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Can the Bogoliubov–de Gennes equation be interpreted as a ‘one-particle’ wave equation?
Superlattices and Microstructures, Vol. 25, No. 5/6, 1999 Article No. spmi.1999.0747 Available online at http://www.idealibrary.com on
Can the Bogoliubov–de Gennes equation be interpreted as a ‘one-particle’ wave equation? S UPRIYO DATTA , P HILIP F. BAGWELL School of Electrical Engineering and the MRSEC for Technology Enabling Heterostructure Materials, Purdue University, West Lafayette, IN 47907-1285, U.S.A. (Received 24 March 1999)
The solutions of the Bogoliubov–de Gennes (BdG) equation are usually interpreted as the excitations from the superconducting ground state. This viewpoint is not easily applied to a strongly coupled heterojunction since the ground state changes across the interface and it is not clear how the ground state should be connected across the heterointerface. In this paper, we present a different viewpoint that does not suffer from this conceptual drawback. We show that the BdG equation can be viewed as a ‘one-particle’ wave equation whose eigenstates (including the negative energy states) can be filled up systematically to describe the superconducting state, in much the same way that we fill the eigenstates of the Schr¨odinger equation to describe normal conductors. The only difference is that we need to start from a special vacuum |V i, consisting of a full band of down-spin electrons, instead of the usual vacuum devoid of all particles. Any quantity of interest, A (such as the charge density or the current density), can be interpreted as the sum of a ‘vacuum contribution’ AVAC due to the vacuum |V i and a one-particle contribution ABdG due to the filled eigenstates of the BdG equation. This picture is easily applied even to strongly coupled heterojunctions since the vacuum |V i is the same on both sides of a heterointerface. As such, we believe it puts the scattering theory of transport for superconductors on a firmer conceptual basis. c 1999 Academic Press
Key words: Bogoliubov–deGennes (BdG) equation, inhomogeneous superconductors, scattering theory of transport.
1. Introduction The self-consistent field method has been widely used to describe inhomogeneous superconductors [1]. This method is based on the Bogoliubov–de Gennes (BdG) equation (E − µS ) 0749–6036/99/051233 + 18 $30.00/0
u H +U = v 1∗
1 −(H ∗ + U )
u , v
(1.1) c 1999 Academic Press
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where H is the one-electron Hamiltonian ( p ≡ −i~∇) minus the electrochemical potential µS † H≡
( p − e A)2 + eV − µS , 2m
(1.2)
while the potentials U and 1 are determined self-consistently from the eigenstates. It can be shown that the eigenstates of the BdG equation occur in pairs, one of which has an energy greater than µS , while the other has an energy less than µS . We will refer to the former as the α-states, and the latter as the β-states (see Fig. 1A). Their eigenfunctions and eigenenergies can be shown to be related as follows (see Appendix A) E mβ = 2µS − E mα ,
∗ u mβ = −vmα ,
vmβ = u ∗mα .
(1.3)
In the usual formulation, the β-states having energy E mβ < µS , do not appear explicitly. All quantities of interest (such as the charge density, current density and the self-consistent fields U and 1) are expressed solely in terms of the α-states having energy E mα > µS . This apparent disregard of the β-states raises conceptual difficulties, if we wish to view the BdG wavefunction as a probability amplitude, since the α-states do not form a complete set by themselves. Scattering processes can give rise to matrix elements that connect the ‘allowed’ α-states and the ‘unallowed’ β-states thus leading to a disappearance of quasiparticles. Such a transition into a β-state represents a return to the ground state as emphasized by Kuemmel and co-workers [2], but the physical picture is not immediately evident in the usual formulation. The purpose of this paper is to present an interpretation that treats the α- and β-states on an equal footing and makes the physical significance of the β-states evident. Specifically, we will show that we can view the BdG equation as a ‘one-particle’ wave equation, just like the Schr¨odinger equation for normal systems, whose eigenstates (both α- and β-) are filled according to the Fermi function at any given temperature. At zero temperature, for example, we know that for normal systems the many-body ground state is obtained by filling up all the eigenstates of the Schr¨odinger equation having energies less than the Fermi energy. We can construct the superconducting ground state |Gi in much the same way by filling up all the eigenstates of the BdG equation having energy less than µS (that is, the β-states) starting from an appropriate vacuum state |V i: Y Y + + |Gi = γM |V i = γmβ |V i, (1.4) E M <µS
m
+ where γ M is the creation operator for the eigenstate ‘M’ (= mα or mβ). Even at nonzero temperatures, the appropriate many-body state is obtained simply by filling up the ‘one-particle’ states of the BdG equation according to the Fermi function at that temperature. One might feel uneasy about the fact that the β-states extend in energy down to negative infinity, so that the ground state appears to have an infinite number of particles. However, the vacuum state |V i is not the usual one devoid of particles; |V i consists of a completely full band of down-spin electrons and an empty band of up-spin electrons. Every time we fill an eigenstate from the BdG equation, we create holes in the full band and/or electrons in the empty band. Far below the Fermi energy the β-states are essentially down-spin holes, which simply serve to empty out the down-spin electrons that fill the vacuum |V i. A formal justification for this interpretation of the BdG equation is provided in Section 2 (and Appendix B). The prescription given in eqn (1.4) yields the standard BCS ground state for a homogeneous superconductor; if the pair potential 1(r ) varies spatially, it yields a generalized BCS-type ground state (see Appendix C). † Note that we have written the eigenenergy as (E − µ ) in eqn (1.1). We could just as well write it as E, but then the S
energy scale gets shifted downwards by µS , because it has been subtracted off from the Hamiltonian H (see eqn (1.2)). The Fermi energy would then lie at E = 0; with our convention it lies at E = µS .
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A, Eigenstates of the BdG equation k E ‘α’ states E µS + 1 µS
k
µS − 1
‘β’ states
B, Excitation spectrum E
¯ states ‘β’
‘α’ states
E µS + 1 µS
k
Fig. 1. A, Eigenstates of the BdG equation for a homogeneous superconductor with an electrochemical potential µS and with 1 = |1|eiq x . The β-states have been flipped in ‘k’ relative to the usual convention, but this makes no difference if there is no superfluid flow (q = 0) since the bands are symmetrical about k = 0. For inhomogeneous conductors, the eigenstates are no longer plane waves and cannot be labeled by ‘k’, but still occur in pairs with E mα > µS and E mβ < µS . B, Excitation spectrum obtained from A by leaving the α-states intact and flipping the β-states.
1.1. Excitation picture It is easy to see that the above interpretation is consistent with the usual picture in terms of the quasiparticle excitation spectrum. In many-body problems it is common to treat the ground state |Gi as the ‘vacuum’ from which excitations are created. The ground state |Gi consists of a full β-band and an empty α-band (see eqn (1.4)) and excitations can be obtained either by creating α-particles or by removing β-particles. Thus the creation operator for an α-state is the same as the creation operator for the corresponding excitation, but it is the annihilation operator for a β-state that is equal to the creation operator for the corresponding excitation. Hence + + γm↑ = γmα + γm↑
+ γm↓
and
+ γm↓ = γmβ ,
(1.5)
where and are the usual quasiparticle creation operators. The excitation spectrum can be obtained from the eigenstates of the BdG equation by leaving the α-band intact and flipping the β-band about E = µS to obtain a β-band as shown in Fig. 1B. Usually, the β-states
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look like a mirror image of the α-states (see eqn (1.3)) so that the β-band is identical to the α-band. Their occupation factors are also normally identical, even away from equilibrium, f mβ = 1 − f mβ = f mα
(1.6)
as shown in Appendix A. Consequently, one can simply assume a doubly degenerate α-band, as is commonly done. But it is important to recognize that the β- states are not just being ignored as unphysical, as emphasized in Ref. [2]. 1.2. Interband transitions The relation between the one-particle picture (Fig. 1A) and the excitation picture (Fig. 1B) is best understood if we consider interband transitions (between the α- and β-states). In the one-particle picture the α- and β-bands are like the conduction and valence bands of an ordinary semiconductor, the former being completely empty and the latter completely full at zero temperature. A transition from a β- state into an α- state is like a electron–hole generation process. In the excitation picture on the other hand this process leads to the simultaneous appearance of two quasiparticles. From the one-particle picture it is straightforward to calculate the matrix element for a transition from a β-state to an α-state due to microwave absorption. Noting that the perturbation in the BdG Hamiltonian (see eqn (1.1)) due to the electromagnetic wave is given by e e p.A + A. p 0 p.A + A. p 0 = 0 −( p.A + A. p)∗ 0 p.A + A. p 2m 2m we can write, using the standard rules of elementary quantum mechanics, e p.A + A. p 0 u mβ ∗ ∗ . Mm 0 α;mβ = (u 0 vm 0 α ) 0 p.A + A. p vmβ 2m m α Making use of eqn (1.3) we can rewrite the matrix element in terms of the β-states only (note that it is the u’s and v’s of the β-states that corresponds to the u’s and v’s appearing in the BCS wavefunction): e p.A + A. p 0 u mβ Mm 0 α;mβ = . ( vm 0 β −u m 0 β ) 0 p.A + A. p vmβ 2m Assuming the eigenstates to be plane waves as in a homogeneous superconductor we obtain e~(k + k 0 ).A (u k vk 0 − vk u k 0 ) (1.7) 2m in agreement with the standard result [3]. The coherence factors for ultrasonic attenuation and spin relaxation rates (which involve intraband rather than interband transitions) are also obtained correctly from this oneparticle picture. Mk 0 α;kβ =
1.3. One-particle versus excitation picture A one-particle picture (Fig. 1A) is possible only within a mean field scheme, while the excitation picture (Fig. 1B) is more generally applicable to many-body problems. But as long as a mean field theory is adequate, the two pictures are equivalent and depending on the problem at hand one may be more convenient or illuminating than the other. Normally the excitation picture makes the book-keeping simpler since we have to worry about one (doubly degenerate) band rather than two. Even under nonequilibrium conditions the relations given in eqns (1.3) and (1.6) usually remain valid (see Appendix A), so that the α- and β- states are in a sense not truly
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independent. This is why we can express all quantities of interest solely in terms of the α- (or the β-) states, and it may seem that we are unnecessarily complicating the picture by explicitly including both the α- and the β-states. However, the one-particle picture has several advantages, one of which we have seen already in the derivation of the matrix element for microwave absorption (see eqn (1.7)). Using elementary quantum mechanics we obtained the correct coherence factor (u k vk 0 −vk u k 0 )—a result that is usually derived using the second quantized formalism. Indeed, from the excitation picture (Fig. 1B), one might even wrongly guess the microwave absorption gap to be 1. The correct result is double this value because two quasiparticles must appear simultaneously [3]—a fact that is difficult to explain in simple terms. But in the one-particle picture the microwave absorption gap is obviously 21, since this is the minimum gap between the α- and β-bands. Another advantage of the one-particle picture is that we work directly with the eigenstates of the BdG equation, in much the same way that one works with the eigenstates of the Schr¨odinger equation in the + normal case. Consequently, the transformation to the Bogoliubov operators γ M represents a standard unitary transformation making the associated algebra relatively straightforward (see Section 2). In the one-particle picture one can interpret any quantity of interest, A (such as the charge or the current density), as the sum of a ‘vacuum contribution’ due to the vacuum |V i and a one-particle contribution due to the filled eigenstates of the BdG equation: A = AVAC + ABdG .
(1.8a)
In the excitation picture, on the other hand, it is common to interpret different quantities of interest as the sum of a ‘ground state contribution’ due to |Gi and a quasiparticle contribution due to the excitations: A = A G + AQP .
(1.8b)
We illustrate these two viewpoints (eqns (1.8a)) and (1.8b)) by considering a number of different quantities, A, namely, the charge density (see Section 3), the current density (see Section 4) and the self-consistent fields (see Section 5). Either viewpoint provides a satisfactory description if used consistently. However, there is an important distinction between the ground state |Gi (see eqn (1.4)) and our special vacuum |V i which becomes apparent if we consider a heterointerface (Fig. 2). In the excitation picture, there are two different ground states |G 1 i and |G 2 i (and hence two different ‘vacuum’ states) on the two sides of the heterointerface. A complete theory should describe the spatial variation of the ground state as well as the excitations across the interface, but the BdG equation describes only the latter leaving one with the feeling that something else is needed. But in the one-particle picture there is a single vacuum |V i (a full band of down spin electrons) on both sides of the interface and one can construct scattering eigenstates for the entire structure based on the BdG equation. As such it puts the scattering theory of transport on a firmer conceptual basis, as described in Datta, Bagwell and Anantram, Phys. Low Dim. Struct. 3 (1996).
2. BdG equation as a one-particle wave equation In this section, we will provide a formal justification for the one-particle interpretation of the BdG equation described above. It is well known that the BCS Hamiltonian (s =↑, ↓)‡ Z Z X HBCS = dr 9s+ (r )H 9s (r ) − dr g(r )9↑+ (r )9↓+ (r )9↓ (r )9↑ (r ) (2.1) s
‡ See Ref. [1]. We are letting the real function g(r ) vary spatially to account for inhomogeneities in the electron–phonon interaction. We believe that our discussion can be extended in a straightforward manner to include nonlocal interactions described by a function of the form g(r, r 0 ). Also, we are neglecting expectation values of the form h9↑+ (r )9↓ (r )i, assuming that there are no magnetic impurities or paramagnetic effects.
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INTERFACE
A, Excitation picture
|G 1 i
|G 2 i
FACE FACE
B, One-particle picture
|V i INTER
|V i
Fig. 2. A, In the excitation picture there are two different ground states |G 1 i and |G 2 i (and hence two different ‘vacuum’ states) on the two sides of the heterointerface. A complete theory should describe the spatial variation of the ground state as well as the excitations across the interface, but the BdG equation describes only the latter. B, In the one-particle picture there is a single vacuum |V i and one can construct scattering eigenstates for the entire structure based on the BdG equation.
can be approximated as HBCS ≈ H1 + Hmf , where Z Z HI = dr [U↑ (r )U↓ (r )/g(r )] + dr [|1(r )|2 /g(r )]
(2.2)
and Hmf =
Z
dr
+
Z
X
9s+ (r )[H + Us ]9s (r )
s
dr [1(r )9↑+ (r )9↓+ (r ) + 1∗ (r )9↓ (r )9↑ (r )]
(2.3)
and the self-consistent fields Us and 1 are defined as U↑ (r ) ≡ −g(r )h9↓+ (r )9↓ (r )i = U↑∗ (r )
(2.4a)
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U↓ (r ) ≡ −g(r )h9↑+ (r )9↑ (r )i = U↓∗ (r )
(2.4b)
1(r ) ≡ −g(r )h9↓ (r )9↑ (r )i.
(2.5)
and Hmf is the mean-field Hamiltonian and HI is the negative of the interaction energy which has to be added because the self-consistent field method double-counts the interaction energy. It is a constant that plays no role in the dynamics and we will ignore it in the following discussion. Our objective is to transform Hmf into a form that looks just like the second quantized Hamiltonian for a set of noninteracting particles obeying the BdG equation. This is achieved if we use a ‘particle–hole’ transformation for the down-spin operators, leaving the up-spin operators intact: + 8+ ↑ (r ) ≡ 9↑ (r )
and
8+ ↓ (r ) ≡ 9↓ (r ).
(2.6)
It can be shown (see Appendix B) that in terms of these transformed operators Hmf is written as (HVAC + HBdG ) where HVAC represents the energy of the vacuum consisting of a full band of down-spin electrons (see eqn (B.5)). It is a constant which plays no role in the dynamics. All the dynamics is contained in HBdG Z Z ∗ HBdG = dr 8+ (r )[H + U ]8 (r ) − dr 8+ ↑ ↑ ↑ ↓ (r )[H + U↓ ]8↓ (r ) Z + ∗ + dr [1(r )8+ (2.7) ↑ (r )8↓ (r ) + 1 (r )8↓ (r )8↑ (r )], which has the same form as the second quantized Hamiltonian for a set of noninteracting particles obeying the one-particle wave equation u H + U↑ 1 u (E − µS ) = . (2.8) v 1∗ −(H ∗ + U↓ ) v This is slightly different from the usual BdG equation (see eqn (1.1)) since the fields U↑ and U↓ need not be equal in general under nonequilibrium conditions. However, in our subsequent discussion we will assume spin-independent systems such that U↑ = U↓ ≡ U (see Appendix A). Since HBdG looks just like the second quantized Hamiltonian for a set of noninteracting particles obeying + the BdG equation, it can be diagonalized simply by defining a new operator γ M that creates a ‘particle’ in an eigenstate (u M , v M ) of eqn (2.8): Z + + γM = dr [u M (r )8+ (2.9a) ↑ (r ) + v M (r )8↓ (r )] or using eqn (2.6): + γM
=
Z
dr [u M (r )9↑+ (r ) + v M (r )9↓ (r )].
In terms of this new quasiparticle operator the mean-field Hamiltonian Hmf takes the form X + Hmf = HVAC + (E M − µS )γ M γM .
(2.9b)
(2.10)
M
HVAC is the energy of the ‘vacuum’ consisting of a completely full band of down-spin electrons (see eqn (B.5)). On top of this vacuum we create particles in eigenstates of the BdG equation with energies E M . A BdG quasiparticle represents a superposition of an up-spin electron and a down-spin hole, as evident from eqn (2.9b). The ground state is obtained by filling all the quasiparticle states below µS , that is, the β-states as indicated in eqn (1.4). In Appendix C, we show that this procedure yields a ground state wavefunction that is identical to the familiar BCS wavefunction for homogeneous superconductors; but if the pair potential 1(r ) varies spatially, it yields a generalized BCS-type ground state.
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2.1. Inverse transformation One advantage of this viewpoint is that the Bogoliubov transformation appears like an ordinary unitary transformation in a Hilbert space spanned by (r, s) that is double the usual size. We can rewrite eqn (2.9a) compactly as XZ + γM = dr w M (r, s)8+ (2.11) s (r ), s
where w M (r, ↑) ≡ u M (r )
and
w M (r, ↓) ≡ v M (r ).
(2.12)
The functions w M (r, s), being eigenvectors of a Hermitian operator (the one on the right of eqn (2.8)), obey the orthogonality and completeness relations: XZ dr w ∗M 0 (r, s)w M (r, s) = δ M 0 ,M (2.13) s
and X
w ∗M (r 0 , s 0 )w M (r, s) = δ(r 0 − r )δs 0 ,s .
(2.14)
M
Using these properties it is straightforward to invert eqn (2.11) X + 8+ w ∗M (r, s)γ M s (r ) =
(2.15)
M
that is, using eqns (2.6) and (2.12), X + ψ↑+ (r ) = u ∗M (r )γ M
and
ψ↓ (r ) =
M
X
+ v ∗M (r )γ M .
(2.16)
M
To make contact with the formulation in [1], we note that the summation over ‘M’ includes both α-states and β- states, which can be written out explicitly as (making use of eqn (1.3)) X + + ψ↑+ (r ) = u ∗mα (r )γmα − vmα (r )γmβ (2.17a) m
ψ↓ (r ) =
X
+ ∗ + vmα (r )γmα + u mα (r )γmβ .
(2.17b)
m
These are precisely the relations given in Ref. [1], if we make the association + + γmα → γm↑
and
+ γmβ → γm↓
which was justified earlier (see eqn (1.5)). Also we prefer the notation ‘α’ and ‘β’ instead of ‘↑’ and ‘↓’, since these states reduce to pure spin states only far from the Fermi energy.
3. Charge density We can obtain expressions for different quantities of interest like the charge density, the current density or the self-consistent fields in terms of the quasiparticle occupation probabilities by a straightforward + transformation from the usual field operators 9 + (r ) to the new operators γ M (see Appendix D). However, we can write down the correct answers intuitively simply by adding the vacuum contribution due to |V i and
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the one-particle contribution arising from the filled eigenstates of the BdG equation. For example, the charge density associated with the vacuum is given by (: normalization volume) X 1 , (3.1) ρVAC = e k
since every down-spin ‘k’ state is occupied. Writing the functions v M (r ) in the form 1 X v M (r ) = √ v M (k)eik.r k and making use of the completeness relation (cf. eqn (2.14)) X v ∗M (k 0 )v M (k) = δk 0 ,k , M
we can rewrite eqn (3.1) in the form ρVAC ≡ e
X
|v M |2 .
(3.2a)
M
The one-particle contribution due to the filled states of the BdG equation is given by X ρBdG = e [|u M |2 − |v M |2 ] f M ,
(3.2b)
M
since ‘u’ denotes the electron-like component and ‘v’ denotes the hole-like component of the quasiparticle wavefunction (see eqn (2.9b)). Adding eqns (3.2a) and (3.2b) we obtain the total charge density X ρ = ρVAC + ρBdG = e |u M |2 f M + |v M |2 (1 − f M ). (3.3a) M
The summations in the above expressions extend over both α- and β-states. We could make use of eqns (1.3) and (1.6) to write these as summations over just the α-states (or just the β-states) multiplied by two X ρ = 2e |u mα |2 f mα + |vmα |2 (1 − f mα ) (3.3b) m
= 2e
X
|u mβ |2 f mβ + |vmβ |2 (1 − f mβ )
(3.3c)
m
as is commonly done in the literature. 3.1. Excitation picture Note that ρVAC is the charge due to our special vacuum |V i which consists of a completely full band of down-spin electrons (and an empty band of up-spin electrons). It is different from the charge due to the ground state |Gi (see eqn (1.4)) which is obtained from eqn (3.3b) by setting f mα = 0, or from eqn (3.3c) by setting f mβ = 1: X X ρG = 2e |vmα |2 = 2e |u mβ |2 . (3.4) m
m
In the excitation picture, one can interpret the total charge density as the sum of the ground state and quasiparticle contributions (ρ = ρG + ρQP ), where the latter is given by X ρQP = 2e (|u mα |2 − |vmα |2 ) f mα . (3.5) m
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4. Current density The electrical current can also be written as the sum of a one-particle current and a vacuum current, where X u v JBdG = e [J M − JM ] fM (4.1) M
JVAC =
X
v JM .
(4.2)
M
u and J v are the particle fluxes associated with the ‘u’ and ‘v’ components of the quasiparticle JM M u is given by the usual expression for the one-particle current density in elementary quantum wavefunction. J M mechanics: p − eA u ∗ J M (r ) ≡ Re u M (r ) u M (r ) . (4.3a) m v is similar but with ‘v’ and ‘v ∗ ’ interchanged: The expression for JM p − eA ∗ v J M (r ) ≡ Re v M (r ) v M (r ) . (4.3b) m
This interchange is necessary in order that the continuity equation is satisfied. Specifically, if we start from the time-dependent version of the BdG eqn (1.1) n o ∂ nu o u H +U 1 i~ = , (4.4) 1∗ −(H ∗ + U ) ∂t v v we obtain the following continuity equation for the probability u ∂ u∗u J +S + ∇. = , ∂t v ∗ v Jv −S A v (r ) ≡ Re v(r ) p−e A v ∗ (r ) . where J u (r ) ≡ Re u ∗ (r ) p−e u(r ) and J m m
(4.5)
Note that the roles of v and v ∗ are interchanged relative to u and u ∗ , in accordance with eqns (4.3a) and (4.3b). The significance of the source term S≡
2 I m[1(r )u ∗ (r )v(r )] ~
(4.6)
will be discussed shortly. The total current is obtained by adding the one-particle and vacuum contributions (eqns (4.1) and (4.2)) X u v J = JBdG + JVAC = e JM f M + JM (1 − f M ) (4.7a) M
and can be expressed solely in terms of the α- (or the β-) states X u v J = 2e Jmα f mα + Jmα (1 − f mα )
(4.7b)
m
= 2e
X
u v Jmβ f mβ + Jmβ (1 − f mβ )
(4.7c)
m
by making use of eqns (1.3) and (1.6). As before we can show that the term JVAC arises from a full band of down-spin electrons: X X ~k − e A(r ) v JVAC = JM =e . m M
k
(4.8)
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This ‘vacuum current’ JVAC is often zero but not always (such as problems involving the ac Josephson effect). In any case it is usually advantageous to retain it in eqn (4.1), because at large negative energies, it cancels out the contribution from JBdG term by term for each ‘M’. This allows us to truncate the summation at a suitably large negative value of E M . 4.1. Excitation picture In the excitation picture, it is common to interpret the total current density as the sum of the ground state current or ‘supercurrent’ and the quasiparticle current (J = JG + JQP ), where the former is obtained from eqn (4.7b) by setting f mα = 0 or from eqn (4.7c) by setting f mβ = 1: JG = 2e
X
v Jmα = 2e
X
u Jmβ ,
(4.9)
u v [Jmα − Jmα ] f mα .
(4.10)
m
m
while the latter is given by JQP = 2e
X m
It is important to note that in the excitation picture, both the supercurrent and the quasiparticle current are non-trivial quantities that can be different from zero. By contrast, in the one-particle picture the vacuum current is usually zero, and all the physics is contained in JBdG . We believe that this is an advantage for it could lead to a unified treatment of supercurrents and quasiparticle currents. 4.2. Current conservation From eqn (4.5) it follows that u v ∇.[JM ] = −∇.[JM ]=
2 I m[1(r )u ∗M (r )v M (r )]. ~
(4.11)
u + J v ) is conserved: This means that the quasiparticle probability current (J M M u v ∇.[J M + JM ]=0
as we would expect since the operator in the BdG equation (see eqn (1.1)) is Hermitian. But the electrical u − J v ) which is not necessarily conserved [4–7]: current (see eqn (4.1)) is proportional to (J M M u v ∇.[J M − JM ]=
4 I m[1(r )u ∗M (r )v M (r )]. ~
Using eqn (4.1) we can write ∇.[JBdG ] =
4e X I m[1(r )u ∗M (r )v M (r ) f M ]. ~
(4.12)
M
The right-hand side can be shown to be zero using the expression for the self-consistent field 1 (see eqn (5.2a)), proving that the electrical current is indeed conserved, if 1 is determined self-consistently, but not otherwise.
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5. Self-consistent fields Finally, we consider the self-consistent fields defined in eqns (2.4a, b) and (2.5). It is evident from eqns (2.4a) and (2.4b) that the field Us (r ) is determined by the density of electrons with the opposite spin. Under the spin-independent conditions assumed in this paper (see Appendix A) this is simply half the total electron density given in eqns (3.3a, b, c), so that gX U =− [|u M |2 f M + |v M |2 (1 − f M )] (5.1a) 2 M X = −g [|u mα |2 f mα + |vmα |2 (1 − f mα ))] (5.1b) m
= −g
X
[|u mβ |2 f mβ + |vmβ |2 (1 − f mβ ))].
(5.1c)
m
The self-consistent field 1 (see eqn (2.5)) is determined by the correlation u M v ∗M between the two components of a quasiparticle state, so that X 1 = −g v ∗M u M f M . (5.2a) M
Once again we can use the relations given in eqns (1.3) and (1.6) to write this expression solely in terms of the α- (or β-) states: X ∗ 1 = −g [vmα u mα (2 f mα − 1)] (5.2b) m
X ∗ = −g [vmβ u mβ (2 f mβ − 1)].
(5.2c)
m
In the excitation picture we could interpret the self-consistent fields as the sum of a ground state contribution X X UG = −g |vmα |2 = −g |u mβ |2 (5.3a) m
1G = +g
m
X
∗ vmα u mα
= −g
m
X
∗ vmβ u mβ
(5.3b)
m
and a quasiparticle contribution: UQP = −g
X
(|u mα |2 − |vmα |2 ) f mα
(5.4a)
m
1QP = −2g
X ∗ [vmα u mα f mα ].
(5.4b)
m
6. Summary The basic point of this paper is that the BdG equation can be interpreted as a ‘one-particle’ wave equation just like the Schr¨odinger equation for normal systems, if we treat the β-states seriously on an equal footing with the α-states. The eigenstates of the BdG equation can be filled up systematically to obtain the superconducting state just as one fills up the eigenstates of the Schr¨odinger equation for the normal state. The only difference is that we start from a special vacuum |V i and not the usual vacuum |0i. Any quantity of interest, A, can be interpreted as the sum of a ‘vacuum contribution’ AVAC due to the vacuum |V i and a one-particle contribution ABdG due to the filled eigenstates of the BdG equation. This is contrasted with
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the commonly used excitation picture in which one interprets different quantities of interest as the sum of a ‘ground state contribution’ A G due to |Gi and a quasiparticle contribution AQP due to excitations. The excitation picture is difficult to apply to a strongly coupled heterojunction since the ground state |Gi changes across the interface. By contrast, the vacuum |V i is the same on both sides of a heterointerface making it conceptually straightforward to apply the one-particle picture to heterojunctions. Acknowledgement—This work was supported by the MRSEC program of the National Science Foundation under Award No. DMR-9400415.
References [1] P. G. de Gennes, Superconductivity of Metals and Alloys (Addison-Wesley, 1989) Chap. 5. [2] C. Virgilio Nino, R. Kuemmel Phys. Rev. B29, 3957 (1984); U. Gunsenheimer, U. Schussler, and R. Kuemmel, Phys. Rev. B49, 6111 (1994). [3] M. Tinkham, Superconductivity (Gordon and Breach, 1965) p. 58. [4] G. E. Blonder, M. Tinkham, and T. M. Klapwijk, Phys. Rev. B25, 4515 (1982). [5] A. Furusaki and M. Tsukada, Solid State Commun. 78, 299 (1991). [6] P. F. Bagwell, Phys. Rev. B49, 6841 (1994). [7] F. Sols and J. Ferrer, Phys. Rev. B49, 15913 (1994). [8] M. Ma and P. A. Lee, Phys. Rev. B32, 5658 (1985).
Appendix A. Symmetry of α - and β -states In most of our discussions, we have assumed the following relations to hold between the α- and β-states: E mβ = 2µS − E mα ,
∗ u mβ = −vmα ,
vmβ = u ∗mα
(1.3)
f mβ = 1 − f mβ = f mα . (1.6) In this appendix, we will show that these relations hold even under nonequilibrium conditions, as long as there are no spin-dependent processes. However, it should be noted that if we include the Zeeman splitting due to a z-directed magnetic field, eqn (1.1) would be modified to n o nu o u H + U + µ B Bz 1 (E − µS ) = . 1∗ −(H ∗ + U − µ B Bz ) v v Since the [22] element is no longer just the negative complex conjugate of the [11] element, eqn (1.3) would not hold (this is easy to see in the special case when 1 is zero). Even if the [22] element is equal to the negative complex conjugate of the [11] element and eqn (1.3) holds, their occupation factors may not obey eqn (1.6), if we have a contact that injects spin-polarized electrons (like a ferromagnetic contact). However, a general treatment of such problems with arbitrarily directed magnetic fields requires one to go beyond a simple (2 × 2) BdG equation (see, for example, Chap. 9 of Ref. [1]) and is outside the scope of this paper. We will now assume spin-independent interactions and prove eqns (1.3) and (1.6). Equation (1.3).
Suppose we have a solution that satisfies the BdG equation (eqn (1.1)) u mα u mα H +U 1 (E mα − µS ) = 1∗ −(H ∗ + U ) vmα vmα
with E mα > µS . A little straightforward algebra shows that ∗ ∗ −vmα H +U 1 −vmα − (E mα − µS ) = . u ∗mα u ∗mα 1∗ −(H ∗ + U )
(A.1)
(A.2)
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m m0
µS + 1 µS µS − 1
m0 m ‘β’ states
B
‘α’ states E µS + 1 µS µS − 1
m m0
m0 m ‘β’ states
Fig. 3. Pairs of A, intraband and B, interband transitions that have identical transition rates (see eqns (A.4a, b)).
But u mβ H +U 1 u mβ (E mβ − µS ) = vmβ 1∗ −(H ∗ + U ) vmβ by definition. Comparing eqns (A.2) and (A.3) we obtain eqn (1.3).
(A.3)
Equation (1.6). Assume that we start from a condition in which eqn (1.6) is satisfied and turn on some interactions that cause the particles to scatter from one state to another. It can be shown that eqn (1.6) will continue to be satisfied provided the scattering rates obey the following relations (see Fig. 3): R(mα → m 0 α) = R(m 0 β → mβ)
(A.4a)
R(mα → m 0 β) = R(m 0 α → mβ).
(A.4b)
and
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These relations ensure that any process leading to a decrease in f mα , is accompanied by an identical one that leads to an increase in f mβ , so that ( f mα + f mβ ) remains unchanged. To prove eqns (A.4a, b), we assume an arbitrary perturbation of the form h δ 0 H = δ ∗ −h ∗ and make use of eqn (1.3) to show that the resulting matrix elements satisfy the relations: hm 0 α|H 0 |mαi = −hmβ|H 0 |m 0 βi and hm 0 β|H 0 |mαi = hmβ|H 0 |m 0 αi. Equations (A.4a, b) then follow readily from the golden rule (the indices γ and γ 0 take on the values α, β) 2π |hm 0 γ 0 |H 0 |mγ i|2 δ[E m 0 γ 0 − E mγ ± ~ω]. ~ To show that ( f mα + f mβ ) remains unchanged by scattering processes, we can start from a master equation of the form X d f mγ = R(m 0 γ 0 → mγ ) f m 0 γ 0 (1 − f mγ ) − R(mγ → m 0 γ 0 ) f mγ (1 − f m 0 γ 0 ) dt 0 0 R(mγ → m 0 γ 0 ) =
mγ
and make use of eqns (1.6) and (A.4a, b) to show that d( f mα + f mβ )/dt = 0. This means that if eqn (1.6) holds as an initial condition, it holds for all time.
Appendix B. Particle–hole transformation of Hmf In this appendix, we will show how eqn (2.7) is obtained from eqn (2.3) using the transformation indicated in eqn (2.6). For this purpose it is convenient to transform from the position representation to a discrete basis using any complete basis set φµ (r ) that spans the one-particle, one-spin Hilbert space. X X 9↑ (r ) = aµ φµ (r ) and 9↓ (r ) = bµ φµ∗ (r ). (B.1) µ
µ
The mean-field Hamiltonian then takes the form X X Hmf = [H + U↑ ]µν aµ+ aν + [H ∗ + U↓ ]µν bν+ bµ µ,ν
+
µ,ν
X µ,ν
[1µν aµ+ bν+ + (1∗ )νµ bν aµ ],
(B.2)
where [H + Us ]µν ≡
Z
dr φµ∗ (r )([H + Us ]φν (r ))
(B.3a)
[1]µν ≡
Z
dr φµ∗ (r )1(r )φν (r ).
(B.3b)
and
Note that (see eqn (B.1)) we are expanding the up-spin field 9↑ operator using the set {φµ } and the down-spin field operator 9↓ using the ‘time-reversed set’ {φµ∗ }. As a result we obtain the ‘normal’ matrix elements 1µν (see eqn (B.3b)). If instead we were R to use the same set {φµ } to expand both 9↑ and 9↓ we would obtain ‘abnormal’ quantities of the form dr φµ∗ (r )1(r )φν∗ (r ).
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To make Hmf look more like a ‘normal’ Hamiltonian we use a ‘particle–hole’ transformation for the downspin operators, leaving the up-spin operators intact: + + cµ,↑ ≡ aµ+ , cµ,↓ ≡ bµ .
(B.4)
Making use of the anti-commutation property bν+ bµ + bµ bν+ = δµν , we can write Hmf as (HVAC + HBdG ) where X HVAC = [H ∗ + U↓ ]µµ = Trace [H ∗ + U↓ ] (B.5) µ
and HBdG =
X µ,ν
+
+ [H + U↑ ]µν cµ,↑ cν,↑ −
X µ,ν
+ [1µν cµ,↑ cν,↓
X µ,ν
+ [H ∗ + U↓ ]µν cµ,↓ cν,↓
+ + (1 )νµ cν,↓ cµ,↑ ]. ∗
(B.6)
We can obtain eqn (2.7) by transforming eqn (B.6) back to the position representation, noting that from eqns (2.6), (B.1) and (B.4): X 8s (r ) = cµ,s φµ (r ) (B.7a) µ
so that the inverse transformation is given by cµ,s =
Z
dr φµ∗ (r )8s (r ).
(B.7b)
Appendix C. Ground state wavefunction In this section we will see how we can construct the ground state wavefunction starting from the special vacuum, |V i, consisting of a full band of down-spin electrons Y + |V i ≡ bµ |0i (C.1) µ
and filling up all the β-eigenstates whose energies are less than µS : Y + |Gi = γmβ |V i.
(C.2)
m
Noting that + γmβ =
X µ
u mβ,µ aµ+ + vmβ,µ bµ
(C.3)
we can write down the ground state wavefunction from eqns (C.1) and (C.2): " #! Y X Y + |Gi = u mβ,µ aµ + vmβ,µ bµ bν+ |0i. m
µ
(C.4)
ν
Equation (C.4) looks rather complicated, but we can simplify it if we assume that the pair potential 1 is a constant. Each eigenstate of the BdG equation (see eqn (2.1)) then corresponds to an eigenstate of [H + U ]. For example, in the position representation we can write u mβ (r ) Umβ = φm (r ) (C.5) vmβ (r ) Vmβ
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assuming that the functions φm (r ) represent eigenstates of [H + U ] with eigenenergies εm . Umβ and Vmβ are constants that are evaluated by solving the (2 × 2) eigenvalue problem: Um εm 1 Um (E m − µS ) = . (C.6) Vm 1∗ −εm Vm This yields two eigenvectors for each ‘m’, which we denote as (Umα , Vmα ) and (Umβ , Vmβ ); to construct the ground state from eqn (C.4) we only need the β-state having energy less than µS . Using eqn (C.5) we can write u mβ,µ = Umβ δm,µ and vmβ,µ = Vmβ δm,µ so that from eqn (C.4), |9G i =
Y
+ + [Vmβ + Umβ am bm ]|0i.
(C.7)
m
For homogeneous superconductors the functions φm (r ) that diagonalize [H + U ] are simply the set of plane waves eik.r · |Gi in eqns (4.7a, b, c) then reduces to the familiar BCS wavefunction: Y |Gi = [Vkβ + Ukβ ak+ bk+ ]|0i ⇒ |9BCS i. (C.8) k
It might appear from eqn (C.8) that we are pairing an up-spin ‘+k’ with a down-spin ‘+k’ state rather than a down-spin ‘−k’ state. However, this is purely a matter of notation. From eqn (B.1) we can write Z + ak = dr e+ik.r 9↑+ (r ) and bk+ =
Z
dr e−ik.r 9↓+ (r )
so that the operator bk+ creates a down-spin electron in a ‘−k’ state, although the operator ak+ creates an up-spin electron in a ‘+k’ state. This difference arises because we expanded the down-spin operator in terms of the time-reversed set (see eqn (B.1)). The wavefunction in eqn (C.7) is often assumed for the ground state of inhomogeneous superconductors, with the functions φm (r ) chosen so as to diagonalize [H + U ]. However, as noted in Refs. [1] and [8], this solution does not diagonalize the BdG equation, since a self-consistent calculation yields a pair potential 1 that varies spatially. It is only by neglecting this spatial variation in 1 that we are able to simplify eqn (C.4) to eqn (C.7). A more accurate solution that takes into account the spatial variation in 1 could lead to a lower energy ground state wavefunction.
Appendix D. Expectation values of field operators Using eqn (2.16) it is straightforward to show that X + h9↑+ 9↑ i = u ∗M (r )u M 0 (r )hγ M γM 0 i
(D.1a)
M,M 0
h9↓+ 9↓ i =
X
M,M 0
+ v ∗M (r )v M 0 (r )hγ M 0 γ M i
(D.1b)
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h9↓ 9↑ i =
X
+ v ∗M (r )u M 0 (r )hγ M γ M 0 i = h9↑+ 9↓+ i∗ .
(D.2)
M,M 0
For a general nonequilibrium state, the quasiparticle states can be occupied arbitrarily, which will be reflected + in arbitrary values of the quantity hγ M γ M 0 i. Usually the density matrix is assumed to be diagonal as we have done in this paper; that is, + hγ M γ M 0 i = f M δ M,M 0
and
+ hγ M 0 γ M i = (1 − f M )δ M,M 0 ,
(D.3)
where f M denotes the occupation factor for the eigenstate ‘M’. The self-consistent fields that we have written down intuitively in eqns (5.1a, b, c) and (5.2a, b, c) can be obtained formally by combining eqns (D.1a, b) through (D.3) with eqns (2.4a, b) and (2.5). We can also use these expressions to obtain eqns (3.3a, b, c) and (4.7a, b, c) for the charge and current densities (which were obtained using intuitive arguments) if we note that the charge density is given by X ρ(r ) = eh9s+ (r )9s (r )i (D.4) s
while the current density is given by J (r ) =
X e Reh9s+ (r )( p − e A)9s (r )i. m s
(D.5)
Can the Bogoliubov–de Gennes equation be interpreted as a ‘one-particle’ wave equation? S UPRIYO DATTA , P HILIP F. BAGWELL School of Electrical Engineering and the MRSEC for Technology Enabling Heterostructure Materials, Purdue University, West Lafayette, IN 47907-1285, U.S.A. (Received 24 March 1999)
The solutions of the Bogoliubov–de Gennes (BdG) equation are usually interpreted as the excitations from the superconducting ground state. This viewpoint is not easily applied to a strongly coupled heterojunction since the ground state changes across the interface and it is not clear how the ground state should be connected across the heterointerface. In this paper, we present a different viewpoint that does not suffer from this conceptual drawback. We show that the BdG equation can be viewed as a ‘one-particle’ wave equation whose eigenstates (including the negative energy states) can be filled up systematically to describe the superconducting state, in much the same way that we fill the eigenstates of the Schr¨odinger equation to describe normal conductors. The only difference is that we need to start from a special vacuum |V i, consisting of a full band of down-spin electrons, instead of the usual vacuum devoid of all particles. Any quantity of interest, A (such as the charge density or the current density), can be interpreted as the sum of a ‘vacuum contribution’ AVAC due to the vacuum |V i and a one-particle contribution ABdG due to the filled eigenstates of the BdG equation. This picture is easily applied even to strongly coupled heterojunctions since the vacuum |V i is the same on both sides of a heterointerface. As such, we believe it puts the scattering theory of transport for superconductors on a firmer conceptual basis. c 1999 Academic Press
Key words: Bogoliubov–deGennes (BdG) equation, inhomogeneous superconductors, scattering theory of transport.
1. Introduction The self-consistent field method has been widely used to describe inhomogeneous superconductors [1]. This method is based on the Bogoliubov–de Gennes (BdG) equation (E − µS ) 0749–6036/99/051233 + 18 $30.00/0
u H +U = v 1∗
1 −(H ∗ + U )
u , v
(1.1) c 1999 Academic Press
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where H is the one-electron Hamiltonian ( p ≡ −i~∇) minus the electrochemical potential µS † H≡
( p − e A)2 + eV − µS , 2m
(1.2)
while the potentials U and 1 are determined self-consistently from the eigenstates. It can be shown that the eigenstates of the BdG equation occur in pairs, one of which has an energy greater than µS , while the other has an energy less than µS . We will refer to the former as the α-states, and the latter as the β-states (see Fig. 1A). Their eigenfunctions and eigenenergies can be shown to be related as follows (see Appendix A) E mβ = 2µS − E mα ,
∗ u mβ = −vmα ,
vmβ = u ∗mα .
(1.3)
In the usual formulation, the β-states having energy E mβ < µS , do not appear explicitly. All quantities of interest (such as the charge density, current density and the self-consistent fields U and 1) are expressed solely in terms of the α-states having energy E mα > µS . This apparent disregard of the β-states raises conceptual difficulties, if we wish to view the BdG wavefunction as a probability amplitude, since the α-states do not form a complete set by themselves. Scattering processes can give rise to matrix elements that connect the ‘allowed’ α-states and the ‘unallowed’ β-states thus leading to a disappearance of quasiparticles. Such a transition into a β-state represents a return to the ground state as emphasized by Kuemmel and co-workers [2], but the physical picture is not immediately evident in the usual formulation. The purpose of this paper is to present an interpretation that treats the α- and β-states on an equal footing and makes the physical significance of the β-states evident. Specifically, we will show that we can view the BdG equation as a ‘one-particle’ wave equation, just like the Schr¨odinger equation for normal systems, whose eigenstates (both α- and β-) are filled according to the Fermi function at any given temperature. At zero temperature, for example, we know that for normal systems the many-body ground state is obtained by filling up all the eigenstates of the Schr¨odinger equation having energies less than the Fermi energy. We can construct the superconducting ground state |Gi in much the same way by filling up all the eigenstates of the BdG equation having energy less than µS (that is, the β-states) starting from an appropriate vacuum state |V i: Y Y + + |Gi = γM |V i = γmβ |V i, (1.4) E M <µS
m
+ where γ M is the creation operator for the eigenstate ‘M’ (= mα or mβ). Even at nonzero temperatures, the appropriate many-body state is obtained simply by filling up the ‘one-particle’ states of the BdG equation according to the Fermi function at that temperature. One might feel uneasy about the fact that the β-states extend in energy down to negative infinity, so that the ground state appears to have an infinite number of particles. However, the vacuum state |V i is not the usual one devoid of particles; |V i consists of a completely full band of down-spin electrons and an empty band of up-spin electrons. Every time we fill an eigenstate from the BdG equation, we create holes in the full band and/or electrons in the empty band. Far below the Fermi energy the β-states are essentially down-spin holes, which simply serve to empty out the down-spin electrons that fill the vacuum |V i. A formal justification for this interpretation of the BdG equation is provided in Section 2 (and Appendix B). The prescription given in eqn (1.4) yields the standard BCS ground state for a homogeneous superconductor; if the pair potential 1(r ) varies spatially, it yields a generalized BCS-type ground state (see Appendix C). † Note that we have written the eigenenergy as (E − µ ) in eqn (1.1). We could just as well write it as E, but then the S
energy scale gets shifted downwards by µS , because it has been subtracted off from the Hamiltonian H (see eqn (1.2)). The Fermi energy would then lie at E = 0; with our convention it lies at E = µS .
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A, Eigenstates of the BdG equation k E ‘α’ states E µS + 1 µS
k
µS − 1
‘β’ states
B, Excitation spectrum E
¯ states ‘β’
‘α’ states
E µS + 1 µS
k
Fig. 1. A, Eigenstates of the BdG equation for a homogeneous superconductor with an electrochemical potential µS and with 1 = |1|eiq x . The β-states have been flipped in ‘k’ relative to the usual convention, but this makes no difference if there is no superfluid flow (q = 0) since the bands are symmetrical about k = 0. For inhomogeneous conductors, the eigenstates are no longer plane waves and cannot be labeled by ‘k’, but still occur in pairs with E mα > µS and E mβ < µS . B, Excitation spectrum obtained from A by leaving the α-states intact and flipping the β-states.
1.1. Excitation picture It is easy to see that the above interpretation is consistent with the usual picture in terms of the quasiparticle excitation spectrum. In many-body problems it is common to treat the ground state |Gi as the ‘vacuum’ from which excitations are created. The ground state |Gi consists of a full β-band and an empty α-band (see eqn (1.4)) and excitations can be obtained either by creating α-particles or by removing β-particles. Thus the creation operator for an α-state is the same as the creation operator for the corresponding excitation, but it is the annihilation operator for a β-state that is equal to the creation operator for the corresponding excitation. Hence + + γm↑ = γmα + γm↑
+ γm↓
and
+ γm↓ = γmβ ,
(1.5)
where and are the usual quasiparticle creation operators. The excitation spectrum can be obtained from the eigenstates of the BdG equation by leaving the α-band intact and flipping the β-band about E = µS to obtain a β-band as shown in Fig. 1B. Usually, the β-states
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look like a mirror image of the α-states (see eqn (1.3)) so that the β-band is identical to the α-band. Their occupation factors are also normally identical, even away from equilibrium, f mβ = 1 − f mβ = f mα
(1.6)
as shown in Appendix A. Consequently, one can simply assume a doubly degenerate α-band, as is commonly done. But it is important to recognize that the β- states are not just being ignored as unphysical, as emphasized in Ref. [2]. 1.2. Interband transitions The relation between the one-particle picture (Fig. 1A) and the excitation picture (Fig. 1B) is best understood if we consider interband transitions (between the α- and β-states). In the one-particle picture the α- and β-bands are like the conduction and valence bands of an ordinary semiconductor, the former being completely empty and the latter completely full at zero temperature. A transition from a β- state into an α- state is like a electron–hole generation process. In the excitation picture on the other hand this process leads to the simultaneous appearance of two quasiparticles. From the one-particle picture it is straightforward to calculate the matrix element for a transition from a β-state to an α-state due to microwave absorption. Noting that the perturbation in the BdG Hamiltonian (see eqn (1.1)) due to the electromagnetic wave is given by e e p.A + A. p 0 p.A + A. p 0 = 0 −( p.A + A. p)∗ 0 p.A + A. p 2m 2m we can write, using the standard rules of elementary quantum mechanics, e p.A + A. p 0 u mβ ∗ ∗ . Mm 0 α;mβ = (u 0 vm 0 α ) 0 p.A + A. p vmβ 2m m α Making use of eqn (1.3) we can rewrite the matrix element in terms of the β-states only (note that it is the u’s and v’s of the β-states that corresponds to the u’s and v’s appearing in the BCS wavefunction): e p.A + A. p 0 u mβ Mm 0 α;mβ = . ( vm 0 β −u m 0 β ) 0 p.A + A. p vmβ 2m Assuming the eigenstates to be plane waves as in a homogeneous superconductor we obtain e~(k + k 0 ).A (u k vk 0 − vk u k 0 ) (1.7) 2m in agreement with the standard result [3]. The coherence factors for ultrasonic attenuation and spin relaxation rates (which involve intraband rather than interband transitions) are also obtained correctly from this oneparticle picture. Mk 0 α;kβ =
1.3. One-particle versus excitation picture A one-particle picture (Fig. 1A) is possible only within a mean field scheme, while the excitation picture (Fig. 1B) is more generally applicable to many-body problems. But as long as a mean field theory is adequate, the two pictures are equivalent and depending on the problem at hand one may be more convenient or illuminating than the other. Normally the excitation picture makes the book-keeping simpler since we have to worry about one (doubly degenerate) band rather than two. Even under nonequilibrium conditions the relations given in eqns (1.3) and (1.6) usually remain valid (see Appendix A), so that the α- and β- states are in a sense not truly
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independent. This is why we can express all quantities of interest solely in terms of the α- (or the β-) states, and it may seem that we are unnecessarily complicating the picture by explicitly including both the α- and the β-states. However, the one-particle picture has several advantages, one of which we have seen already in the derivation of the matrix element for microwave absorption (see eqn (1.7)). Using elementary quantum mechanics we obtained the correct coherence factor (u k vk 0 −vk u k 0 )—a result that is usually derived using the second quantized formalism. Indeed, from the excitation picture (Fig. 1B), one might even wrongly guess the microwave absorption gap to be 1. The correct result is double this value because two quasiparticles must appear simultaneously [3]—a fact that is difficult to explain in simple terms. But in the one-particle picture the microwave absorption gap is obviously 21, since this is the minimum gap between the α- and β-bands. Another advantage of the one-particle picture is that we work directly with the eigenstates of the BdG equation, in much the same way that one works with the eigenstates of the Schr¨odinger equation in the + normal case. Consequently, the transformation to the Bogoliubov operators γ M represents a standard unitary transformation making the associated algebra relatively straightforward (see Section 2). In the one-particle picture one can interpret any quantity of interest, A (such as the charge or the current density), as the sum of a ‘vacuum contribution’ due to the vacuum |V i and a one-particle contribution due to the filled eigenstates of the BdG equation: A = AVAC + ABdG .
(1.8a)
In the excitation picture, on the other hand, it is common to interpret different quantities of interest as the sum of a ‘ground state contribution’ due to |Gi and a quasiparticle contribution due to the excitations: A = A G + AQP .
(1.8b)
We illustrate these two viewpoints (eqns (1.8a)) and (1.8b)) by considering a number of different quantities, A, namely, the charge density (see Section 3), the current density (see Section 4) and the self-consistent fields (see Section 5). Either viewpoint provides a satisfactory description if used consistently. However, there is an important distinction between the ground state |Gi (see eqn (1.4)) and our special vacuum |V i which becomes apparent if we consider a heterointerface (Fig. 2). In the excitation picture, there are two different ground states |G 1 i and |G 2 i (and hence two different ‘vacuum’ states) on the two sides of the heterointerface. A complete theory should describe the spatial variation of the ground state as well as the excitations across the interface, but the BdG equation describes only the latter leaving one with the feeling that something else is needed. But in the one-particle picture there is a single vacuum |V i (a full band of down spin electrons) on both sides of the interface and one can construct scattering eigenstates for the entire structure based on the BdG equation. As such it puts the scattering theory of transport on a firmer conceptual basis, as described in Datta, Bagwell and Anantram, Phys. Low Dim. Struct. 3 (1996).
2. BdG equation as a one-particle wave equation In this section, we will provide a formal justification for the one-particle interpretation of the BdG equation described above. It is well known that the BCS Hamiltonian (s =↑, ↓)‡ Z Z X HBCS = dr 9s+ (r )H 9s (r ) − dr g(r )9↑+ (r )9↓+ (r )9↓ (r )9↑ (r ) (2.1) s
‡ See Ref. [1]. We are letting the real function g(r ) vary spatially to account for inhomogeneities in the electron–phonon interaction. We believe that our discussion can be extended in a straightforward manner to include nonlocal interactions described by a function of the form g(r, r 0 ). Also, we are neglecting expectation values of the form h9↑+ (r )9↓ (r )i, assuming that there are no magnetic impurities or paramagnetic effects.
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INTERFACE
A, Excitation picture
|G 1 i
|G 2 i
FACE FACE
B, One-particle picture
|V i INTER
|V i
Fig. 2. A, In the excitation picture there are two different ground states |G 1 i and |G 2 i (and hence two different ‘vacuum’ states) on the two sides of the heterointerface. A complete theory should describe the spatial variation of the ground state as well as the excitations across the interface, but the BdG equation describes only the latter. B, In the one-particle picture there is a single vacuum |V i and one can construct scattering eigenstates for the entire structure based on the BdG equation.
can be approximated as HBCS ≈ H1 + Hmf , where Z Z HI = dr [U↑ (r )U↓ (r )/g(r )] + dr [|1(r )|2 /g(r )]
(2.2)
and Hmf =
Z
dr
+
Z
X
9s+ (r )[H + Us ]9s (r )
s
dr [1(r )9↑+ (r )9↓+ (r ) + 1∗ (r )9↓ (r )9↑ (r )]
(2.3)
and the self-consistent fields Us and 1 are defined as U↑ (r ) ≡ −g(r )h9↓+ (r )9↓ (r )i = U↑∗ (r )
(2.4a)
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U↓ (r ) ≡ −g(r )h9↑+ (r )9↑ (r )i = U↓∗ (r )
(2.4b)
1(r ) ≡ −g(r )h9↓ (r )9↑ (r )i.
(2.5)
and Hmf is the mean-field Hamiltonian and HI is the negative of the interaction energy which has to be added because the self-consistent field method double-counts the interaction energy. It is a constant that plays no role in the dynamics and we will ignore it in the following discussion. Our objective is to transform Hmf into a form that looks just like the second quantized Hamiltonian for a set of noninteracting particles obeying the BdG equation. This is achieved if we use a ‘particle–hole’ transformation for the down-spin operators, leaving the up-spin operators intact: + 8+ ↑ (r ) ≡ 9↑ (r )
and
8+ ↓ (r ) ≡ 9↓ (r ).
(2.6)
It can be shown (see Appendix B) that in terms of these transformed operators Hmf is written as (HVAC + HBdG ) where HVAC represents the energy of the vacuum consisting of a full band of down-spin electrons (see eqn (B.5)). It is a constant which plays no role in the dynamics. All the dynamics is contained in HBdG Z Z ∗ HBdG = dr 8+ (r )[H + U ]8 (r ) − dr 8+ ↑ ↑ ↑ ↓ (r )[H + U↓ ]8↓ (r ) Z + ∗ + dr [1(r )8+ (2.7) ↑ (r )8↓ (r ) + 1 (r )8↓ (r )8↑ (r )], which has the same form as the second quantized Hamiltonian for a set of noninteracting particles obeying the one-particle wave equation u H + U↑ 1 u (E − µS ) = . (2.8) v 1∗ −(H ∗ + U↓ ) v This is slightly different from the usual BdG equation (see eqn (1.1)) since the fields U↑ and U↓ need not be equal in general under nonequilibrium conditions. However, in our subsequent discussion we will assume spin-independent systems such that U↑ = U↓ ≡ U (see Appendix A). Since HBdG looks just like the second quantized Hamiltonian for a set of noninteracting particles obeying + the BdG equation, it can be diagonalized simply by defining a new operator γ M that creates a ‘particle’ in an eigenstate (u M , v M ) of eqn (2.8): Z + + γM = dr [u M (r )8+ (2.9a) ↑ (r ) + v M (r )8↓ (r )] or using eqn (2.6): + γM
=
Z
dr [u M (r )9↑+ (r ) + v M (r )9↓ (r )].
In terms of this new quasiparticle operator the mean-field Hamiltonian Hmf takes the form X + Hmf = HVAC + (E M − µS )γ M γM .
(2.9b)
(2.10)
M
HVAC is the energy of the ‘vacuum’ consisting of a completely full band of down-spin electrons (see eqn (B.5)). On top of this vacuum we create particles in eigenstates of the BdG equation with energies E M . A BdG quasiparticle represents a superposition of an up-spin electron and a down-spin hole, as evident from eqn (2.9b). The ground state is obtained by filling all the quasiparticle states below µS , that is, the β-states as indicated in eqn (1.4). In Appendix C, we show that this procedure yields a ground state wavefunction that is identical to the familiar BCS wavefunction for homogeneous superconductors; but if the pair potential 1(r ) varies spatially, it yields a generalized BCS-type ground state.
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2.1. Inverse transformation One advantage of this viewpoint is that the Bogoliubov transformation appears like an ordinary unitary transformation in a Hilbert space spanned by (r, s) that is double the usual size. We can rewrite eqn (2.9a) compactly as XZ + γM = dr w M (r, s)8+ (2.11) s (r ), s
where w M (r, ↑) ≡ u M (r )
and
w M (r, ↓) ≡ v M (r ).
(2.12)
The functions w M (r, s), being eigenvectors of a Hermitian operator (the one on the right of eqn (2.8)), obey the orthogonality and completeness relations: XZ dr w ∗M 0 (r, s)w M (r, s) = δ M 0 ,M (2.13) s
and X
w ∗M (r 0 , s 0 )w M (r, s) = δ(r 0 − r )δs 0 ,s .
(2.14)
M
Using these properties it is straightforward to invert eqn (2.11) X + 8+ w ∗M (r, s)γ M s (r ) =
(2.15)
M
that is, using eqns (2.6) and (2.12), X + ψ↑+ (r ) = u ∗M (r )γ M
and
ψ↓ (r ) =
M
X
+ v ∗M (r )γ M .
(2.16)
M
To make contact with the formulation in [1], we note that the summation over ‘M’ includes both α-states and β- states, which can be written out explicitly as (making use of eqn (1.3)) X + + ψ↑+ (r ) = u ∗mα (r )γmα − vmα (r )γmβ (2.17a) m
ψ↓ (r ) =
X
+ ∗ + vmα (r )γmα + u mα (r )γmβ .
(2.17b)
m
These are precisely the relations given in Ref. [1], if we make the association + + γmα → γm↑
and
+ γmβ → γm↓
which was justified earlier (see eqn (1.5)). Also we prefer the notation ‘α’ and ‘β’ instead of ‘↑’ and ‘↓’, since these states reduce to pure spin states only far from the Fermi energy.
3. Charge density We can obtain expressions for different quantities of interest like the charge density, the current density or the self-consistent fields in terms of the quasiparticle occupation probabilities by a straightforward + transformation from the usual field operators 9 + (r ) to the new operators γ M (see Appendix D). However, we can write down the correct answers intuitively simply by adding the vacuum contribution due to |V i and
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the one-particle contribution arising from the filled eigenstates of the BdG equation. For example, the charge density associated with the vacuum is given by (: normalization volume) X 1 , (3.1) ρVAC = e k
since every down-spin ‘k’ state is occupied. Writing the functions v M (r ) in the form 1 X v M (r ) = √ v M (k)eik.r k and making use of the completeness relation (cf. eqn (2.14)) X v ∗M (k 0 )v M (k) = δk 0 ,k , M
we can rewrite eqn (3.1) in the form ρVAC ≡ e
X
|v M |2 .
(3.2a)
M
The one-particle contribution due to the filled states of the BdG equation is given by X ρBdG = e [|u M |2 − |v M |2 ] f M ,
(3.2b)
M
since ‘u’ denotes the electron-like component and ‘v’ denotes the hole-like component of the quasiparticle wavefunction (see eqn (2.9b)). Adding eqns (3.2a) and (3.2b) we obtain the total charge density X ρ = ρVAC + ρBdG = e |u M |2 f M + |v M |2 (1 − f M ). (3.3a) M
The summations in the above expressions extend over both α- and β-states. We could make use of eqns (1.3) and (1.6) to write these as summations over just the α-states (or just the β-states) multiplied by two X ρ = 2e |u mα |2 f mα + |vmα |2 (1 − f mα ) (3.3b) m
= 2e
X
|u mβ |2 f mβ + |vmβ |2 (1 − f mβ )
(3.3c)
m
as is commonly done in the literature. 3.1. Excitation picture Note that ρVAC is the charge due to our special vacuum |V i which consists of a completely full band of down-spin electrons (and an empty band of up-spin electrons). It is different from the charge due to the ground state |Gi (see eqn (1.4)) which is obtained from eqn (3.3b) by setting f mα = 0, or from eqn (3.3c) by setting f mβ = 1: X X ρG = 2e |vmα |2 = 2e |u mβ |2 . (3.4) m
m
In the excitation picture, one can interpret the total charge density as the sum of the ground state and quasiparticle contributions (ρ = ρG + ρQP ), where the latter is given by X ρQP = 2e (|u mα |2 − |vmα |2 ) f mα . (3.5) m
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4. Current density The electrical current can also be written as the sum of a one-particle current and a vacuum current, where X u v JBdG = e [J M − JM ] fM (4.1) M
JVAC =
X
v JM .
(4.2)
M
u and J v are the particle fluxes associated with the ‘u’ and ‘v’ components of the quasiparticle JM M u is given by the usual expression for the one-particle current density in elementary quantum wavefunction. J M mechanics: p − eA u ∗ J M (r ) ≡ Re u M (r ) u M (r ) . (4.3a) m v is similar but with ‘v’ and ‘v ∗ ’ interchanged: The expression for JM p − eA ∗ v J M (r ) ≡ Re v M (r ) v M (r ) . (4.3b) m
This interchange is necessary in order that the continuity equation is satisfied. Specifically, if we start from the time-dependent version of the BdG eqn (1.1) n o ∂ nu o u H +U 1 i~ = , (4.4) 1∗ −(H ∗ + U ) ∂t v v we obtain the following continuity equation for the probability u ∂ u∗u J +S + ∇. = , ∂t v ∗ v Jv −S A v (r ) ≡ Re v(r ) p−e A v ∗ (r ) . where J u (r ) ≡ Re u ∗ (r ) p−e u(r ) and J m m
(4.5)
Note that the roles of v and v ∗ are interchanged relative to u and u ∗ , in accordance with eqns (4.3a) and (4.3b). The significance of the source term S≡
2 I m[1(r )u ∗ (r )v(r )] ~
(4.6)
will be discussed shortly. The total current is obtained by adding the one-particle and vacuum contributions (eqns (4.1) and (4.2)) X u v J = JBdG + JVAC = e JM f M + JM (1 − f M ) (4.7a) M
and can be expressed solely in terms of the α- (or the β-) states X u v J = 2e Jmα f mα + Jmα (1 − f mα )
(4.7b)
m
= 2e
X
u v Jmβ f mβ + Jmβ (1 − f mβ )
(4.7c)
m
by making use of eqns (1.3) and (1.6). As before we can show that the term JVAC arises from a full band of down-spin electrons: X X ~k − e A(r ) v JVAC = JM =e . m M
k
(4.8)
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This ‘vacuum current’ JVAC is often zero but not always (such as problems involving the ac Josephson effect). In any case it is usually advantageous to retain it in eqn (4.1), because at large negative energies, it cancels out the contribution from JBdG term by term for each ‘M’. This allows us to truncate the summation at a suitably large negative value of E M . 4.1. Excitation picture In the excitation picture, it is common to interpret the total current density as the sum of the ground state current or ‘supercurrent’ and the quasiparticle current (J = JG + JQP ), where the former is obtained from eqn (4.7b) by setting f mα = 0 or from eqn (4.7c) by setting f mβ = 1: JG = 2e
X
v Jmα = 2e
X
u Jmβ ,
(4.9)
u v [Jmα − Jmα ] f mα .
(4.10)
m
m
while the latter is given by JQP = 2e
X m
It is important to note that in the excitation picture, both the supercurrent and the quasiparticle current are non-trivial quantities that can be different from zero. By contrast, in the one-particle picture the vacuum current is usually zero, and all the physics is contained in JBdG . We believe that this is an advantage for it could lead to a unified treatment of supercurrents and quasiparticle currents. 4.2. Current conservation From eqn (4.5) it follows that u v ∇.[JM ] = −∇.[JM ]=
2 I m[1(r )u ∗M (r )v M (r )]. ~
(4.11)
u + J v ) is conserved: This means that the quasiparticle probability current (J M M u v ∇.[J M + JM ]=0
as we would expect since the operator in the BdG equation (see eqn (1.1)) is Hermitian. But the electrical u − J v ) which is not necessarily conserved [4–7]: current (see eqn (4.1)) is proportional to (J M M u v ∇.[J M − JM ]=
4 I m[1(r )u ∗M (r )v M (r )]. ~
Using eqn (4.1) we can write ∇.[JBdG ] =
4e X I m[1(r )u ∗M (r )v M (r ) f M ]. ~
(4.12)
M
The right-hand side can be shown to be zero using the expression for the self-consistent field 1 (see eqn (5.2a)), proving that the electrical current is indeed conserved, if 1 is determined self-consistently, but not otherwise.
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5. Self-consistent fields Finally, we consider the self-consistent fields defined in eqns (2.4a, b) and (2.5). It is evident from eqns (2.4a) and (2.4b) that the field Us (r ) is determined by the density of electrons with the opposite spin. Under the spin-independent conditions assumed in this paper (see Appendix A) this is simply half the total electron density given in eqns (3.3a, b, c), so that gX U =− [|u M |2 f M + |v M |2 (1 − f M )] (5.1a) 2 M X = −g [|u mα |2 f mα + |vmα |2 (1 − f mα ))] (5.1b) m
= −g
X
[|u mβ |2 f mβ + |vmβ |2 (1 − f mβ ))].
(5.1c)
m
The self-consistent field 1 (see eqn (2.5)) is determined by the correlation u M v ∗M between the two components of a quasiparticle state, so that X 1 = −g v ∗M u M f M . (5.2a) M
Once again we can use the relations given in eqns (1.3) and (1.6) to write this expression solely in terms of the α- (or β-) states: X ∗ 1 = −g [vmα u mα (2 f mα − 1)] (5.2b) m
X ∗ = −g [vmβ u mβ (2 f mβ − 1)].
(5.2c)
m
In the excitation picture we could interpret the self-consistent fields as the sum of a ground state contribution X X UG = −g |vmα |2 = −g |u mβ |2 (5.3a) m
1G = +g
m
X
∗ vmα u mα
= −g
m
X
∗ vmβ u mβ
(5.3b)
m
and a quasiparticle contribution: UQP = −g
X
(|u mα |2 − |vmα |2 ) f mα
(5.4a)
m
1QP = −2g
X ∗ [vmα u mα f mα ].
(5.4b)
m
6. Summary The basic point of this paper is that the BdG equation can be interpreted as a ‘one-particle’ wave equation just like the Schr¨odinger equation for normal systems, if we treat the β-states seriously on an equal footing with the α-states. The eigenstates of the BdG equation can be filled up systematically to obtain the superconducting state just as one fills up the eigenstates of the Schr¨odinger equation for the normal state. The only difference is that we start from a special vacuum |V i and not the usual vacuum |0i. Any quantity of interest, A, can be interpreted as the sum of a ‘vacuum contribution’ AVAC due to the vacuum |V i and a one-particle contribution ABdG due to the filled eigenstates of the BdG equation. This is contrasted with
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the commonly used excitation picture in which one interprets different quantities of interest as the sum of a ‘ground state contribution’ A G due to |Gi and a quasiparticle contribution AQP due to excitations. The excitation picture is difficult to apply to a strongly coupled heterojunction since the ground state |Gi changes across the interface. By contrast, the vacuum |V i is the same on both sides of a heterointerface making it conceptually straightforward to apply the one-particle picture to heterojunctions. Acknowledgement—This work was supported by the MRSEC program of the National Science Foundation under Award No. DMR-9400415.
References [1] P. G. de Gennes, Superconductivity of Metals and Alloys (Addison-Wesley, 1989) Chap. 5. [2] C. Virgilio Nino, R. Kuemmel Phys. Rev. B29, 3957 (1984); U. Gunsenheimer, U. Schussler, and R. Kuemmel, Phys. Rev. B49, 6111 (1994). [3] M. Tinkham, Superconductivity (Gordon and Breach, 1965) p. 58. [4] G. E. Blonder, M. Tinkham, and T. M. Klapwijk, Phys. Rev. B25, 4515 (1982). [5] A. Furusaki and M. Tsukada, Solid State Commun. 78, 299 (1991). [6] P. F. Bagwell, Phys. Rev. B49, 6841 (1994). [7] F. Sols and J. Ferrer, Phys. Rev. B49, 15913 (1994). [8] M. Ma and P. A. Lee, Phys. Rev. B32, 5658 (1985).
Appendix A. Symmetry of α - and β -states In most of our discussions, we have assumed the following relations to hold between the α- and β-states: E mβ = 2µS − E mα ,
∗ u mβ = −vmα ,
vmβ = u ∗mα
(1.3)
f mβ = 1 − f mβ = f mα . (1.6) In this appendix, we will show that these relations hold even under nonequilibrium conditions, as long as there are no spin-dependent processes. However, it should be noted that if we include the Zeeman splitting due to a z-directed magnetic field, eqn (1.1) would be modified to n o nu o u H + U + µ B Bz 1 (E − µS ) = . 1∗ −(H ∗ + U − µ B Bz ) v v Since the [22] element is no longer just the negative complex conjugate of the [11] element, eqn (1.3) would not hold (this is easy to see in the special case when 1 is zero). Even if the [22] element is equal to the negative complex conjugate of the [11] element and eqn (1.3) holds, their occupation factors may not obey eqn (1.6), if we have a contact that injects spin-polarized electrons (like a ferromagnetic contact). However, a general treatment of such problems with arbitrarily directed magnetic fields requires one to go beyond a simple (2 × 2) BdG equation (see, for example, Chap. 9 of Ref. [1]) and is outside the scope of this paper. We will now assume spin-independent interactions and prove eqns (1.3) and (1.6). Equation (1.3).
Suppose we have a solution that satisfies the BdG equation (eqn (1.1)) u mα u mα H +U 1 (E mα − µS ) = 1∗ −(H ∗ + U ) vmα vmα
with E mα > µS . A little straightforward algebra shows that ∗ ∗ −vmα H +U 1 −vmα − (E mα − µS ) = . u ∗mα u ∗mα 1∗ −(H ∗ + U )
(A.1)
(A.2)
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m m0
µS + 1 µS µS − 1
m0 m ‘β’ states
B
‘α’ states E µS + 1 µS µS − 1
m m0
m0 m ‘β’ states
Fig. 3. Pairs of A, intraband and B, interband transitions that have identical transition rates (see eqns (A.4a, b)).
But u mβ H +U 1 u mβ (E mβ − µS ) = vmβ 1∗ −(H ∗ + U ) vmβ by definition. Comparing eqns (A.2) and (A.3) we obtain eqn (1.3).
(A.3)
Equation (1.6). Assume that we start from a condition in which eqn (1.6) is satisfied and turn on some interactions that cause the particles to scatter from one state to another. It can be shown that eqn (1.6) will continue to be satisfied provided the scattering rates obey the following relations (see Fig. 3): R(mα → m 0 α) = R(m 0 β → mβ)
(A.4a)
R(mα → m 0 β) = R(m 0 α → mβ).
(A.4b)
and
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These relations ensure that any process leading to a decrease in f mα , is accompanied by an identical one that leads to an increase in f mβ , so that ( f mα + f mβ ) remains unchanged. To prove eqns (A.4a, b), we assume an arbitrary perturbation of the form h δ 0 H = δ ∗ −h ∗ and make use of eqn (1.3) to show that the resulting matrix elements satisfy the relations: hm 0 α|H 0 |mαi = −hmβ|H 0 |m 0 βi and hm 0 β|H 0 |mαi = hmβ|H 0 |m 0 αi. Equations (A.4a, b) then follow readily from the golden rule (the indices γ and γ 0 take on the values α, β) 2π |hm 0 γ 0 |H 0 |mγ i|2 δ[E m 0 γ 0 − E mγ ± ~ω]. ~ To show that ( f mα + f mβ ) remains unchanged by scattering processes, we can start from a master equation of the form X d f mγ = R(m 0 γ 0 → mγ ) f m 0 γ 0 (1 − f mγ ) − R(mγ → m 0 γ 0 ) f mγ (1 − f m 0 γ 0 ) dt 0 0 R(mγ → m 0 γ 0 ) =
mγ
and make use of eqns (1.6) and (A.4a, b) to show that d( f mα + f mβ )/dt = 0. This means that if eqn (1.6) holds as an initial condition, it holds for all time.
Appendix B. Particle–hole transformation of Hmf In this appendix, we will show how eqn (2.7) is obtained from eqn (2.3) using the transformation indicated in eqn (2.6). For this purpose it is convenient to transform from the position representation to a discrete basis using any complete basis set φµ (r ) that spans the one-particle, one-spin Hilbert space. X X 9↑ (r ) = aµ φµ (r ) and 9↓ (r ) = bµ φµ∗ (r ). (B.1) µ
µ
The mean-field Hamiltonian then takes the form X X Hmf = [H + U↑ ]µν aµ+ aν + [H ∗ + U↓ ]µν bν+ bµ µ,ν
+
µ,ν
X µ,ν
[1µν aµ+ bν+ + (1∗ )νµ bν aµ ],
(B.2)
where [H + Us ]µν ≡
Z
dr φµ∗ (r )([H + Us ]φν (r ))
(B.3a)
[1]µν ≡
Z
dr φµ∗ (r )1(r )φν (r ).
(B.3b)
and
Note that (see eqn (B.1)) we are expanding the up-spin field 9↑ operator using the set {φµ } and the down-spin field operator 9↓ using the ‘time-reversed set’ {φµ∗ }. As a result we obtain the ‘normal’ matrix elements 1µν (see eqn (B.3b)). If instead we were R to use the same set {φµ } to expand both 9↑ and 9↓ we would obtain ‘abnormal’ quantities of the form dr φµ∗ (r )1(r )φν∗ (r ).
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To make Hmf look more like a ‘normal’ Hamiltonian we use a ‘particle–hole’ transformation for the downspin operators, leaving the up-spin operators intact: + + cµ,↑ ≡ aµ+ , cµ,↓ ≡ bµ .
(B.4)
Making use of the anti-commutation property bν+ bµ + bµ bν+ = δµν , we can write Hmf as (HVAC + HBdG ) where X HVAC = [H ∗ + U↓ ]µµ = Trace [H ∗ + U↓ ] (B.5) µ
and HBdG =
X µ,ν
+
+ [H + U↑ ]µν cµ,↑ cν,↑ −
X µ,ν
+ [1µν cµ,↑ cν,↓
X µ,ν
+ [H ∗ + U↓ ]µν cµ,↓ cν,↓
+ + (1 )νµ cν,↓ cµ,↑ ]. ∗
(B.6)
We can obtain eqn (2.7) by transforming eqn (B.6) back to the position representation, noting that from eqns (2.6), (B.1) and (B.4): X 8s (r ) = cµ,s φµ (r ) (B.7a) µ
so that the inverse transformation is given by cµ,s =
Z
dr φµ∗ (r )8s (r ).
(B.7b)
Appendix C. Ground state wavefunction In this section we will see how we can construct the ground state wavefunction starting from the special vacuum, |V i, consisting of a full band of down-spin electrons Y + |V i ≡ bµ |0i (C.1) µ
and filling up all the β-eigenstates whose energies are less than µS : Y + |Gi = γmβ |V i.
(C.2)
m
Noting that + γmβ =
X µ
u mβ,µ aµ+ + vmβ,µ bµ
(C.3)
we can write down the ground state wavefunction from eqns (C.1) and (C.2): " #! Y X Y + |Gi = u mβ,µ aµ + vmβ,µ bµ bν+ |0i. m
µ
(C.4)
ν
Equation (C.4) looks rather complicated, but we can simplify it if we assume that the pair potential 1 is a constant. Each eigenstate of the BdG equation (see eqn (2.1)) then corresponds to an eigenstate of [H + U ]. For example, in the position representation we can write u mβ (r ) Umβ = φm (r ) (C.5) vmβ (r ) Vmβ
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assuming that the functions φm (r ) represent eigenstates of [H + U ] with eigenenergies εm . Umβ and Vmβ are constants that are evaluated by solving the (2 × 2) eigenvalue problem: Um εm 1 Um (E m − µS ) = . (C.6) Vm 1∗ −εm Vm This yields two eigenvectors for each ‘m’, which we denote as (Umα , Vmα ) and (Umβ , Vmβ ); to construct the ground state from eqn (C.4) we only need the β-state having energy less than µS . Using eqn (C.5) we can write u mβ,µ = Umβ δm,µ and vmβ,µ = Vmβ δm,µ so that from eqn (C.4), |9G i =
Y
+ + [Vmβ + Umβ am bm ]|0i.
(C.7)
m
For homogeneous superconductors the functions φm (r ) that diagonalize [H + U ] are simply the set of plane waves eik.r · |Gi in eqns (4.7a, b, c) then reduces to the familiar BCS wavefunction: Y |Gi = [Vkβ + Ukβ ak+ bk+ ]|0i ⇒ |9BCS i. (C.8) k
It might appear from eqn (C.8) that we are pairing an up-spin ‘+k’ with a down-spin ‘+k’ state rather than a down-spin ‘−k’ state. However, this is purely a matter of notation. From eqn (B.1) we can write Z + ak = dr e+ik.r 9↑+ (r ) and bk+ =
Z
dr e−ik.r 9↓+ (r )
so that the operator bk+ creates a down-spin electron in a ‘−k’ state, although the operator ak+ creates an up-spin electron in a ‘+k’ state. This difference arises because we expanded the down-spin operator in terms of the time-reversed set (see eqn (B.1)). The wavefunction in eqn (C.7) is often assumed for the ground state of inhomogeneous superconductors, with the functions φm (r ) chosen so as to diagonalize [H + U ]. However, as noted in Refs. [1] and [8], this solution does not diagonalize the BdG equation, since a self-consistent calculation yields a pair potential 1 that varies spatially. It is only by neglecting this spatial variation in 1 that we are able to simplify eqn (C.4) to eqn (C.7). A more accurate solution that takes into account the spatial variation in 1 could lead to a lower energy ground state wavefunction.
Appendix D. Expectation values of field operators Using eqn (2.16) it is straightforward to show that X + h9↑+ 9↑ i = u ∗M (r )u M 0 (r )hγ M γM 0 i
(D.1a)
M,M 0
h9↓+ 9↓ i =
X
M,M 0
+ v ∗M (r )v M 0 (r )hγ M 0 γ M i
(D.1b)
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Superlattices and Microstructures, Vol. 25, No. 5/6, 1999
h9↓ 9↑ i =
X
+ v ∗M (r )u M 0 (r )hγ M γ M 0 i = h9↑+ 9↓+ i∗ .
(D.2)
M,M 0
For a general nonequilibrium state, the quasiparticle states can be occupied arbitrarily, which will be reflected + in arbitrary values of the quantity hγ M γ M 0 i. Usually the density matrix is assumed to be diagonal as we have done in this paper; that is, + hγ M γ M 0 i = f M δ M,M 0
and
+ hγ M 0 γ M i = (1 − f M )δ M,M 0 ,
(D.3)
where f M denotes the occupation factor for the eigenstate ‘M’. The self-consistent fields that we have written down intuitively in eqns (5.1a, b, c) and (5.2a, b, c) can be obtained formally by combining eqns (D.1a, b) through (D.3) with eqns (2.4a, b) and (2.5). We can also use these expressions to obtain eqns (3.3a, b, c) and (4.7a, b, c) for the charge and current densities (which were obtained using intuitive arguments) if we note that the charge density is given by X ρ(r ) = eh9s+ (r )9s (r )i (D.4) s
while the current density is given by J (r ) =
X e Reh9s+ (r )( p − e A)9s (r )i. m s
(D.5)