Summary

Although the dynamical screening on the CCS is a complex Coulomb interaction, the dimensional dependence can be understood in a simple picture and the picture is also valid on the Fröhlich interaction and the static-screened interaction. The memory effect is as a result of quantum coherence and would cause a burning hole on the carrier’s distribution at the early stage. As the time further goes, the two hills aside the burning hole is expected to continue to evolve to two burning holes and three hills due to the memory effect and go on until the carrier dephases or thermalizes. The non-equilibrium carrier’s evolution is normal in the presence of 2D cold electron and the Knox’s result should be due to the effect of wafer’s preparation.

References

1. J. Shah, Ultrafast Spectroscopy of Semiconductors and Semiconductor Nanostructures (Springer, Berlin, 1996).

2. J. –Y. Bigot, M. T. Portella, R. W. Schoenlein, J. E. Cunningham, and C. V. Shank, Phys. Rev. Lett. 67, 636 (1991).

3. P. C. Becker, H. L. Fragnito, C. H. Brito Cruz, R. L. Fork, J. E. Cunningham, J. E.

Henry, and C. V. Shank, Phys. Rev. Lett. 61, 1647 (1988).

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62, 2686 (2000).

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49, 7337 (1994).

6. H. C. Lee, K. W. Sun, andC. P. Lee, Solid State Comm. 128, 245 (2003).

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8. M. Betz, G. Göger, A. Laubereau, P. Gartner, L. Bányai, H. Haug, K. Ortner, C. R.

Becker, and A. Leitenstorfer, Phys. Rev. Lett. 86, 4684 (2001).

9. C. Fürst, A. Leitenstorfer, A. Laubereau, and R. Zimmermann, Phys. Rev. Lett. 78, 3733 (1997).

10. L. Bányai, D. B. Tran Thoai, E. Reitsamer, H. Haug, D. Steinbach, M. U. Wehner, M. Wegener, T. Marschner and W. Stolz, Phys. Rev. Lett. 75, 2188 (1995).

11. W. H. Knox, D. S. Chemla, G. Livescu, J. E. Cunningham, and J. E. Henry, Phys.

Rev. Lett. 61, 1290 (1988).

12. M. G. Kane, Phys. Rev. B 54, 16345 (1996) and Carrier-Carrier Scattering among Photoexcited Nonequilibrium Carriers in GaAs (Dissertation of Princeton University, 1994).

13. L. P. Kadanoff and G. Baym, Quantum Statistical Mechanics (Benjamin, New

Tork, 1962).

14. P. Lipavský, V. Špička, and B. Velický, Phys. Rev. B 34, 6933 (1986).

15. G. Mahan, “Many Particle Physics” (Plenum, New York, 2000).

16. H. Haug and A. -P Jauho, Quantum Kinetics in Transport and Optics of Semiconductors, 2^nd ed. (Springer, Berlin, 1998).

17. D. C. Langreth, Linear and Nonlinear Electron Transport in Solids, ed. By J. T.

Devreese and E. Van Doren (Plenum, New York, 1976).

18. L. Bányai, Q. T. Vu, B. Mieck, and H. Haug, Phys. Rev. Lett. 81, 882 (1998).

19. The form factor is given by

where ϕ(z) denotes the electron’s wave function and lower subscript indicates the subband of initial and final states. The form factor of intrasubband scattering in the ground state can be expressed as

20. S. Adachi, J. Appl. Phys. 58, R1 (1985).

21. M. Abramowitz and A. Stegun, Handbook of Mathematical Functions, p.916, 917 (Dover, New York, 1964).

22. D. W. Snoke, Phys. Rev. B 50, 11583 (1994).

∫

∫

^∞

∞

−

−

∞ −

∞

−

ϕ ϕ ϕ ϕ

= ^' _{i j} ^{' *}_m ^*_n ^{' qz z'}

ijmn(q) dz dz (z) (z ) (z) (z)e F

[ ] ^⎥ _⎦ ^⎤

⎢ ⎣

⎡ +

π +

−

− π π π

= +

⁻_{2 2 w}

w 2 w

qL 4

w 2 2 2

w

1111

qL

2 1 ) 2 ( ) qL ( ) qL (

) e 1 ( 4 qL

) 2 ( ) qL ( ) 8 q (

F

^w

Chapter 3

Hot Carrier Relaxation

3.1 Introduction

Although hot carrier relaxations in a bulk GaAs and quantum wells have been studied experimentally^1-11 and theoretically^12-14 for more than one decade, the dependence of the dynamical screening in hot carrier relaxations on the sample’s dimensionality is still not well understood. The screening behavior caught less attention on hot carrier relaxations in GaAs probably attributes that the hot phonon effect was primarily considered to be responsible for the great drop of energy-loss rates via Fröhlich interaction^2,5,11, and hot carrier relaxations seem not to depend on the dimensionality experimentally^4,5,6. However, the deduction could not hold on the overall carrier densities. Because more recent experimental results indicated that there is a clear difference in energy-loss rates between a bulk GaAs and quantum wells when the carrier density is above a certain critical value^7,8,9, shown in Fig. 3.1 and 3.2. Though the critical carrier densities determined in those experiments are not consistent, the results imply that the dimensionality and the dynamical screening may have a significant effect on hot carrier relaxations in a bulk GaAs and quantum wells.

To theoretically study the difference of hot carrier relaxations between the two different dimensional systems, it is important to consider the optical phonon modes in a quantum well. Many improved models were developed to give a better description for atomic vibrations and the interaction Hamiltonians with electrons in the quasi two-dimensional structure^15-26. In our calculations, we use the dielectric continuum

Figure 3.1: Time-resolved luminescence spectra at room temperature from Pelouch’s results. (i) for 400nm bulk GaAs. (ii) for multiple quantum well. square: 10¹⁹cm^-3;

18 -3 18 -3

(i)

(ii)

Figure 3.2: Comparison of carrier temperature versus initial carrier density between different sample’s structures. (quoted from Phys. Rev. B 45, 1450 (1992))

model^15-17,20 (DCM) because the model has provided a good agreement with many earlier experimental results^27-33. Although the interaction Hamiltonians of phonon modes in a quantum well are strongly dependent on the well width, many experimental results^4,5,6 demonstrated the less well-width dependence of hot carrier relaxations except few report shown the contrary results^34,35. This discrepancy also stimulates us to study the structural dependence of energy-loss rates in a quantum well.

In the thesis, the significance of the dimensionality and the dynamical screening on hot carrier relaxations in a bulk GaAs and quantum wells is investigated. The distinct dimensionality and the dynamical screening indeed cause that hot carriers in quantum wells relax significantly slower than that in a bulk GaAs above the critical carrier density of 2×10¹⁸cm^-3. We attribute this to the smaller density of state in quantum wells and the strong 2D dynamical screening. The dynamical screening in quantum wells appears to be much stronger than that in the bulk and considerable as compared to the hot phonon effect. The critical carrier density determined in our studied is in very good agreement with the earlier experiments of Pelouch and co-researchers⁷. We also found that the average energy-loss rate in quantum wells depends on the well width more appreciably when Al compositions are high.

3.2 Semiclassical Boltzmann Equation

The average energy-loss rate (AELR) is calculated in order to compare the difference of hot carrier relaxations between the two different dimensional systems. In this section, we describe the derivations of the AELR in a bulk GaAs and a quantum well where the net phonon generation rate, and the treatments of hot phonon effect and the dynamical screening are included. The dynamical screening is dealt with the RPA³⁶.

dynamical screenings. In our calculations, the electron-phonon scattering is through Fröhlich interaction and only intrasubband scattering is considered in the calculation of the AELR in quantum wells. The hole-phonon interaction is neglected. The plasmon-phonon coupling (PPC) is not considered here because the significant enhancement of energy-loss rates^37,38 induced by the PPC does not appear above the critical carrier density of 2×10¹⁸cm^-3.

The AELR is determined by the net phonon generation rate and the phonon energy.

The net phonon generation rate represents the subtracting difference between phonon’s generation rates and absorption rates. In a bulk GaAs, the 3D net phonon generation rate is given by³⁹

where q and ω denote phonon’s wave vector and phonon’s energy, respectively. _q )

(

f ζ is the electron’s distribution. With the thermalized assumption for carriers, Fermi-Dirac distribution is used where T is the carrier temperature. _C N represents _q

phonon population.

k

_B and V have their usual meanings. The quantity N_q(T_C) can be written as³⁹

ζ , a dimensionless quantity, represents the normalized energy (energy divided by thermal energy k_BT_C).

ζ

_min is the minimum normalized energy required for an electron to kick out a phonon of wave vector. It is given by³⁹

2

M_q 2, Fröhlich interaction strength, is given by³⁹

where ε_∞ and ε_S are high frequency and static dielectric constants.

In a quantum well, based on the DCM^15-17,20, the confined (C), the symmetric plus interface (S+), symmetric minus (S-) interface and the half-space (HS) phonon modes are considered in our calculations. Anti-symmetric interface modes are excluded due to the selection rule for the intra-subband scattering. The dispersion relations for the S+ and the S- interface modes are shown in the sec. 5.2⁴⁰. The noun

“half-space” in double heterojunctions originates from the report of Mori and Ando²⁰ where the same name as the case of a single heterojunction is used. The 2D net phonon generation rate can be written as

where

q

_// and ^{(C,S ,HS)}

//

ω

_q ± denote the in-plane phonon wave vector and the phonon energies of various modes. m is the n_en ^th layer effective electron’s mass while 1 represents GaAs and 2 represents AlGaAs layers. A denotes the area.

ζ

₁ is the normalized ground state energy to the thermal energy. ^{(C,S ,HS) 2}

M

_q// ^± represents the electron-phonon interaction strength of various modes shown in Table I. The used Hamiltonians are taken from the report of Mori and Ando²⁰.

The quantity of ^(C₍^,S₎^,HS)

N

_qq//^± is to be determined. When the hot phonon effect is excluded, the phonon population satisfies the Bose-Einstein relation with a lattice temperature

T

_L. In general, the hot phonon effect plays an important role in hot

(3.4)

TABLE I. The electron-optical-phonon interaction strengths in a quantum well structure

Half-space mode

^a,b,c

a

n

GS_± is <ϕ₁|φ_S± |ϕ₁>, where

ϕ

₁ is the electron’s ground state and φ_S± is potential for interface modes, and the factors G_C^p,G_HSare the overlap integral for the p^th confined mode and the half-space mode respectively. Their expressions and hn(．) are shown in the sec. 5.2.

b

Sn n,ε

ε_∞ are n^th layer high frequency and static dielectric constants.

c

q

z is the phonon wave vector paralleled to the crystal’s growth direction.

carrier relaxations. The phonon dynamics can be governed by the phonon Boltzmann equation. At the steady state, the phonon’s population can be given by the following equation with using eq.(3.1) for bulk (eq.(3.5) for quantum wells)³⁹.

where τ is the phonon life time. _ph

The dynamical screening on hot carrier relaxations is handled with the electronic dielectric function. Based on the RPA, the dielectric function is given by³⁶

the effective screened electron-phonon interaction strength

M

^eff_q ²can be expressed as

2

The 3D and 2D zero-temperature dielectric functionsare, respectively, given by⁴³ (3.6)

where

E

_F,

k

_Fand

v

_F represent Fermi energy, Fermi wave vector and Fermi

3.3 Results and Discussion

The material’s parameters and the used assumptions are referred^40,44-46. Our calculations of the reduced dimensionality on hot carrier relaxations are performed on a bulk GaAs and a 10nm-width single GaAs/Al_0.3Ga_0.7As quantum well where the band-offset ratio of ∆E_C:∆E_V= 65 : 35 is used⁴⁴. The average phonon energy is approximated in AlGaAs layers to simplify the two-mode behaviors of the GaAs-like and the AlAs-like phonons⁴⁰. The material’s parameters are quoted from Adachi’s report⁴⁵. The electron’s distribution function is assumed to satisfy with Fermi-Dirac relation. We use 300K as the electron’s temperature except the section reported the structural dependence in quantum wells, where 600K is taken. An initial lattice temperature is chosen to be 15K. We quote 7ps to be the phonon life times in both bulk GaAs and quantum wells⁴⁶. Only first-order mode²⁰ of confined phonons is considered in our calculation because of the very less contribution to the AELR from

(3.9b)

higher-order modes. In general, the AELRs in quantum wells are summed over S±

interface and the confined modes.

3.3.1 Reduced Dimensionality

In Fig. 3.3 we show the dependence of the AELR on the carrier density in bulk GaAs and a quantum well where the sheet carrier densities are transferred byn_2D/L_w. Three sets of curves are shown in the figure representing three different conditions: (1) in the absence of the hot phonon effect and the dynamical screening (denoted by None in the plot), (2) in the presence of the hot phonon effect alone (denoted by HP), and (3) in the presence of both the hot phonon effect and the dynamical screening (denoted by HP+DS). For the first case, the AELRs in the bulk are shown to be higher than that in a quantum well. Above the carrier density of 10¹⁸cm^-3, the AELR in a quantum well drops much faster than that in a bulk. Because the energy-loss rate in materials is equal to the product of the AELR and the carrier density, this implies that there is a considerable difference in the energy-loss rate between bulk GaAs and a quantum well as the carrier density is increased. In the absence of hot phonon effect and the dynamical screening, the rapid deviation of the AELRs between the 2D and the 3D structures is only attributed to the difference in density of states. Due to the smaller density of state, hot carriers in quantum wells are shown to relax considerably slower than that in the bulk above the critical carrier density. When the hot phonon effect is considered, although the AELRs greatly reduce in both sample’s structures, the rapid deviation of the AELR between bulk GaAs and a quantum well still appears while the critical carrier density is shifted to the higher one (toward 2×10¹⁸cm^-3).

In Fig. 3.4 we show the difference AELR³_HP^D−AELR²_HP^D on the left axis and the

ratio _2D

D 2 HP D

3 HP

AELR AELR

AELR −

on the right, where the lower and upper symbols of the

1E17 1E18 1E19 1

10 100

3D None 2D None 3D HP 2D HP 3D HP+DS 2D HP+DS

Average energy loss rates AELR (m eV/ p s)

Carrier density n_3D (cm^-3)

Figure 3.3: Average energy-loss rate with distinct conditions in a bulk GaAs and a 10nm GaAs/Al_0.24Ga_0.76As quantum well. The symbols “None”, “HP”, and “HP+DS”, respectively, denote the AELRs in the absence of the hot phonon effect and the dynamical screening, in the presence of the hot phonon effect, and in the presence of the hot phonon effect and the dynamical screening. The confined and the S± interface modes were considered in the AELR of a quantum well. The carrier temperature of 300K and the initial lattice temperature of 15K were used.

1E17 1E18 1E19 0.0

0.3 0.6 0.9 1.2

Carrier density n

3D (cm^-3)

0.0 0.3 0.6 0.9 1.2 1.5

Figure 3.4: Illustration of reduced dimensionality on hot carrier relaxation. The left axis shows the absolute difference of the AELR with the hot phonon effect between a bulk GaAs and a 10nm GaAs/Al_0.24Ga_0.76As quantum well while the right axis shows the percentage change of AELR³_HP^D −AELR²_HP^D per AELR²_HP^D. The AELR of the quantum well was obtained by summing the confined and the S

±

interface modes. The carrier temperature of 300K and the initial lattice temperature of 15K were used.

AELR represent the considered effect and dimensions, respectively. Below the carrier density of 10¹⁸cm^-3, slightly higher 2D AELRs are demonstrated and this is because the hot phonon effect in quantum wells is weaker than that in the bulk. But, above the carrier density, the effect of the smaller density of state in quantum wells overcomes the hot phonon effect so that 2D AELRs recover to be lower than the 3D case, and the threshold curve clearly shows the significance effect of the distinct density of state on hot carrier relaxations between a bulk GaAs to quantum wells.

3.3.2 Dynamical screening versus Hot Phonon Effect

In our investigation, the 2D dynamical screening is also found to be an important role on hot carrier relaxations at a high carrier density. Due to the great difference of the dynamical screening between the two different dimensions, the calculated results with the HP+DS in Fig. 3.3 shows the more rapid deviation of AELRs between the two sample’s structures. In order to compare the screening strength between a bulk GaAs and quantum wells, we plot Fig. 3.5 where the right and left axes, respectively, show reduction factors due to the dynamical screening and the hot phonon effect. The symbols were mentioned earlier. The dynamical screening in quantum wells is shown to be much stronger than that in the bulk and more quickly increased when the carrier density is above 10¹⁸ cm^-3. The quicker increase for the 2D dynamical screening is the consequence of the chemical potential in quantum wells, which is raised faster than that in the bulk as the carrier density is increased. The 2D dynamical screening is also shown to be as important as the hot phonon effect at a high carrier density. To our best knowledge, the earlier investigations^2,4,5 usually omit the effect of the dynamical screening on hot carrier relaxations. In a short summary, because of the fewer density of state and the strong 2D dynamical screening, hot carriers in quantum wells relax significantly slower than that in the bulk at a carrier density above the critical value of

1E17 1E18 1E19 0.60

0.65 0.70 0.75 0.80 0.85 0.90

2D

3D 2D

3D

Carrier density n

3D

(cm

^-3

)

0.0 0.1 0.2 0.3 0.4 0.5 0.6

Figure 3.5: Comparison of dynamical screening and hot phonon effect on average energy-loss rate between a bulk GaAs and a 10nm GaAs/Al_0.24Ga_0.76As quantum well.

The left axis shows the hot phonon effect while the right axis shows the dynamical screening. The AELR of a quantum well was obtained by summing the confined and the S± interface modes. The carrier temperature of 300K and the initial lattice temperature of 15K were used.

2×10¹⁸ cm^-3. The threshold behavior and the critical carrier density are in very good agreement with the earlier experimental results of Leo⁵, Pelouch⁷, and their co-researchers.

3.3.3 Well-Width Dependence

Next, we interpret the calculated results for the well-width dependence of hot carrier relaxations in quantum wells. The sheet density of 5×10¹¹cm^-2 is fixed for all calculated results with different structural parameters. Firstly, we show the dependence of the AELR on the well width for various phonon modes in Fig. 3.6(a), where the S+, the S-, the confined, and the half-space modes are considered. The AELRs for the confined mode always increases until to the well width of 10nm. This is the consequence of the electrons better confined in the wider well and the decreased phonon wave vector parallel to the crystal’s growth direction. For the S+ interface modes, because the electron’s spatial distribution departs from the double interfaces and the decreased Hamiltonian, the AELRs are shown to quickly decrease with the increased well widths. The S- interface and the half-space modes are less noticeable because of the flatter dependence and much smaller AELR as compared to the other modes. In Fig. 3.6(b), we show the AELRs as functions of the well width and the Al composition by summing over all phonon modes. The opposite dependence on the well width between the confined and the S+ interface modes compensates with each other and brings the protruding well-width dependence for various Al compositions.

The protruding behavior was also ever found in the earlier experiment of Ryan and Tatham³⁵. As the Al composition is increased, the well-width dependence of the AELR becomes more appreciable and the maximum AELR moves toward the shorter well width. The reason is the increasingly stronger effect of the S+ interface phonon mode on the hot carrier relaxations with the increased Al composition. The slightly

2 4 6 8 10 12 0

8 16 24 32 40 48

HS S-S+

C

Average e nerg y -lo s s rates AELR (meV/ps)

Well width L

w

(nm)

2 4 6 8 10 12

70 80 90 100 110

x=0.3 x=0.5

x=0.7 x=1

A v era ge en erg y -loss rates AE LR (me V/ps)

Well width L

_w

(nm)

0.0 0.2 0.4 0.6

0.0 0.3 0.6 0.9

HS

S+

S-q// (10⁹m^-1)

dN

q//

/dt (10

11

se c

-1

)

C

Figure 3.6: Well-width dependence of average energy-loss rate in a GaAs/Al_xGa_1-xAs quantum well. (a) for the confined, S± interface, and the half-space modes at x=0.3. (b) for the total AELR at x=0.3, 0.5, 0.7, and 1. The inset figure shows the spectrum of the net phonon generation rates in a quantum well with L_w of 10nm and x=0.3. The hot phonon effect is included in the AELR. The carrier temperature of 600K, the initial lattice temperature of 15K, and the carrier density of 5×10¹¹ cm^-2 were used.

(a)

(b)

roughness in AELR’s curves is the consequence of the numerical inaccuracy from the 2D net phonon generation rate where the finite spike at a given in-plane phonon wave vector is shown in the inset of Fig. 3.6(b).

3.4 Summary

We clarify the discrepancies of the earlier experimental results on hot carrier relaxations in bulk GaAs and quantum wells. In contrast to the results in a bulk GaAs, both the dimensionality and the dynamical screening have a significant effect on hot carrier relaxations in quantum wells. The smaller density of state in quantum wells and the strong 2D dynamical screening cause hot carriers in quantum wells to relax significantly slower than that in a bulk GaAs when the carrier density is above 2×10¹⁸ cm^-3. The influence of the 2D dynamical screening on hot carrier relaxations is considerable and is as important as the hot phonon effect when the carrier density is high. As the Al composition is increased, the AELR in quantum wells has a more appreciable dependence on the well width.

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