy in GaAs well and TO energy in AlGaAs barrier. At the short wave length limit, the S+ and the A+ modes go to the

(5.1)

TABLE II. Material parameters for GaAs, AlAs and AlxGa1-xAs used in the dissertation

Parameter GaAs AlAs Al_xGa_1-xAs LO-phonon energy hω_LO(meV)

GaAs-type 36.25 36.25-6.55x+1.79x² AlAs type 50.09 44.63+8.78x-3.32x² TO-phonon energy hω_TO (meV)

GaAs-type 33.29 33.29-0.64x-1.16x² AlAl type 44.88 44.63+0.05x-0.30x² Relative dielectric constant

Static κ _S 13.18 10.06 13.18-3.12x High frequency

κ

_∞ 10.89 8.16 10.89-2.73x Band-gap energy

E

_g^Γ

( 15 K )

(eV) 1.519 3.13 1.519+1.611x Electron’s effective mass m (m_{e 0})^a 0.067 0.15 0.067+0.083x

a m₀ denotes the free electron mass.

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 0.033

0.034 0.035 0.036 0.037

0.038 ω

S+

ω_A+

ω

A-ω

S-Phonon energy ω (eV)

In-plane phonon wave vector q

//

(10

⁹

m

^-1

)

Figure 5.2: Dispersion curves of symmetric and anti-symmetric interface phonon modes in a 50nm GaAs/Al0.3Ga0.7As quantum well.

same phonon energy while the S- and the A- go to the same energy. The long-wave-length phonons play a more important role on the scattering with electrons because the Fröhlich interaction inversely proportional to the phonon’s wave vector, which favors the small-angle scattering. The anti-symmetric interface phonons become important on the interaction with electrons when the electron has intersubband scatterings.

5.2.2 Electron-Phonon Scattering Rates

With the average phonon energy in AlxGa1-xAs alloy shown in eq.(5.3), intrasubband electron-optical-phonon scattering rates in the lowest subband can be calculated using the Fermi’s golden rule. The scattering rates are obtained by integrating over all possible states using the two-dimensional density of state function with states restricted by energy and momentum conservations. Scattering rates of the interface modes, W , the confined mode, W_{S± C}, and the half-space mode, W_HS, are respectively written as

q are phonon occupation numbers of the interface

(5.4)

modes, the confined mode, and the half-space mode respectively.

ϕ

_i and

ϕ

_f are the electron’s wave functions of the initial and the final states in the quantum well.

and ,

, _{C HS}

S φ φ

φ _± , given in Table III, are the potential functions of the interface, the confined, and the half-space modes respectively. The function, hn(ω_S±), is expressed as

The minimum and maximum in-plane phonon wave vector can be expressed as

In order to clearly explain the dependence of scattering rates on the structure parameters for the interface modes, we introduce the H factor, defined as

It appears in the eq. (5.4) for the interface modes. In addition, we call the overlap integral,

〈 ϕ

_f

| φ | ϕ

_i

〉

, for the electric potential G factors for the phonon modes. For S+ and S- modes, the G factors in the well and the barrier are respectively

⎥ ⎥

The G factor for the p_th confined mode and the half-space mode are respectively

where Ee1 is the electron’s ground-state energy, and ∆E_C is the barrier height of the quantum well. The G factor for the intersubband scattering is in Appendix C.

We used the DCM instead of the HZM⁸ for the boundaries’ treatment. It is because the scattering rate calculated by HZM gives an unreasonably large scattering rate even with very narrow well width due to the slow convergence of the higher order’s modes. The RPA was used for the dynamical screening^23,24.

)

TABLE III. The potential in a quantum well for various phonon modes Optical-phonon mode Potential for three distinct regions

2

z≤−L^w

2 z L 2

L_{w w}

≤

<

− 2 z>L^w

Symmetric interface φ_S± ^{2 )}

z L ( q_// ^w

e

⁺ cosh(q_//z)/cosh(q_//L_w/2) ^{2 )}

z L ( q_// ^w

e

^{− −}

Anti-symmetric interface

φ

_A± - ^{2 )}

z L ( q_// ^w

e

⁺ sinh(q_//z)/sinh(q_//L_w/2) ^{2 )}

z L ( q_// ^w

e

^{− −}

Confined φ _c ) L

z cos(n

w

π , n: odd

) L

z sin(n

w

π , n: even

Half-Space φ _HS L )]

2 z 1 ( q

sin[ _z + _w L )]

2 z 1 ( q

sin[ _z − _w

5.3 Intrasubband Scattering

In our calculations, band-offset ratio ∆E_c :∆E_v in GaAs/AlxGa1-xAs quantum wells was chosen to be 65:35. The electrons were given an excess energy of 50meV so that the intersubband transition can be neglected. The sheet-charge density was chosen to be 5x10¹⁰cm^-2.

In Fig. 5.3 we show the dependence of the phonon energy of the S+ mode and the S- mode on the Al composition in the barrier at the minimum q_//minand the maximum q_//max in-plane phonon wave vectors with a well width of 5nm. For the S+

mode, the phonon energy increases quickly with the Al composition for both q_//min and q_//max. It approaches the LO phonon energy in the barrier layer when q _//

approaches zero. The increase of the calculated S+ mode energy with Al composition at q_//min agrees with the increased LO phonon energy in AlxGa1-xAs layer as Al composition is increased. For the S- mode, the phonon energy has a weak dependence on the Al composition. It approaches the TO phonon energy of the well when q _//

approaches zero. The weak dependence on the Al composition is easily understood because there is no Al in the well.

In Fig. 5.4 we show the calculated dependence of electron-optical phonon scattering rates on the Al composition for various types of phonon modes in a 5nm wide GaAs/Al_xGa_1-xAs quantum well with a lattice temperature of 15K. For the S+

mode, the scattering rate increases from 4.1ps^-1 to 6.9ps^-1 as the Al composition, x, is increased from 0.2 to 1. In order to interpret the results, we show in Fig. 5.5 (a) the dependence of the H factor and the G factor on q . As we can see, both the H factor _//

and the G factor increase with the Al composition at small q . Since the S+ mode _//

favors the small-angle scattering, this dependence follows the behavior of the H and the G factors at small q . For the S- mode, the scattering rate increases from 0.32ps_// ^-1 to 1.1ps^-1 as the Al composition is increased from 0.2 to 1. The strong dependence on

0.0 0.2 0.4 0.6 0.8 1.0 33

36 39 42 45 48 51

S+ mode at q_//min S+ mode at q_//max S- mode at q_//max S- mode at q_//min

Phonon ener gy ω (me V )

Al composition x

Figure 5.3: Dependence of phonon energy of S+ and S- interface modes on the Al composition at q_//minand q_//max.

0.2 0.4 0.6 0.8 1.0 0

1 2 3 4 5 6 7

W

_S+

without screening W

S+

with screening W

C

without screening W

C

mode with screening W

S-

without screening W

S-

with screening

Scatteri ng rates W (ps

-1

)

Al composition x

Figure 5.4: Dependence of electron- phonon scattering rate of S+ mode, confined mode, and S- mode on the Al composition. The well width is 5nm, the lattice temperature is 15K, and the amount of the excess kinetic energy of the electron is 50meV.

0 1 2 3 4 5 6 0.004

0.006 0.008 0.010 0.012 0.014 0.016 0.018

H factor at Al composition x=1 H factor at Al composition x=0.7 H factor at Al composition x=0.3

H factor

In-plane phonon wave vector q

_//

(10

⁸

m

^-1

)

0.5 0.6 0.7 0.8 0.9 G factor at Al composition x=1 1.0

G factor at Al composition x=0.7 G factor at Al composition x=0.3

G fa cto r

1 2 3 4 5

0.0010 0.0015 0.0020 0.0025 0.0030 0.0035

H factor at Al composition x=1 H factor at Al composition x=0.7 H factor at Al composition x=0.3

H factor

In-plane phonon wave vector q

_//

(10

⁸

m

^-1

)

Figure 5.5: Dependence of H and G factors on q_// at three Al compositions in the barrier, 0.3, 0.7, and 1. (a) for the S+ mode. (b) for the S- mode.

(a)

(b)

the Al composition is mostly due to the H factor. In Fig. 5.5(b) we show the dependence of the H factor on q . As _// q decreases toward zero, _// ωS- approaches the TO phonon energy in the well. This leads to the decrease of the H factor. Because of this, the small-angle scattering for the S- mode is not as important as that for other phonon modes. In Fig. 5.4 we have also found that the screening effect for the S+

mode and the S- mode is not significant.

Comparing to the S+ and the S- modes, the scattering rate of the confined phonon mode does not show strong dependence with the Al composition in the range that we have investigated. It is because that the G factor in the expression of the scattering rate equation for the confined phonon mode is less sensitive to the Al composition. The screening effect for the confined mode is stronger than that of the S+ and the S- interface modes.

For the 5nm well the electron wave function does not penetrate deep into the barriers. Therefore, the half-space mode’s contribution to the scattering rate is insignificant in comparing to the other three types of phonon modes and is not considered here.

The calculated results were compared with the experimental results²⁰ performed by hot-electron neutral-acceptor luminescence for GaAs/Al_xGa_1-xAs quantum wells with various Al compositions. The calculated effective phonon energy (ω

_eff

) is given by the following equation.

In Fig. 5.6 we show the dependence of the effective phonon energy on the Al composition of both the experimental result²⁰ and our calculations. There are two calculated curves in the figure. The solid line represents the results without screening

(5.14)

C S S

C C S S S S

eff W W W

W W

W

+ +

ω + ω +

= ω ω

− +

−

− + +

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 36

38 40 42 44 46 48 50

Experimental results

Calculated results with screening Calculated results without screening

Eff e ct ive phonon ener gy ω

eff

(meV)

Al composition x

Figure 5.6: Comparison of experimental and calculated results for the dependence of the effective phonon energy on the Al composition. The well width is 5nm, the lattice temperature is 15K, and the amount of the excess kinetic energy of the electron is 180meV.

and the dash one with screening. Since the S+ mode plays the dominant role in the calculated scattering rate among all phonon modes, the calculated effective phonon energy basically follows the behavior of the S+ mode. The tendency of the calculations is in good agreement with the experiments.

The minor difference between the measured result and the calculated result on the effective phonon energy is attributed to the assumptions that we made in the calculations of the average phonon energy in Al_xGa_1-xAs alloy, which probably simplified the complexity of the phonon spectrum in the ternary compound.

In Fig. 5.7 we show the dependence of scattering rates on the well width for various types of phonon modes with an Al composition x=0.3 in the barriers. Other parameters are kept the same as in previous calculations. For the S+ mode, the scattering rate decreases considerably from 5.3ps^-1 to 1.4ps^-1 as the well width is increased from 4nm to 12nm. We attribute this to the decrease of the H factor and the G factor as the well width is increased. When the wells move toward wider wells, the electron wave functions centered at the middle of the well do not spread deep into the interfaces as in the narrower wells. The interface is the place where the strongest electron-phonon interaction took place. Thus, it leads to the decrease of the G factor.

But, the tendency on the decrease of the G factor does not hold for extremely small q . As the well gets narrower, the increasing of the G factor has been canceled out by //

the decreasing of the H Factor and results in the weak dependence for well width narrower than 4nm. This behavior was not found in the earlier calculated results²⁵ where the assumption of an infinite quantum well was made.

For the S- mode, due to the small H factor, the scattering rate is much smaller than the rate of the S+ mode. In addition, the increased H factor with the well width compensates the decreased G factor each other. This results in a weak dependence of scattering rates on the well width.

2 4 6 8 10 12 0

1 2 3 4 5 6 7

Total W

_S+

W

_C

W

W

_HS

Scatteri ng rates W (ps

-1

)

Well width L_w (nm)

Figure 5.7: Dependence of electron-optical phonon scattering rate of S+ mode, confined mode, S- mode, half-space mode, and the total rate contributed by all types of phonon modes on the well width. The Al composition is 0.3, the lattice temperature is 15K, and the amount of the excess kinetic energy of the electron is 50meV.

The scattering rate of the confined mode increases from 0.27 ps^-1 to 2.2 ps^-1 while the rate of the half-space mode decreases sharply from 0.13 ps^-1 to 0.12 ns^-1 as the well width is increased from 2nm to 12nm. The increase of the scattering rate of the

The scattering rate of the confined mode increases from 0.27 ps^-1 to 2.2 ps^-1 while the rate of the half-space mode decreases sharply from 0.13 ps^-1 to 0.12 ns^-1 as the well width is increased from 2nm to 12nm. The increase of the scattering rate of the

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