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Thermoelectric power of superlattices
Surface Science 142 (1984)241-245 North-Holland. Amsterdam
THERMOELECTRIC Lionel
POWER
241
OF SUPERLATTICES
FRIEDMAN
GTE L.uhoratortes, Inc.. 40 Sylvan Rood, Waltham, Received
6 July 1983; accepted
for publication
Massachusetts
19 August
02254, USA
1983
The low temperature thermoelectric power of a doped superlattice as a function of the Fermi energy (band filling) locates the positions of the edges of the minibands. The thermoelectric power is anisotropic according to whether the temperature gradient is along or perpendicular to the superlattice axis: for momentum relaxation rates proportional to the density of final states and for the Fermi energy in the gap between minibands, it vanishes in the former case, but not in the latter. For the Fermi level near the tops of the minibands, a sign reversal is predicted indicative of hole-like behaviour.
Layered structures with precise interfacial and dimensional control made possible by molecular beam epitaxy (MBE) are the subject of much current interests. The spatial quantization of the electronic energy levels in such single or multiple quantum well structures has been observed in optical absorption [l]. When the quantum wells are sufficiently close that there is a spatially periodic, finite overlap of the electronic wave functions of adjacent wells, one has a bona-fide superlattice. The resulting zone folding and broading of the discrete electronic levels into-narrow minibands was the basis for the early predictions of negative differential resistance and Bloch oscillations by Esaki and Tsu [2], and later predictions of nonlinear optical properties of superlattices [3]. While analogous phenomena have been observed for the phonon spectra [4], it has not yet been seen for electrons (or holes). However, with continued improvement in growth and interfacial quality made possible by MBE, the electronic miniband structure will likely be realized [5]. For this case, it is shown in the present paper that the low temperature thermoelectric power as a function of the Fermi level (band filling) maps out the location in energy and widths of the minibands, thereby providing basic information about the band structure and density of states distribution of a superlattice. Additionally, information is provided about the scattering mechanisms operative. The standard form of the electron’s dispersion in a superlattice is free-electron-like parallel to the layers, and tight-binding-like perpendicular to the layers, viz:
E(k,, ,
k,)=Z +t(l
- cos k,d),
0039-6028/84/$03.00 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
242
L. Frredman / Thermoelectrrc power of superlattices
where m” is the conduction band effective mass, t is the transfer integral (half the bandwidth) in the perpendicular direction, and d is the superlattice period. The density of states corresponding to eq. (1) is (6)
%Lb~=~q;s-‘(1-c/t),
O
(2)
i>21.
At T = 0, impurities introduced by uniform or modulation doping yield carriers which fill the density of available states. For partial filling of the lowest miniband, the carrier density N corresponding to a Fermi energy [ is given by
The thermoelectric
power S of a degenerate
electron
gas may be written
[7] (4)
where n(e) is the density of states, 3 is the square of the electron’s velocity in the direction of the external temperature gradient averaged over the Fermi surface c = {, and T(C) is the energy-dependent momentum relaxation time. Eq. (4) requires that: (1) k,T < 2 and (2) the energy dependent quantities be slowly varying on the scale of k,T. These conditions will be investigated immediately below.
Fig. 1. Logarithmic
derivative
of superlattice
density
of states versus energy
L. Friedman / Thermoelecfric power of superlattices
243
The logarithmic derivative of 11s’(e) is found to vary smoothly through the miniband, but to diverge at the band extrema as
1 ansL --= ns’
i3C
i
(t2S)_‘,
se
(t?7-‘(2s,)-“‘,
6’<1,
0,
s>2
with
e
1,
(5)
where S = C/T, 6’ = (2 - 6). The behaviour through the miniband is shown in fig. 1. However, when the energy variation is sufficient that condition (2) above is violated, the quantity is averaged over an energy interval - k,T and does not diverge. But for layers sufficiently thin to show superlattice banding, e.g. a typical GaAssGaAl.,As’ _-x superlattices (d - 50-70 A), Kronig-Penney calculations yield bandwidths 2 t - 60-120 meV [3]. Since k,T in low temperature thermoelectric power measurements can be made significantly smaller than this, most of the variation through the band will be discernable, except very close to the miniband edges. Of course for 6 > 2 in the “gap”, 8 In n,,/& = 0. Thus the positions of the miniband edges are indicated. We also find that for carrier densities 1016 to 5 x 10” cmm3, condition (1) is also satisfied for typical superlattice configurations. Turning next to the second term in the brackets of eq. (4) there are two cases according to whether the temperature gradient is parallel or perpendicular to the layers. For the first case, u, = Ak,,/m*, it is found that
I au, --= -2 ae -3
7 (c-qcos-
(1 - 8) + (28 - 82)“2
(26
\t-‘(S -
-
S2)“2
1
0’
-
2
ac
I
(6)
6 < 2, I
’
sin kLd,
1 - S) - cos-‘(1
x
-q2_((2&p)“*_
b,
- S)
s>2,
2t (2s - s*)“2(1
1
cos-‘(1
l),
while in the second case, u, = (td/A)
1aU: --=
1
(1-6)
“I
(2&sy(1
- S) 1--E
coss’(1
-S)
I’ s<2 s>2
(7) Both eq. (6) and eq. (7) tend to diverge as S-’ at the bottom f the band, and as - (Sf)‘/2 at the top of the band, i.e. a sign reversal indicative of hole-like behaviour is found here. Their behaviour are very similar and is shown in fig.
244
L. Friedman
/ Thermoelectnc
power of superluttices
2. The removal of the divergences just at the band edges occurs for the same reasons as for the density of states. Finally, we consider the third term in the brackets of eq. (4). In contrast to the bulk or an isolated quantum well, the momentum relaxation rates are not known for all the scattering mechanisms of interest in the case of a true superlattice. Unlike the quantum well case, there is a density of final states with momenta along the superlattice axis into which the carrier can scatter. One result derived for a superlattice is that of deformation potential scattering under the usual assumptions of elastic scattering and phonon equipartition [6]. For this case r&(e) - n,,(e), the logarithmic derivative of T& exactly cancels the first term of eq. (4) and the entire contribution to S comes from eqs. (6) and (7), exhibiting the sign reversals at the tops of the minibands. On the other hand, if energy independent rates characteristic of a quantum well (infinite potential barriers) are assumed [8], the logarithmic derivative of 7-l vanishes, and the contributions to S are the sums of eqs. (5) (6) and (7). In this case, the sign reversals at the tops of the minibands do not occur. Also, as shown by eqs. (5) (6), and (7) for the case 6 > 2 (Fermi energy above the top of the lowest miniband), these contributions to S vanish when the temperature gradient is along the superlattice axis, but not when it is parallel to the layers. The reason is that there is no coherent transport of carrier kinetic energy perpendicular to the layers when the Fermi level is in the gap between minibands. If rates proportional to the density of final states multiplied by the carrier energy to some power e” are assumed (the latter characterizing the energy dependence of the matrix element), then an additional slowly varying term -p/c contributes for both orientations of the thermal gradient, and S never vanishes for the
Fig. 2. Logarithmic
derivative
of energy of transport
versus energy
L. Friedman
/ Thermoelectric
power of superlattices
245
Fermi level in the “gap”. However, the tendency of S to “diverge” at the miniband extrema occurs in all cases, indicating the location in energy of the miniband edges. A more detailed description of this work will appear in J. Phys. C, and a critique of the tight-binding approximation (eq. (11)) will appear elsewhere [9].
References [l] R. Dingle, in: Festk&perprobleme (Advances in Solid State Physics), Vol. 15, Ed. H.J. Queisser (Pergamon/Vieweg, Braunschweig, 1975) p. 21. [2] L. Esaki and R. Tsu, IBM J. Res. Develop. 14 (1970) 61,’ [3] W.L. Bloss and L. Friedman, Appl. Phys. Letters 41 (1982) 1023. [4] C. Colvard, R. Merlin, M.V. Klein and A.C. Gossard, Phys. Rev. Letters 45 (1980) 298. [5] L.L. Chang and L. Esaki, Surface Sci. 98 (1980) 70. (61 J.F. Palmier and Y. Ballini, J. Physique 41 (1980) L539. [7] D.K.C. MacDonald, Thermoelectricity: An Introduction to the Principles (Wiley, New York, 1962). [8] B.K. Ridley. J. Phys. Cl5 (1982) 5899. [9] G. Cooperman, L. Friedman and W.L. Bless, Appl. Phys. Letters, to be published.
THERMOELECTRIC Lionel
POWER
241
OF SUPERLATTICES
FRIEDMAN
GTE L.uhoratortes, Inc.. 40 Sylvan Rood, Waltham, Received
6 July 1983; accepted
for publication
Massachusetts
19 August
02254, USA
1983
The low temperature thermoelectric power of a doped superlattice as a function of the Fermi energy (band filling) locates the positions of the edges of the minibands. The thermoelectric power is anisotropic according to whether the temperature gradient is along or perpendicular to the superlattice axis: for momentum relaxation rates proportional to the density of final states and for the Fermi energy in the gap between minibands, it vanishes in the former case, but not in the latter. For the Fermi level near the tops of the minibands, a sign reversal is predicted indicative of hole-like behaviour.
Layered structures with precise interfacial and dimensional control made possible by molecular beam epitaxy (MBE) are the subject of much current interests. The spatial quantization of the electronic energy levels in such single or multiple quantum well structures has been observed in optical absorption [l]. When the quantum wells are sufficiently close that there is a spatially periodic, finite overlap of the electronic wave functions of adjacent wells, one has a bona-fide superlattice. The resulting zone folding and broading of the discrete electronic levels into-narrow minibands was the basis for the early predictions of negative differential resistance and Bloch oscillations by Esaki and Tsu [2], and later predictions of nonlinear optical properties of superlattices [3]. While analogous phenomena have been observed for the phonon spectra [4], it has not yet been seen for electrons (or holes). However, with continued improvement in growth and interfacial quality made possible by MBE, the electronic miniband structure will likely be realized [5]. For this case, it is shown in the present paper that the low temperature thermoelectric power as a function of the Fermi level (band filling) maps out the location in energy and widths of the minibands, thereby providing basic information about the band structure and density of states distribution of a superlattice. Additionally, information is provided about the scattering mechanisms operative. The standard form of the electron’s dispersion in a superlattice is free-electron-like parallel to the layers, and tight-binding-like perpendicular to the layers, viz:
E(k,, ,
k,)=Z +t(l
- cos k,d),
0039-6028/84/$03.00 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
242
L. Frredman / Thermoelectrrc power of superlattices
where m” is the conduction band effective mass, t is the transfer integral (half the bandwidth) in the perpendicular direction, and d is the superlattice period. The density of states corresponding to eq. (1) is (6)
%Lb~=~q;s-‘(1-c/t),
O
(2)
i>21.
At T = 0, impurities introduced by uniform or modulation doping yield carriers which fill the density of available states. For partial filling of the lowest miniband, the carrier density N corresponding to a Fermi energy [ is given by
The thermoelectric
power S of a degenerate
electron
gas may be written
[7] (4)
where n(e) is the density of states, 3 is the square of the electron’s velocity in the direction of the external temperature gradient averaged over the Fermi surface c = {, and T(C) is the energy-dependent momentum relaxation time. Eq. (4) requires that: (1) k,T < 2 and (2) the energy dependent quantities be slowly varying on the scale of k,T. These conditions will be investigated immediately below.
Fig. 1. Logarithmic
derivative
of superlattice
density
of states versus energy
L. Friedman / Thermoelecfric power of superlattices
243
The logarithmic derivative of 11s’(e) is found to vary smoothly through the miniband, but to diverge at the band extrema as
1 ansL --= ns’
i3C
i
(t2S)_‘,
se
(t?7-‘(2s,)-“‘,
6’<1,
0,
s>2
with
e
1,
(5)
where S = C/T, 6’ = (2 - 6). The behaviour through the miniband is shown in fig. 1. However, when the energy variation is sufficient that condition (2) above is violated, the quantity is averaged over an energy interval - k,T and does not diverge. But for layers sufficiently thin to show superlattice banding, e.g. a typical GaAssGaAl.,As’ _-x superlattices (d - 50-70 A), Kronig-Penney calculations yield bandwidths 2 t - 60-120 meV [3]. Since k,T in low temperature thermoelectric power measurements can be made significantly smaller than this, most of the variation through the band will be discernable, except very close to the miniband edges. Of course for 6 > 2 in the “gap”, 8 In n,,/& = 0. Thus the positions of the miniband edges are indicated. We also find that for carrier densities 1016 to 5 x 10” cmm3, condition (1) is also satisfied for typical superlattice configurations. Turning next to the second term in the brackets of eq. (4) there are two cases according to whether the temperature gradient is parallel or perpendicular to the layers. For the first case, u, = Ak,,/m*, it is found that
I au, --= -2 ae -3
7 (c-qcos-
(1 - 8) + (28 - 82)“2
(26
\t-‘(S -
-
S2)“2
1
0’
-
2
ac
I
(6)
6 < 2, I
’
sin kLd,
1 - S) - cos-‘(1
x
-q2_((2&p)“*_
b,
- S)
s>2,
2t (2s - s*)“2(1
1
cos-‘(1
l),
while in the second case, u, = (td/A)
1aU: --=
1
(1-6)
“I
(2&sy(1
- S) 1--E
coss’(1
-S)
I’ s<2 s>2
(7) Both eq. (6) and eq. (7) tend to diverge as S-’ at the bottom f the band, and as - (Sf)‘/2 at the top of the band, i.e. a sign reversal indicative of hole-like behaviour is found here. Their behaviour are very similar and is shown in fig.
244
L. Friedman
/ Thermoelectnc
power of superluttices
2. The removal of the divergences just at the band edges occurs for the same reasons as for the density of states. Finally, we consider the third term in the brackets of eq. (4). In contrast to the bulk or an isolated quantum well, the momentum relaxation rates are not known for all the scattering mechanisms of interest in the case of a true superlattice. Unlike the quantum well case, there is a density of final states with momenta along the superlattice axis into which the carrier can scatter. One result derived for a superlattice is that of deformation potential scattering under the usual assumptions of elastic scattering and phonon equipartition [6]. For this case r&(e) - n,,(e), the logarithmic derivative of T& exactly cancels the first term of eq. (4) and the entire contribution to S comes from eqs. (6) and (7), exhibiting the sign reversals at the tops of the minibands. On the other hand, if energy independent rates characteristic of a quantum well (infinite potential barriers) are assumed [8], the logarithmic derivative of 7-l vanishes, and the contributions to S are the sums of eqs. (5) (6) and (7). In this case, the sign reversals at the tops of the minibands do not occur. Also, as shown by eqs. (5) (6), and (7) for the case 6 > 2 (Fermi energy above the top of the lowest miniband), these contributions to S vanish when the temperature gradient is along the superlattice axis, but not when it is parallel to the layers. The reason is that there is no coherent transport of carrier kinetic energy perpendicular to the layers when the Fermi level is in the gap between minibands. If rates proportional to the density of final states multiplied by the carrier energy to some power e” are assumed (the latter characterizing the energy dependence of the matrix element), then an additional slowly varying term -p/c contributes for both orientations of the thermal gradient, and S never vanishes for the
Fig. 2. Logarithmic
derivative
of energy of transport
versus energy
L. Friedman
/ Thermoelectric
power of superlattices
245
Fermi level in the “gap”. However, the tendency of S to “diverge” at the miniband extrema occurs in all cases, indicating the location in energy of the miniband edges. A more detailed description of this work will appear in J. Phys. C, and a critique of the tight-binding approximation (eq. (11)) will appear elsewhere [9].
References [l] R. Dingle, in: Festk&perprobleme (Advances in Solid State Physics), Vol. 15, Ed. H.J. Queisser (Pergamon/Vieweg, Braunschweig, 1975) p. 21. [2] L. Esaki and R. Tsu, IBM J. Res. Develop. 14 (1970) 61,’ [3] W.L. Bloss and L. Friedman, Appl. Phys. Letters 41 (1982) 1023. [4] C. Colvard, R. Merlin, M.V. Klein and A.C. Gossard, Phys. Rev. Letters 45 (1980) 298. [5] L.L. Chang and L. Esaki, Surface Sci. 98 (1980) 70. (61 J.F. Palmier and Y. Ballini, J. Physique 41 (1980) L539. [7] D.K.C. MacDonald, Thermoelectricity: An Introduction to the Principles (Wiley, New York, 1962). [8] B.K. Ridley. J. Phys. Cl5 (1982) 5899. [9] G. Cooperman, L. Friedman and W.L. Bless, Appl. Phys. Letters, to be published.