- Email: [email protected]
Resonant wavefunctions and exchange effect in nanostructures
Superlattices and Microstructures, Vol. 26, No. 6,1999 Article No. spmi.1999.0793 Available online at http://www.idealibrary.com on
Resonant wavefunctions and exchange effect in nanostructures P ETER J. P RICE† IBM T.J. Watson Research Center, Yorktown Heights, New York 10598, U.S.A. (Received 12 October 1999) It is shown that itinerant electrons in a nanostructure have the property, in a resonance energy range, that the wavefunctions exhibiting a resonance peak all have substantially the same position dependence within the ‘quantum dot’. In the resonance energy range, they differ only by a Lorentzian function of energy including a phase factor. Consequently, for electron states with parallel spins there should be an exchange cancellation of their Coulomb interaction. c 1999 Academic Press
Key words: nanostructure, exchange effect, resonant wavefunctions.
In nanostructures that incompletely confine electrons, with the electrons propagating into and out of the structure through one or more ‘waveguide’ channels, for a variety of nanostructure geometries there are resonances in the coherent quantum states of these propagating electrons. The amplitudes of the outward channel waves, relative to the inward wave amplitudes, vary sharply over a small range, 0, of electron energy E, both in their phases and in the transmission (reflection) probabilities. Also, at a resonance there is a ‘bulge’ in the wavefunction ψ(r) within the structure, so that electron density ρ(r) = |ψ|2 peaks in the resonance energy range. The sum of this density ρ(r, E) (for ψ normalized in the channels) over all electron states in a resonance energy range equals just one electron, for each of the two spin states. Thus a resonance corresponds—generically—to a ‘quasi-level’ (being equivalent to the decaying local state in the Gamow theory of the alpha decay of atomic nuclei). The Coulomb interaction within this resonant space charge will displace the propagating single-electron states upward in energy, and the resulting effect on conductance properties can be considerable. It has been argued [1, 2] (a) that the ‘bulge’ wavefunctions within the nanostructure should be substantially of the form ψ(r, E) = φ(r)F(E)
(1)
over a resonance energy range, and (b) that consequently an exchange effect will substantially cancel the Coulomb interaction for resonant electron states with parallel spins, so that the interaction energy will be reduced to approximately half the Hartree value. The present communication will substantiate this heuristic idea, by means of an analytical treatment of the orthogonality factor, Z I1 2 = ψ1∗ ψ2 d 3 r (2) † E-mail: [email protected]
0749–6036/99/120363 + 03 $30.00/0
c 1999 Academic Press
364
Superlattices and Microstructures, Vol. 26, No. 6,1999
where the integration domain is the nanostructure confinement space, while the energy eigenvalues E 1 and E 2 vary over a resonance range ∼0 around E 0 . (The equivalent orthogonality factor for eigenstates integrated over the whole domain of the wavefunctions including the ‘waveguides’ would be zero.) It is found that, to sufficient approximation, |I1 2 | = |F(E 1 )F(E 2 )|, so that ψ varies only by the Lorentzian factor F(E) and by a phase angle. (The latter does not affect the exchange cancellation, as will be noted below.) The starting point is the identity Z 0 dψ2∗ ~2 ∗ ∗ dψ1 (E 2 − E 1 ) ψ2 ψ1 d x = ψ − ψ1 (3) 2m ∗ 2 d x d x x=0 which is a consequence of the single-electron wave equation. Here the recourse to a one-dimensional variable, x, is for convenience and is not a significant restriction. The resonant nanostructure is represented by the domain x ≤ 0, and the attached ‘waveguide’ region by x > 0. The restriction in (3) to a single propagation channel, so that the resonance is a property of a reflection, is for simplicity; and the multi-channel generalization is straightforward [2]. The ψ(x) are meant to be envelope functions, with an electron effective mass m ∗ . The implicit restriction to the parabolic range of Bloch energy function E(k) is a real meaningful limitation. (It is expressed in eqn (3) by the form of current operator applied to the ψ functions, the underlying formula being current conservation.) It is convenient to take the channel wavefunctions, for x > 0, as normalized in unit length: ψ = e−ikx + r (E)eikx
(4)
where the first term is the incident wave and the second the reflected wave. In the multi-channel generalization, the coefficient r (E) becomes the S matrix, which for real E is unitary as a consequence of current conservation. Here, in particular, r = exp iθ(E) (5) where θ is real for real E. Substituting (4) and (5) into (3), and writing the integral on the left there as I2 1 to correspond to (2), we obtain m∗ i (E 1 − E 2 )I2 1 = {(k1 + k2 )(1 − r1r2∗ ) + (k1 − k2 )(r2∗ − r1 )}. 2 ~2
(6)
That is, m∗ (E 1 − E 2 )I2 1 = ei(θ1 −θ2 )/2 {(k1 + k2 ) sin(θ1 − θ2 )/2 + (k1 − k2 ) sin(θ1 + θ2 )/2}. (7) ~2 An immediate consequence of (7) is a formula for resonant space charge density in terms of θ (E) and the R0 dwell time tdwell = I /ν where I = |ψ|2 d x, with the normalization of eqn (4), and where ~ν = d E/dk. When E 1 → E 2 then (7) reduces to (m ∗ /~2 )I = kdθ/d E + (k/2E) sin θ . Since we have E proportional to k 2 , the result is 1 dθ sin θ tdwell = + . (8) ~ dE 2E The first term of (8) is dominant at a sharp resonance, where θ(E) increases by 2π over the resonance width, while the second term is a version of the Friedel oscillation [3]. We may also note an inequality: that the right-hand side of (8) must be ≥ 0 (since the left-hand side cannot be negative), which restricts the possible forms of θ versus E. As with eqn (8), at a resonance the first term on the right of eqns (6) and (7) dominates over the second term, because θ (E) varies rapidly while k(E) does not. We therefore drop this second term now. At a simple resonance (the Fano type being a little more complicated) we have r = eiθ =
E − E 0 − i0 E − E 0 + i0
(9)
Superlattices and Microstructures, Vol. 26, No. 6,1999
365
where E 0 is the quasi-level energy and 0 is the Lorentzian half-width. On substituting (9) into the first term, only, of (6) we obtain 0 m∗ 02 I = . 2 1 k 1 + k 2 ~2 (E 1 − E 0 )(E 2 − E 0 ) − i0(E 1 − E 2 ) + 0 2
(10)
For E 1 = E 2 , the right-hand side of eqn (10) becomes the usual Lorentzian factor 3(E) = 0 2 /[(E − E 0 )2 + 0 2 ] which is given by the first term dθ/d E of (8). Otherwise, we have from (10) 0 m∗ 2 |I1 2 |2 = 3(E 1 )3(E 2 ). k 1 + k 2 ~2
(11)
(12)
Since k1 = k2 to sufficient approximation within the resonant range of E, this verifies (1). The denominator on the right of (10) is equal to (E 1 − E 0 + i0)(E 2 − E 0 − i0). Hence the phase angle of the resonant wavefunction ≡ arg F(E) in (1) is given by tan = −0/(E − E 0 ).
(13)
Thus the phase angle of the single-electron wavefunction swings through π as P E increases through the resonant range. Because the sum of the phases of the component electron states, n (E n ), is unchanged by the permutations in a Slater determinant, the exchange effect does not involve these phases. Thus we have verified, by the foregoing analysis, that, as was previously argued heuristically, [1] for electrons with parallel spins and with energies in a resonance range there is an exchange cancellation of the Coulomb interaction within the ‘quantum dot’. This would normally leave about half the ‘Hartree’ interaction, remaining from states with opposite spins.
References [1] P. J. Price, J. Appl. Phys. 79, 7379 (1996). [2] P. J. Price, Superlatt. Microstruct. 20, 253 (1996). [3] J. Friedel, Phil. Mag. 43, 153 (1952). This paper contains (in terms of the expansion in spherical harmonics of standard scattering theory) both eqn (8) here and the derivation of it from eqn (3) here by taking the limit E 1 − E 2 → 0. This prior source was overlooked in my previous publications.
Resonant wavefunctions and exchange effect in nanostructures P ETER J. P RICE† IBM T.J. Watson Research Center, Yorktown Heights, New York 10598, U.S.A. (Received 12 October 1999) It is shown that itinerant electrons in a nanostructure have the property, in a resonance energy range, that the wavefunctions exhibiting a resonance peak all have substantially the same position dependence within the ‘quantum dot’. In the resonance energy range, they differ only by a Lorentzian function of energy including a phase factor. Consequently, for electron states with parallel spins there should be an exchange cancellation of their Coulomb interaction. c 1999 Academic Press
Key words: nanostructure, exchange effect, resonant wavefunctions.
In nanostructures that incompletely confine electrons, with the electrons propagating into and out of the structure through one or more ‘waveguide’ channels, for a variety of nanostructure geometries there are resonances in the coherent quantum states of these propagating electrons. The amplitudes of the outward channel waves, relative to the inward wave amplitudes, vary sharply over a small range, 0, of electron energy E, both in their phases and in the transmission (reflection) probabilities. Also, at a resonance there is a ‘bulge’ in the wavefunction ψ(r) within the structure, so that electron density ρ(r) = |ψ|2 peaks in the resonance energy range. The sum of this density ρ(r, E) (for ψ normalized in the channels) over all electron states in a resonance energy range equals just one electron, for each of the two spin states. Thus a resonance corresponds—generically—to a ‘quasi-level’ (being equivalent to the decaying local state in the Gamow theory of the alpha decay of atomic nuclei). The Coulomb interaction within this resonant space charge will displace the propagating single-electron states upward in energy, and the resulting effect on conductance properties can be considerable. It has been argued [1, 2] (a) that the ‘bulge’ wavefunctions within the nanostructure should be substantially of the form ψ(r, E) = φ(r)F(E)
(1)
over a resonance energy range, and (b) that consequently an exchange effect will substantially cancel the Coulomb interaction for resonant electron states with parallel spins, so that the interaction energy will be reduced to approximately half the Hartree value. The present communication will substantiate this heuristic idea, by means of an analytical treatment of the orthogonality factor, Z I1 2 = ψ1∗ ψ2 d 3 r (2) † E-mail: [email protected]
0749–6036/99/120363 + 03 $30.00/0
c 1999 Academic Press
364
Superlattices and Microstructures, Vol. 26, No. 6,1999
where the integration domain is the nanostructure confinement space, while the energy eigenvalues E 1 and E 2 vary over a resonance range ∼0 around E 0 . (The equivalent orthogonality factor for eigenstates integrated over the whole domain of the wavefunctions including the ‘waveguides’ would be zero.) It is found that, to sufficient approximation, |I1 2 | = |F(E 1 )F(E 2 )|, so that ψ varies only by the Lorentzian factor F(E) and by a phase angle. (The latter does not affect the exchange cancellation, as will be noted below.) The starting point is the identity Z 0 dψ2∗ ~2 ∗ ∗ dψ1 (E 2 − E 1 ) ψ2 ψ1 d x = ψ − ψ1 (3) 2m ∗ 2 d x d x x=0 which is a consequence of the single-electron wave equation. Here the recourse to a one-dimensional variable, x, is for convenience and is not a significant restriction. The resonant nanostructure is represented by the domain x ≤ 0, and the attached ‘waveguide’ region by x > 0. The restriction in (3) to a single propagation channel, so that the resonance is a property of a reflection, is for simplicity; and the multi-channel generalization is straightforward [2]. The ψ(x) are meant to be envelope functions, with an electron effective mass m ∗ . The implicit restriction to the parabolic range of Bloch energy function E(k) is a real meaningful limitation. (It is expressed in eqn (3) by the form of current operator applied to the ψ functions, the underlying formula being current conservation.) It is convenient to take the channel wavefunctions, for x > 0, as normalized in unit length: ψ = e−ikx + r (E)eikx
(4)
where the first term is the incident wave and the second the reflected wave. In the multi-channel generalization, the coefficient r (E) becomes the S matrix, which for real E is unitary as a consequence of current conservation. Here, in particular, r = exp iθ(E) (5) where θ is real for real E. Substituting (4) and (5) into (3), and writing the integral on the left there as I2 1 to correspond to (2), we obtain m∗ i (E 1 − E 2 )I2 1 = {(k1 + k2 )(1 − r1r2∗ ) + (k1 − k2 )(r2∗ − r1 )}. 2 ~2
(6)
That is, m∗ (E 1 − E 2 )I2 1 = ei(θ1 −θ2 )/2 {(k1 + k2 ) sin(θ1 − θ2 )/2 + (k1 − k2 ) sin(θ1 + θ2 )/2}. (7) ~2 An immediate consequence of (7) is a formula for resonant space charge density in terms of θ (E) and the R0 dwell time tdwell = I /ν where I = |ψ|2 d x, with the normalization of eqn (4), and where ~ν = d E/dk. When E 1 → E 2 then (7) reduces to (m ∗ /~2 )I = kdθ/d E + (k/2E) sin θ . Since we have E proportional to k 2 , the result is 1 dθ sin θ tdwell = + . (8) ~ dE 2E The first term of (8) is dominant at a sharp resonance, where θ(E) increases by 2π over the resonance width, while the second term is a version of the Friedel oscillation [3]. We may also note an inequality: that the right-hand side of (8) must be ≥ 0 (since the left-hand side cannot be negative), which restricts the possible forms of θ versus E. As with eqn (8), at a resonance the first term on the right of eqns (6) and (7) dominates over the second term, because θ (E) varies rapidly while k(E) does not. We therefore drop this second term now. At a simple resonance (the Fano type being a little more complicated) we have r = eiθ =
E − E 0 − i0 E − E 0 + i0
(9)
Superlattices and Microstructures, Vol. 26, No. 6,1999
365
where E 0 is the quasi-level energy and 0 is the Lorentzian half-width. On substituting (9) into the first term, only, of (6) we obtain 0 m∗ 02 I = . 2 1 k 1 + k 2 ~2 (E 1 − E 0 )(E 2 − E 0 ) − i0(E 1 − E 2 ) + 0 2
(10)
For E 1 = E 2 , the right-hand side of eqn (10) becomes the usual Lorentzian factor 3(E) = 0 2 /[(E − E 0 )2 + 0 2 ] which is given by the first term dθ/d E of (8). Otherwise, we have from (10) 0 m∗ 2 |I1 2 |2 = 3(E 1 )3(E 2 ). k 1 + k 2 ~2
(11)
(12)
Since k1 = k2 to sufficient approximation within the resonant range of E, this verifies (1). The denominator on the right of (10) is equal to (E 1 − E 0 + i0)(E 2 − E 0 − i0). Hence the phase angle of the resonant wavefunction ≡ arg F(E) in (1) is given by tan = −0/(E − E 0 ).
(13)
Thus the phase angle of the single-electron wavefunction swings through π as P E increases through the resonant range. Because the sum of the phases of the component electron states, n (E n ), is unchanged by the permutations in a Slater determinant, the exchange effect does not involve these phases. Thus we have verified, by the foregoing analysis, that, as was previously argued heuristically, [1] for electrons with parallel spins and with energies in a resonance range there is an exchange cancellation of the Coulomb interaction within the ‘quantum dot’. This would normally leave about half the ‘Hartree’ interaction, remaining from states with opposite spins.
References [1] P. J. Price, J. Appl. Phys. 79, 7379 (1996). [2] P. J. Price, Superlatt. Microstruct. 20, 253 (1996). [3] J. Friedel, Phil. Mag. 43, 153 (1952). This paper contains (in terms of the expansion in spherical harmonics of standard scattering theory) both eqn (8) here and the derivation of it from eqn (3) here by taking the limit E 1 − E 2 → 0. This prior source was overlooked in my previous publications.