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Quantum well geometrical effects on two-dimensional electron mobility
Solid-State Ektronics Vol. 35, No. Printed
II, pp. 1597-1599, 1992
0038-I101/92 $5.00+ 0.00
in Great Britain
Pergamon Pmss Ltd
QUANTUM WELL GEOMETRICAL EFFECTS ON TWO-DIMENSIONAL ELECTRON MOBILITY TAHUI WANG,
TING-HUA
HSIEH
and YI-TEH CHEN
Institute of Electronics, National Chiao-Tung University, Hsin-Chu, Taiwan, R. 0. China (Received
17 April 1992)
Abstract-A
Monte Carlo analysis to study the quantization effects on the low-field electron mobility in various AlGaAs/GaAs/AlGaAs quantum well structures has been performed. The influence of the electron envelop wave-function and the subband structure on the two-dimensional electron scattering rates was evaluated. Our results showed that a maximum two-dimensional electron mobility can he achieved in a quantum well structure where the energy difference between the first subband and the second subband is about two times the polar optical phonon energy.
Recently, much research effort has been devoted to the use of heterostructures to achieve an enhancement of the electron mobility in ultra-high speed devices. Among the successful demonstrations of the heterojunction field effect transistors are AlGaAs/ GaAs Modulation Doped FETs[ 11,AlGaAs/InGaAs/ GaAs Pseudomorphic MODFETs[2] and Multiple Quantum Well FETs[3]. Although all of the above structures employ the concept of the “two-dimensional electron gas” (2DEG), the electron wavefunctions and subband structures in these 2DEG devices are quite different. Since the electron scattering rate is closely related to the wave-function distribution and the subband structure, the two-dimensional electron mobility may change significantly with the quantum-well geometry. In this study, we investigated the quantum well confinement effects on the 2DEG mobility. It was our intention to find an optimized AlGaAs/GaAs/AlGaAs quantum well structure for the mobility enhancement by tailoring the subband structure and the electron wavefunction. A 2DEG Monte Carlo analysis was utilized to study electron transport in the subbands of the quantum wells. The two-dimensional quantization effects on the scattering mechanisms in the Monte Carlo were evaluated from the Fermi-Golden rule[4]. As far as the low-field mobility is concerned, polar optical phonon (POP) scattering and accoustic phonon (ACP) scattering[5,6] were considered in this work. These are the two dominant scattering mechanisms in III-V semiconductors at low fields. The scattering rate of the 2DEG can be modulated with respect to the bulk electron scattering rate in two aspects, overlap integral and subband structure. The overlap integral is expressed as follows [7]:
s
L,(q) = ~,(z)~,(z)ev(iqz) dz
(1)
where q is the phonon wave-vector in the quantization direction. m and n are the index of the subbands SSE 35/11X
and F,,,(z) and F,(z) are the associated electron wave-functions. For intra-subband scattering (m = n), the overlap integral is our primary consideration in the calculation of the 2DEG scattering rate. Since a broad distribution of the wave-function in coordinate space corresponds to a narrow distribution in momentum space, the spread of the electron wave-function in wide quantum wells leads to a stringent momentum selection rule in eqn (1) and thus a reduced intrasubband scattering rate when q is away from 0. For inter-subband scattering (m #n), the energy separation in a subband structure dictates its significance to the 2DEG mobility. For example, if the second subband is far above the first subband, electrons in the first subband have little chance to be scattered to the second subband at low fields via phonon absorption. Inter-subband scattering is therefore virtually negligible in this situation. Unfortunately, the wavefunction distribution and the subband structure are usually two conflicting factors in the optimization of the 2DEG mobility. In a wide quantum well, the wave-function spreads and the intra-subband scattering rate is lowered. On the other side, the energy separation between the subbands is also reduced in wide quantum wells, which increases the occurrences of inter-subband scattering. The trade-off between these two factors leads to an optimization of the quantum well geometry. Two sorts of quantum-well structures were studied. The first kind has an Al,Ga,_,As/GaAs/ Al,Ga, _,As rectangular quantum well. The GaAs well width was varied in the simulatioin. The bandbending was purposely neglected in this kind of structures so that the spread of the wave-function can be solely determined by the well width. The second kind of structures have a realistic configuration, SOOA n+ Al,Ga,_. As/30 A i - Al,Ga, _ .As/GaAs/ i - Al,Ga, _,As. The doping concentration in then+ AlGaAs layer was 1 x lo’* l/cm3. Aluminum composition x was 0.3. In the second kind of structures, the
TAHUIWANG
1598
Schriidinger and the Poisson equations were solved self-consistently to derive the subband structures and the electron wave-functions. In the simulations, the electric field was 1.0 kV/cm. Polar optical phonon energy, hw, is 35 meV. Figure 1 shows the dependence of the 2DEG POP and ACP scattering rates in the first kind of structures as a function of a well width at an electron energy of 80meV. The scattering rates in bulk GaAs are also plotted as a comparison. The sawtooth-like feature of the 2DEG scattering rates can be understood as follows: as the quantum well size increases, the intra-subband transition matrix element decreases as explained previously. Therefore, the scattering rate declines with the increasing well width until at a certain width the inter-subband transition to the next subband occurs. Figure 1 also indicates that there exists a window of the well width that the 2DEG scattering rate is lower than the bulk electron scattering rate. This suggests that an enhancement of the 2DEG mobility over the bulk electron mobility is achievable in certain geometry quantum wells. Figure 2 shows the 2DEG mobility as a function of a well width. The peak mobility occurs at a width of 120 A, which corresponds to a subband structure with an energy difference between the first subband (E,) and the second subband (E,) equal to 2hw. The second peak appears at a well width of 250 8, corresponding to E2 - E, = 2hw. The 2DEG mobility approaches the bulk value, i.e. 8100 cm*/V-s in intrinsic GaAs, as the well width increases. In the inset of Fig. 2 we redraw the 2DEG mobility against the energy difference between E, and E, . A maximum mobility around E, - E. = 2hw is noticed. In order to explain this result, we calculated the electron distribution functions in a 120 A well at fields of 1.0 and 2.0 kV/cm respectively. The result is shown in Fig. 3. The equilibrium Boltzmann distribution is plotted as a reference, Figure 3 shows that the electron distri-
et al.
r
9000
6000
0
I
I
I
100
200
300
E,-E,(meV) I I 400
I
500
600
Well width (A) Fig. 2. Calculated 2D electron mobility as a function of a well width. The inset of the figure shows the 2D mobility as a function of the energy separation between the first subband E,, and the second subband E, .
butions exhibit a significant drop about Iho above the ground state due to the POP emission. In other words, the majority of electrons are restricted in an energy range from E. to E,, + ho. If the subband structure has E, -E. > 2hw, the majority of the electrons are immune from inter-subband scattering. Only electrons in the portion of the reduced distribution function (electrons with energy above E,, + hw) suffer from inter-subband scattering. When E, -E. = 2hw, the majority of the first subband electrons begin to have inter-subband scattering. As a consequence, the mobility declines at E, - E,, = 2hw. The second peak in the mobility has a similar explanation except that inter-subband transition takes place between the first subband and the third subband. In the second kind of quantum well structures, the 2DEG mobility versus the GaAs layer thickness in Fig. 4 shows a similar result. We find a peak mobility
12 x1012
Boltzmann --...- Field= 1.0 kV/cm ---FieldP 2.0 kV/cm
is ._
0
100
200 Well
300
400
500
width (A)
Fig. 1. POP and ACP scattering rates vs a quantum well width at an electron energy of 80 meV. The solid line is the 2DEG ACP scattering rate, the dashed line is the 2DEG POP scattering rate and the dotted lines represent the bulk electron scattering rates.
g 2
6
5 ._ 5 .g Iz ._ 0
4 2
20
40
60
80
100
Electron energy (meV) Fig. 3. The 2D electron distributions plotted as a function of electron energy at fields of 1 and 2 kV/cm. The solid line represents the equilibrium Boltzmann distribution.
Quantum well geometrical effects
1599
of the 2DEG mobility is expected in the AlGaAs/GaAs/AlGaAs structures over the AlGaAs/GaAs MODFET structures. The disappearante of the second peak at E, -E,, = 2hw in Fig. 4 arises from the same argument that the band-bending effect in a realistic structure limits the spread of the wave-function distribution. As a conclusion, our study shows that the 2DEG mobility varies significantly with the quantum well structures from 7250cmz/V-s in a SOA well to 8600 cm2/V-s in a 120 A well. The maximum mobility is obtained in a quantum well structure where the energy difference between the first and the second subbands is 2hw, or 70meV in GaAs wells.
enhancement
w
0
100
200
300
400
500
600
GaAs layer thickness (A) Fig. 4. 2D electron mobility vs the GaAs layer thickness in realistic n + AlGaAs/GaAs/AlGaAs quantum well structures. The inset illustrates the conduction band-edge profile and the subband structures.
Acknowledgement-Financial support from National Science Council, R.O.C., under contract No. NSCXO-0404E-009-09 is greatly acknowledged.
REFERENCE6
at E, - EO= 2hw again. The self-consistent bandedge diagram and the subband structure are illustrated in the inset of this figure. The 2DEG mobility converges to a value of 7900 cm2/V-s rather than the bulk electron mobility at a large GaAs layer thickness. The reason is that the quantum confinement of electrons always exists due to the band-bending at the n + AlGaAs/GaAs heterojunction without regard to the GaAs layer thickness. As a matter of fact, the convergent value of the mobility is the same as that in an AlGaAs/GaAs quantum well. About 10%
1. T. Mimura, S. Hiyamizu, T. Fujii and K. Nanbu, Japan J. appl. Phys. 19, L225 (1980).
2. L. Nguyen, W. Schaff, P. Tasker, A. Lepore, L. 3. 4. 5. 6 ’ 7.
Palmateer, M. Foisy and L. Eastman, IEEE Trans. Electron. Devices ED-35, 139 (1988). N. Sheng, C. P. Lee, R. T. Chen, D. Miller and S. J. Lee, IEEE Electron. Devices L~II. EDL6, 3078 (1985). B. K. Ridley, Quanmm Processing in Semiconductors, 2nd edn. Oxford Science (1988). K. Yokoyama and K. Hess, Phjs. Rev. 33,5595 (1985). M. Tomizawa. T. Furuta. K. Yokoyama and A. Yoshii, Proc. IEDM, b. 500 (1988). P. J. Price, Ann. Phys. 133, 217 (1981).
II, pp. 1597-1599, 1992
0038-I101/92 $5.00+ 0.00
in Great Britain
Pergamon Pmss Ltd
QUANTUM WELL GEOMETRICAL EFFECTS ON TWO-DIMENSIONAL ELECTRON MOBILITY TAHUI WANG,
TING-HUA
HSIEH
and YI-TEH CHEN
Institute of Electronics, National Chiao-Tung University, Hsin-Chu, Taiwan, R. 0. China (Received
17 April 1992)
Abstract-A
Monte Carlo analysis to study the quantization effects on the low-field electron mobility in various AlGaAs/GaAs/AlGaAs quantum well structures has been performed. The influence of the electron envelop wave-function and the subband structure on the two-dimensional electron scattering rates was evaluated. Our results showed that a maximum two-dimensional electron mobility can he achieved in a quantum well structure where the energy difference between the first subband and the second subband is about two times the polar optical phonon energy.
Recently, much research effort has been devoted to the use of heterostructures to achieve an enhancement of the electron mobility in ultra-high speed devices. Among the successful demonstrations of the heterojunction field effect transistors are AlGaAs/ GaAs Modulation Doped FETs[ 11,AlGaAs/InGaAs/ GaAs Pseudomorphic MODFETs[2] and Multiple Quantum Well FETs[3]. Although all of the above structures employ the concept of the “two-dimensional electron gas” (2DEG), the electron wavefunctions and subband structures in these 2DEG devices are quite different. Since the electron scattering rate is closely related to the wave-function distribution and the subband structure, the two-dimensional electron mobility may change significantly with the quantum-well geometry. In this study, we investigated the quantum well confinement effects on the 2DEG mobility. It was our intention to find an optimized AlGaAs/GaAs/AlGaAs quantum well structure for the mobility enhancement by tailoring the subband structure and the electron wavefunction. A 2DEG Monte Carlo analysis was utilized to study electron transport in the subbands of the quantum wells. The two-dimensional quantization effects on the scattering mechanisms in the Monte Carlo were evaluated from the Fermi-Golden rule[4]. As far as the low-field mobility is concerned, polar optical phonon (POP) scattering and accoustic phonon (ACP) scattering[5,6] were considered in this work. These are the two dominant scattering mechanisms in III-V semiconductors at low fields. The scattering rate of the 2DEG can be modulated with respect to the bulk electron scattering rate in two aspects, overlap integral and subband structure. The overlap integral is expressed as follows [7]:
s
L,(q) = ~,(z)~,(z)ev(iqz) dz
(1)
where q is the phonon wave-vector in the quantization direction. m and n are the index of the subbands SSE 35/11X
and F,,,(z) and F,(z) are the associated electron wave-functions. For intra-subband scattering (m = n), the overlap integral is our primary consideration in the calculation of the 2DEG scattering rate. Since a broad distribution of the wave-function in coordinate space corresponds to a narrow distribution in momentum space, the spread of the electron wave-function in wide quantum wells leads to a stringent momentum selection rule in eqn (1) and thus a reduced intrasubband scattering rate when q is away from 0. For inter-subband scattering (m #n), the energy separation in a subband structure dictates its significance to the 2DEG mobility. For example, if the second subband is far above the first subband, electrons in the first subband have little chance to be scattered to the second subband at low fields via phonon absorption. Inter-subband scattering is therefore virtually negligible in this situation. Unfortunately, the wavefunction distribution and the subband structure are usually two conflicting factors in the optimization of the 2DEG mobility. In a wide quantum well, the wave-function spreads and the intra-subband scattering rate is lowered. On the other side, the energy separation between the subbands is also reduced in wide quantum wells, which increases the occurrences of inter-subband scattering. The trade-off between these two factors leads to an optimization of the quantum well geometry. Two sorts of quantum-well structures were studied. The first kind has an Al,Ga,_,As/GaAs/ Al,Ga, _,As rectangular quantum well. The GaAs well width was varied in the simulatioin. The bandbending was purposely neglected in this kind of structures so that the spread of the wave-function can be solely determined by the well width. The second kind of structures have a realistic configuration, SOOA n+ Al,Ga,_. As/30 A i - Al,Ga, _ .As/GaAs/ i - Al,Ga, _,As. The doping concentration in then+ AlGaAs layer was 1 x lo’* l/cm3. Aluminum composition x was 0.3. In the second kind of structures, the
TAHUIWANG
1598
Schriidinger and the Poisson equations were solved self-consistently to derive the subband structures and the electron wave-functions. In the simulations, the electric field was 1.0 kV/cm. Polar optical phonon energy, hw, is 35 meV. Figure 1 shows the dependence of the 2DEG POP and ACP scattering rates in the first kind of structures as a function of a well width at an electron energy of 80meV. The scattering rates in bulk GaAs are also plotted as a comparison. The sawtooth-like feature of the 2DEG scattering rates can be understood as follows: as the quantum well size increases, the intra-subband transition matrix element decreases as explained previously. Therefore, the scattering rate declines with the increasing well width until at a certain width the inter-subband transition to the next subband occurs. Figure 1 also indicates that there exists a window of the well width that the 2DEG scattering rate is lower than the bulk electron scattering rate. This suggests that an enhancement of the 2DEG mobility over the bulk electron mobility is achievable in certain geometry quantum wells. Figure 2 shows the 2DEG mobility as a function of a well width. The peak mobility occurs at a width of 120 A, which corresponds to a subband structure with an energy difference between the first subband (E,) and the second subband (E,) equal to 2hw. The second peak appears at a well width of 250 8, corresponding to E2 - E, = 2hw. The 2DEG mobility approaches the bulk value, i.e. 8100 cm*/V-s in intrinsic GaAs, as the well width increases. In the inset of Fig. 2 we redraw the 2DEG mobility against the energy difference between E, and E, . A maximum mobility around E, - E. = 2hw is noticed. In order to explain this result, we calculated the electron distribution functions in a 120 A well at fields of 1.0 and 2.0 kV/cm respectively. The result is shown in Fig. 3. The equilibrium Boltzmann distribution is plotted as a reference, Figure 3 shows that the electron distri-
et al.
r
9000
6000
0
I
I
I
100
200
300
E,-E,(meV) I I 400
I
500
600
Well width (A) Fig. 2. Calculated 2D electron mobility as a function of a well width. The inset of the figure shows the 2D mobility as a function of the energy separation between the first subband E,, and the second subband E, .
butions exhibit a significant drop about Iho above the ground state due to the POP emission. In other words, the majority of electrons are restricted in an energy range from E. to E,, + ho. If the subband structure has E, -E. > 2hw, the majority of the electrons are immune from inter-subband scattering. Only electrons in the portion of the reduced distribution function (electrons with energy above E,, + hw) suffer from inter-subband scattering. When E, -E. = 2hw, the majority of the first subband electrons begin to have inter-subband scattering. As a consequence, the mobility declines at E, - E,, = 2hw. The second peak in the mobility has a similar explanation except that inter-subband transition takes place between the first subband and the third subband. In the second kind of quantum well structures, the 2DEG mobility versus the GaAs layer thickness in Fig. 4 shows a similar result. We find a peak mobility
12 x1012
Boltzmann --...- Field= 1.0 kV/cm ---FieldP 2.0 kV/cm
is ._
0
100
200 Well
300
400
500
width (A)
Fig. 1. POP and ACP scattering rates vs a quantum well width at an electron energy of 80 meV. The solid line is the 2DEG ACP scattering rate, the dashed line is the 2DEG POP scattering rate and the dotted lines represent the bulk electron scattering rates.
g 2
6
5 ._ 5 .g Iz ._ 0
4 2
20
40
60
80
100
Electron energy (meV) Fig. 3. The 2D electron distributions plotted as a function of electron energy at fields of 1 and 2 kV/cm. The solid line represents the equilibrium Boltzmann distribution.
Quantum well geometrical effects
1599
of the 2DEG mobility is expected in the AlGaAs/GaAs/AlGaAs structures over the AlGaAs/GaAs MODFET structures. The disappearante of the second peak at E, -E,, = 2hw in Fig. 4 arises from the same argument that the band-bending effect in a realistic structure limits the spread of the wave-function distribution. As a conclusion, our study shows that the 2DEG mobility varies significantly with the quantum well structures from 7250cmz/V-s in a SOA well to 8600 cm2/V-s in a 120 A well. The maximum mobility is obtained in a quantum well structure where the energy difference between the first and the second subbands is 2hw, or 70meV in GaAs wells.
enhancement
w
0
100
200
300
400
500
600
GaAs layer thickness (A) Fig. 4. 2D electron mobility vs the GaAs layer thickness in realistic n + AlGaAs/GaAs/AlGaAs quantum well structures. The inset illustrates the conduction band-edge profile and the subband structures.
Acknowledgement-Financial support from National Science Council, R.O.C., under contract No. NSCXO-0404E-009-09 is greatly acknowledged.
REFERENCE6
at E, - EO= 2hw again. The self-consistent bandedge diagram and the subband structure are illustrated in the inset of this figure. The 2DEG mobility converges to a value of 7900 cm2/V-s rather than the bulk electron mobility at a large GaAs layer thickness. The reason is that the quantum confinement of electrons always exists due to the band-bending at the n + AlGaAs/GaAs heterojunction without regard to the GaAs layer thickness. As a matter of fact, the convergent value of the mobility is the same as that in an AlGaAs/GaAs quantum well. About 10%
1. T. Mimura, S. Hiyamizu, T. Fujii and K. Nanbu, Japan J. appl. Phys. 19, L225 (1980).
2. L. Nguyen, W. Schaff, P. Tasker, A. Lepore, L. 3. 4. 5. 6 ’ 7.
Palmateer, M. Foisy and L. Eastman, IEEE Trans. Electron. Devices ED-35, 139 (1988). N. Sheng, C. P. Lee, R. T. Chen, D. Miller and S. J. Lee, IEEE Electron. Devices L~II. EDL6, 3078 (1985). B. K. Ridley, Quanmm Processing in Semiconductors, 2nd edn. Oxford Science (1988). K. Yokoyama and K. Hess, Phjs. Rev. 33,5595 (1985). M. Tomizawa. T. Furuta. K. Yokoyama and A. Yoshii, Proc. IEDM, b. 500 (1988). P. J. Price, Ann. Phys. 133, 217 (1981).