Response properties of III-V dilute magnetic semiconductors: interplay of disorder, dynamical electron-electron interactions and band-structure effects

0.54 MB·English

by  F. V. Kyrychenko

Response properties of III-V dilute magnetic semiconductors: interplay of disorder,
dynamical electron-electron interactions and band-structure effects
F. V. Kyrychenko and C. A. Ullrich
Department of Physics and Astronomy, University of Missouri, Columbia, Missouri 65211, USA
(Dated: January 31, 2011)
A theory of the electronic response in spin and charge disordered media is developed with the
particularaimtodescribeIII-VdilutemagneticsemiconductorslikeGa1−xMnxAs. Thetheorycom-
bines a detailed k·p description of the valence band, in which the itinerant carriers are assumed
1 toreside,withfirst-principlescalculationsofdisordercontributionsusinganequation-of-motionap-
1 proachforthecurrentresponsefunction. Afullydynamictreatmentofelectron-electroninteraction
0 isachievedbymeansoftime-dependentdensityfunctionaltheory. Itisfoundthatcollectiveexcita-
2 tionswithinthevalencebandsignificantlyincreasethecarrierrelaxation ratebyprovidingeffective

n channels for momentum relaxation. This modification of the relaxation rate, however, only has a
a minor impact on the infrared optical conductivity in Ga1−xMnxAs, which is mostly determined by
J thedetails of thevalence band structureand found to bein agreement with experiment.

8
2 PACSnumbers: 72.80Ey, 75.50Pp,78.20.Bh

]
i I. INTRODUCTION to details of the growth conditions15 and post-growth
c
s annealing16–18 points to the crucial role played by the
- defects and their configurations, and has stimulated in-
l The idea of using both charge and spin of electrons
r tense research on the structure of defects and their in-
t in a new generation of electronic devices constitutes the
m basisofspintronics.1 Themagneticpropertiesofthe ma- fluence on the various properties of the system.19 It is
essential,therefore, to developa theory of electricalcon-
. terial combined with its semiconducting nature makes
t ductivity inDMSs with moreemphasis givento disorder
a dilute magnetic semiconductors (DMSs) potentially ap-
m pealing for various spintronics applications.2 In particu- andelectron-electroninteractions,withoutneglectingthe
intricacies of the electronic band structure.
- lar, the effect of carrier mediated ferromagnetism opens
d up the possibility to control the electron spin and mag- Here we present a comprehensive theory for the elec-
n
netic state of a system or device by means of an electric trondynamics inDMSs whichaccountsforthe complex-
o
field. A lotofattentionis drawnto Ga Mn As due to ity of the valence band structure of the semiconductor
c 1 x x
[ the well developed technology of the co−nventional GaAs host material and treats disorder and electron-electron
basedelectronicsanddiscoveryofitsrelativelyhighferro- interaction on an equal footing. In previous work we
1
magnetic transition temperature,2 with a current record used a simplified treatment of the semiconductor va-
v
of T =185 K.3 lenceband20,21 orconsideredonlystaticpropertiesofthe
8 c
1 Unlike most other III-V DMSs, the nature of the itin- system.22Inthispaperwesimultaneouslyaccountforthe
4 erant carriersin Ga Mn As is still under debate.4,5 It complexity of the valence band, use a first-principles ap-
1 x x
5 is widely acceptedth−at for low-dopedinsulating samples proach to describe disorder contributions, and employ a
1. the Fermi energy lies in a narrow impurity band. For fully dynamic treatment of electron interactions.
0 more heavily doped, high-Tc metallic samples there are To account for the valence band structure we use the
1 strong indications that the impurity band merges with generalized k p approach23 where a certain number of
1 the host semiconductor valence band forming mostly bandsaretrea·tedexactlywhilethecontributionfromthe
:
v host-likestatesattheFermienergywithsomelow-energy remote bands is included up to second order in momen-
i tail of disorder-related localized states.6 First-principles tum. To describe disorder effects we use the equation of
X
calculations7–9 have so far not been fully conclusive re- motion for the paramagnetic current response function
ar gardingthenatureoftheitinerantcarriersinthisregime, of the fully disordered system. This approach has some
and further theoretical studies continue to be necessary. similaritiestomodelsdevelopedearlierusingthememory
Meanwhile, attention has shifted to model Hamiltonian function formalism.24–26 The advantage of our approach
approaches assuming either the valence band10 or impu- ascomparedtothememoryfunctionformalismistherel-
rity band11 picture and their ability to adequately de- ative simplicity and transparency of the derivation and
scribe the experimental results in Ga1 xMnxAs. the straightforwardpossibility to include the spindegree
−
Most theoretical approaches assuming the valence- of freedom. Another advantage is that our formalism is
band nature of itinerant holes in Ga Mn As treat the expressedin terms of a current-currentand a set of den-
1 x x
band structure in detail, while disord−er and many-body sityandspin-densityresponsefunctions. Thisenablesus
effects are only accounted for using simple phenomeno- tousethepowerfulapparatusoftime-dependentdensity-
logicalrelaxationtime approximationsandstatic screen- functionaltheory(TDDFT)27 totreatmany-bodyeffects
ingmodels.12–14 Ontheotherhand,theextremesensitiv- such as dynamic screening and collective excitations of
ityofmagneticandtransportpropertiesofGa Mn As the itinerant carriers in principle exactly.
1 x x
−
2
The paper is divided into two major sections and con- with the components
clusions. Forease of reading,some of the derivationsare
1
presented in appendices. The theory section (Sec. II) ρˆµ(k)= u σµ u aˆ+ aˆ . (5)
is organized as follows. In Sec. IIA we present our V h n′,q−k| | n,qi n′,q−k n,q
q nn′
general formalism based on the equation of motion of XX
the current-current response function of the disordered Here, σµ (µ = 1,z,+, ) is defined via the Pauli matri-
system. In Sec. IIB we describe the evaluation of the ces,whereσ1 isthe2 −2unitmatrix,σ =(σx iσy)/2,
±
current-current, density and spin-density response func- and u arethetwo×-componentBlochfunctio±nspinors
n,q
tions for the multiband system using a generalized k p with|waveivectorqandbandindexn. Thesummationin
·
perturbation approach. Next, in Sec. IIC we show the Eq.(3)isperformedoveralldefects. Notethatthemean
treatment of electron-electron interaction by means of field part of the p-d exchange interactionbetween itiner-
TDDFT. In Section III we first discuss the new fea- ant holes and localized spins is absorbed into the clean
tures that the valence band character of itinerant car- system band structure Hamiltonian Hˆ ; disorder in our
e
riers brings into the system, namely the dominance of modelconsistsoftheCoulombpotentialofchargedefects
the long-wavelengthside of the single-particle excitation and fluctuations of localizedspins aroundthe mean field
spectrum by the interband spin transitions and the ef- value S .
fective suppression of the collective plasmon excitations Thehgeineral case of multiple types of defects, includ-
within the valence band for the whole range of momen- ing defect correlations, was considered in Ref. 20. For
tum. Next, in Sec. IIIB we discuss the effect of mag- simplicity we here include only the most important de-
neticdoping: spinandchargedisorderinthesystemand fect type, namely randomly distributed manganese ions
modification of the band structure in the magnetically in gallium substitutional positions (Mn ). Our model
Ga
orderedphase. Weshowthatthefulldynamictreatment treats localized spins as quantum mechanical operators
of electron-electron interactions allows us to capture the coupledtothe bandcarriersviaacontactHeisenbergin-
effect of collective excitations on the carrier relaxation teraction featuring a momentum-independent exchange
time. We then compareour results alsowith experimen- constant J. We use the value of VJ = 55meVnm3,
tal data on infrared conductivity. Finally, in Sec. IV we whichcorrespondstothe widelyusedDMS−p-dexchange
draw our conclusions. constant N β = 1.2eV.10 The z-axis is chosen along
0
−
the direction of the macroscopic magnetization.
Earlier we developed a theory of transport in charge
II. THEORY
andspindisorderedmediawithemphasisonatreatment
ofdisorderandelectron-electroninteraction.21Itisbased
A. General formalism on an equation of motion28,29 approach for the param-
agnetic current-current response of the full, disordered
We discuss a system described by the Hamiltonian system:
Hˆ =Hˆ +Hˆ +Hˆ , (1)
e m d i
χ (r,r,τ)= Θ(τ) [ˆj (τ,r),ˆj (r)] , (6)
whereHˆ isthecontributionoftheitinerantcarriersand jpαjpβ ′ −¯h h pα pβ ′ iH
e
Hˆ representsthesubsystemoflocalizedmagneticspins.
m where
These two terms constitute the “clean” part of the total
Hamiltonian. The lastterminEq.(1)describesdisorder ˆjpα(τ,r)=eh¯iHˆτˆjpα(r)e−h¯iHˆτ (7)
in the system:
Hˆ =V2 ˆ~(k) ρˆ~( k), (2) is the paramagnetic current-density operator in Heisen-
d U · − bergrepresentationandα,β =x,y,z areCartesiancoor-
k
X dinates. Duringthederivationweassumedoursystemto
wherethefour-componentchargeandspindisorderscat- bemacroscopicallyhomogeneous,whichisjustifiedifthe
tering potential coherence length of the electrons is much shorter than
U (k) the system size. In this case, summing over all electrons
j
willleaveuswithanaveragedeffectofdisorderthatdoes
Uˆ~(k)= V1 Xj −J2 (cid:16)−−SˆJ2J2jzSSˆˆ−j+j−hSi(cid:17) eik·Rj (3) nsatuutorctpbhodaimetnpitoaencrnrddaonesocpdnoenpntoidhctseaoloplnynaltrhyhtoieocmnupolataghrreteindcdeiuisosloautrarsdncsechyreoscti|ocreenm−fiosgfrtu′ph|roteaiontritteohssnper.opaFnenosrder-
is coupled to the four-component vector of charge and
r. Anothermajorapproximationinvolvesthedecoupling
′
spin density operators of the itinerant carriers:
procedure,whereweneglecttheinfluenceoftheitinerant
ρˆ1 nˆ carriers on the localized spins. Therefore, our approach
ρˆ~= ρˆz = sˆz (4) does not include magnetic polaron effects and lacks the
 ρˆ+   sˆ+  microscopicfeaturesofcarriermediatedferromagnetism.
ρˆ sˆ The latter,however,canbe reinstatedto some extentby
 −   − 
   
3
introducing a phenomenological Heisenberg-like term in Equations (10)–(12) will be used in the following.
the magnetic subsystemHamiltonianHˆ . Details ofthe
m
derivation are presented in Ref. 21.
The final expression for the total current response
reads B. Multiband k·p approach
n
χJ (q,ω)=χc (q,ω)+ δ
αβ jpαjpβ m αβ Inordertoobtainthe conductivitythroughEqs.(10)–
V2 (12) we will have to calculate the paramagnetic current
+ m2ω2 Xk kαkβXµν DUˆµ(k)Uˆν(−k)EHm roefstphoensceleaanndsyspstinema.ndTcohparrogpeedrelynsditeyscrreisbpeotnhsee fcuonmcptiloenxs-
χ (q k,ω) χc ( k) , (8) ity of the semiconductor valence band we are going to
× ρµρν − − ρµρν − implement the multiband k p approach.
(cid:16) (cid:17) ·
where χ (k,ω) is the set of charge and spin density First we derive the current and density response func-
ρµρν
response functions with respect to the operators (4)–(5) tions in the formal basis of the Bloch states
and the superscript “c” indicates quantities defined in
1
the clean system. By comparing Eq. (8) with the Drude n,k = eikr u (13)
· n,k
formula in the weak disorder limit ωτ 1, | i √V | i
≫
χJ(ω)= n 1 n in , (9) which diagonalize the clean system Hamiltonian
D m1+i/ωτ ≈ m − mωτ
Hˆ = ε aˆ+ aˆ . (14)
we identify the tensor of Drude-like frequency- and n,k n,k n,k
momentum-dependent relaxation rates of the form Xn,k
V2 Withinsecondquantizationinthebasis(13),theparam-
τα−β1(q,ω)=inmω Xµkν kαkβDUˆµ(−k)Uˆν(k)EHm aisggnievteicncbuyrrent in the system with spin-orbit interaction
χ (q k,ω) χc (k,0) . (10)
× ρµρν − − ρµρν
(cid:16) (cid:17)
Notethattheright-handsideofEqs.(8)and(10)con- 1 ¯h 1
ˆj (q)= k q u u
ttahiensfutlhl,edsiesotrodfersepdinsyasntedmc.haTrgheerreefsoproe,nssetrfiuctnlcytisopnesako-f p V nX,n′,k(cid:20)m0 (cid:18) − 2 (cid:19)h n′,k−q| n,ki
ing, Eq. (8) should be evaluated self-consistently30 with + 1 u ~πˆ u aˆ+ aˆ , (15)
the continuity equations closing the loop. Here we use a m0h n′,k−q| | n,ki(cid:21) n′,k−q n,k
simplified approachbased ontwo approximations. First,
taking the weak disorder limit in the right hand side of
with
Eq. (10) we retain terms up to the second order in com-
ponents of the disorder potential. In other words, the
¯h
spin and charge response functions of the full system in ~πˆ =pˆ+ 4m c2[σˆ×∇ˆUc], (16)
Eq.(10)arereplacedbytheircleansystemcounterparts: 0
χ (q k,ω) χc (q k,ω). (11) whereUcistheperiodiccrystalfieldpotential. Hereafter,
ρµρν − → ρµρν − performingtherealspaceintegrationweassumethatthe
envelop function varies slowly on the scale of the unit
Next we assume that the paramagnetic currentresponse
cell.
function of the full system may be expressed as the
clean system response function with a lifetime broaden- Introducing the time dependence of the creation and
ing given by Eq. (10): destruction operators in (15), the paramagnetic current
response of the multiband system can be directly evalu-
χjpαjpβ(q,ω)≈χcjpαjpβ(q,ω−iτα−β1). (12) ated, and one finds
1 f f
χcjpαjpβ(q,ω)= Vm20 nX,n′,kεn′,k−qn′−,k−εnq,k−+n¯h,kω+iη (17)
q q
α β
¯h k u u + u πˆ u ¯h k u u + u πˆ u .
α n′,k q n,k n′,k q α n,k β n,k n′,k q n,k β n′,k q
× − 2 h − | i h − | | i − 2 h | − i h | | − i
h (cid:16) (cid:17) ih (cid:16) (cid:17) i
4
A similar procedure for the spin and charge density response yields:
1 f f
χcρµρν(q,ω)= V εn′,k qn′,k−εnq,k−+n¯h,kω+iη hun′,k−q|σˆµ|un,kihun,k|σˆν|un′,k−qi. (18)
nX,n′,k − −
AllweneednowforevaluatingEqs.(17)and(18)istodeterminetheformoftheperiodicBlochfunctions u that
n,k
| i
diagonalize the clean system Hamiltonian. The common approach is to diagonalize the multiband k p Hamiltonian
·
that treats certain bands exactly and treats contributions from remote bands up to second order in momentum. The
derivationofsuchaHamiltonianisoutlinedinAppendixA. BydiagonalizingthematrixofthisHamiltonian,however,
we obtain the eigenvectors of the modified Hamiltonian (A7). Before evaluating the matrix elements between Bloch
periodic functions u in Eqs.(17) and(18) we therefore haveto performthe unitary transformation(A4). Details
n,k
| i
of these calculations are presented in Appendix B.
The final expression for the paramagnetic current response function in the long-wave limit q = 0 (since we are
looking for the optical response) is given by
1 f f
χc (ω)= n′,k− n,k (19)
jpαjpβ Vm2 ε ε +¯hω+iη
0 n,n′,k n′,k− n,k
X
m ∂ m ∂
×" Bs∗′(n′,k)Bs(n,k) ¯h0∂kαhs′|H¯|si#" Bs∗(n,k)Bs′(n′,k) ¯h0∂kβhs|H¯|s′i#,
s′s s′s
X X
where H¯ denotes the effective multiband k p Hamiltonian (A7) and B(n,k) is its eigenvector for the state with
·
energy ε . The charge and spin density response is approximated by
n,k
1 f f
χcρµρν(q,ω)≈ V ε n′,k−εq−+n¯h,kω+iη (20)
n′,k q n,k
nX,n′,k − −
B (n,k q)B (n,k q)B (n,k)B (n,k) s σˆµ s τ σˆν τ .
× s∗′ ′ − τ′ ′ − s τ∗ h ′| | ih | | ′i
s′,s,τ,τ′
X
If σˆµ =(σˆν)+, i.e. for χ , χ and χ , the second sum is a real quantity. Then, the imaginary part is
nn szsz s±s∓
2
π
[χc (q,ω)]= d3k(f f )δ[¯hω (ε ε )] B (n,k q)B (n,k) s σˆµ s .
ℑ ρµ(ρµ)+ −(2π)3 n′,k−q− n,k − n,k− n′,k−q (cid:12) s∗′ ′ − s h ′| | i(cid:12)
n,n′Z (cid:12)s′,s′ (cid:12)
X (cid:12)X (cid:12)
(cid:12) ((cid:12)21)
(cid:12) (cid:12)
(cid:12) (cid:12)
It is seen that in the long-wavelength limit (q 0) the system. In this paper we are considering the optical re-
→
imaginary part of the density response (σµ σ1) van- sponse, i.e. the response to transverse perturbations.
≡
ishes as a product of orthogonal states, while the imag- Since transverse perturbations only induce a transverse
inary part of spin response is, in general, finite. We response in a homogeneous system, there are no density
conclude from this that the long-wavelength spectrum fluctuations directly created by an electromagnetic field.
ofsingle-particleexcitationsisdominatedbyspintransi- The total current response of the interacting system in
tions. this case can be expressed as
The calculations were performed within an 8-band
1 1 4πe q2
k p model. The basis functions and explicit form of χJ(q,ω) − = χJ(q,ω) − + + v G ,
th·e Hamiltonian matrix are presented in Appendix C. 0 ω2 c2q2 ω2 q T+
(cid:16) (cid:17) (cid:16) (cid:17) − (22)
whereχJ istheresponseofthenoninteractingsystem,v
0 q
istheCoulombinteraction,andthelocalfieldfactorG
T+
C. Electron-electron interaction
representscorrectionsfromtheexchange-correlation(xc)
part of the electron interaction.
A major advantage of our formalism is that it is ex- The corrections to the transverse current response
pressed in terms of current and density response func- function causedby electron-electroninteractionarerela-
tions. This allows us to use the powerful apparatus of tivistically small in this case and can be neglected. So,
TDDFT to account for the effects of electron-electron for the transverse current response of the clean system
interaction. we will use the noninteracting form.
Let us first examine the current response of the clean The set of the density and spin-density response func-
5
tions of the clean system enters our expression (10) exchange part of f and apply the adiabatic local spin
xc
for the frequency- and momentum-dependent relaxation
density approximation. Explicitexpressionsforthe local
rates. TDDFT allows us to describe all the effect of
field factors of the partially spin polarized system are
electroninteraction,including correlationsandcollective
given in Appendix D.
modes,inprinciple,exactly. WithintheTDDFT formal-
Ingeneral,f isasymmetric4 4matrix. If,however,
ism the charge- and spin-density responses of the inter- xc ×
acting system can be expressed as:32 the z-axis is directed along the average spin, then the
groundstatetransversalspindensitiesvanish,ρ =ρ =
+
0, and the matrix f becomes block-diagonal: −
xc
χ 1(q,ω)=χ 1(q,ω) v(q) f (q,ω), (23)
− 0− xc
− −
f f 0 0
11 1z
f f 0 0
where all quantities are 4 4 matrices and χ0 denotes fxc = 01z 0zz 0 f . (24)
× +
the matrix of response functions of the noninteracting  0 0 f+ 0− 
system, v(q) is the Hartree part of the electron-electron  − 
PerformingthematrixinversioninEq.(23)weobtainthe
interactions,andf representsxccorrectionsintheform
xc tensor of response functions of the interacting system in
of local field factors. As a simplification we use only the the form
χ0 f ∆ χ0 +f ∆
nn− zz nsz 1z 0 0
χnn χnsz χns+ χns−  εLFF εLFF 
 χszn χszsz χszs+ χszs−   χ0szn+f1z∆ χ0szsz −(v(q)+f11)∆ 0 0 
 ε ε 
χ  = LFF LFF ,
≡ χs+n χs+sz χs+s+ χs+s−   χ0 
 χs−n χs−sz χs−s+ χs−s−   00 00 χ0s−0s+ 1−f+s0+−sχ−0s+s− 
   1 f χ0 
 − +− s−s+ 
 (25)
where
ε =1 v(q)+f χ0 (q,ω) f χ0 (q,ω) f χ0 (q,ω)+χ0 (q,ω) + f v(q)+f f2 ∆, (26)
LFF − 11 nn − zz szsz − 1z nsz szn zz 11 − 1z
(cid:0) (cid:1) (cid:0) (cid:1) (cid:16) (cid:0) (cid:1) (cid:17)
and A. Clean p-type GaAs
∆=χ0 χ0 χ0 χ0 =4χ0χ0. (27) Before considering the effects of magnetic impurities
nn szsz − nsz szn ↑ ↓ andassociatedchargeandspindisorderonthetransport
properties, we would like to discuss some new features
that the valence band character of the itinerant carriers
brings into the system. They stem from the complex-
III. RESULTS AND DISCUSSION
ity of the semiconductor valence band: strong spin orbit
interaction and the Γ-point degeneracy of the p-states.
We now discuss applications of our formalism for the The multiband nature of the valence band gives rise
specific case of GaMnAs DMSs. The band structure pa- to a rich single-particle excitation spectrum. In Fig. 1
rameters used in our calculations correspond to those of we show a schematic representation of the valence band
the GaAs host material: the band gap and spin-orbit structureofap-typesemiconductor. Arrowsindicatethe
splittingareE =1.519eVand∆=0.341eV,Luttinger possiblesingle-particleexcitations. Inadditiontothein-
g
parameters are γ =6.97, γ =2.25 and γ = 2.85, con- traband excitations within the heavy hole band (anal-
1 2 3
ductionbandeffective massism =0.065m ,Kanemo- ogous to the excitations within the conduction band of
e 0
mentum matrix element is E =27.86 eV and the static n-doped semiconductors), here we have intra-band exci-
p
dielectricconstantK =13. Thes(p)-dexchangeinterac- tationswithinthelightholebandaswellasinter-valence
tion constants within the conduction and valence bands bandexcitationsbetweenlightandheavyholebandsand
are N α=0.2 eV and N β = 1.2 eV, respectively. between split-off and heavy and/or light hole bands.
0 0
−
6
hh-lh
E
F
hh lh so
FIG. 1. (Color online) Schematic diagram of the possible
single-particle excitations in the valence band of a p-type
FIG. 3. (Color online) Schematic diagram of the excitation
semiconductor. Dashedlinesindicateintra-valencebandexci-
spectrum within the semiconductor valence band. Labels in-
tationswithintheheavyholeband(hh),withinthelighthole
dicate the edges of single-particle excitation regions within
band (lh) and inter-valence band excitations between heave
the heavy hole band (hh), within the light hole band (lh),
hole and light hole bands (hh-lh) and between split-off and
betweenheavyholeandlightholebands(hh-lh)andbetween
heavy hole and light hole bands (so).
the split-off band and heavy and light hole bands (so), see
Fig. 1. As a result, the plasmon mode in the valence band
liesentirelywithinthesingle-particleexcitationspectrumand
0 0
is effectively suppressed dueto Landau damping.
(a)
(b)
-0.01 -0.002
-1-3χ [eV Å]-0.02 q=0.003 Å-1 -1-3χ [eV Å]-0.004 q=0.05 Å-1 weinxathveireb-ivvtaeslcetanocrsetqrbo=anng0d.0p0sep3a˚Aikn−ae1rxotchuitenadltoino0ng.2istubedVeitnwaaelsessnopcihinaetraeevdsypwoanintshde
light hole subbands. The corresponding density excita-
tionsaresuppressedduetotheorthogonalityoftheinitial
-0.03 IImm[[χχnzzn]] -0.006 IImm[[χχnzzn]] andfinalstates,seeEq.(21). As aresult,the density re-
sponseforshortwavevectorsisalmostnonexistent. Ifwe
0 0.2 0.4 0 0.2 0.4 0.6 increase the wave vector to q = 0.05˚A−1, the intraband
energy [eV] energy [eV]
excitations within the heavy hole band become notice-
able in both density and spin responses. The longitudi-
FIG. 2. (Color online) Imaginary part of the noninteracting
density and longitudinal spin response functions in p-doped nal spin response, however, still prevails in the range of
GaAs for different wave vectors q = 0.003˚A−1 (a) and q = inter-valence band transitions.
0.05˚A−1 (b). The hole concentration is p=3.5×1020 cm−3. This leads us to conclude that the long-wavelength
spectrum of the single-particle excitations in p-doped
semiconductors is dominated by the inter-valence band
The variety of the possible single-particle excitations spin excitations. The origin of this effect is in the spin-
substantially modifies the density and spin response of orbit interaction, which mixes spin and orbital degrees
the system. Some ofthe modificationsarenotveryobvi- of freedom. Without the spin-orbit interaction, vertical
ous. AttheendoftheSec.IIBwealreadymentionedthe spin excitations wouldbe prohibiteddue to the orthogo-
significantdifference between spin and density responses nality of the orbital parts of Bloch functions.
inthelongwavelengthlimit. Letusconsiderthisinmore Anotherinterestingfeatureofp-dopedsemiconductors
detail. The spin response of the noninteracting electron is the effective suppressionofthe collective modes in the
gas coincides with the density response and can be ex- valence band. In the conventional picture of the con-
pressed through the Lindhard function. The spin-orbit duction band, collective plasmon excitations are wellde-
interactionwithinthevalencebandbreaksdownthiscor- fined in the long-wavelength side of the excitation spec-
respondence. trum. With increasing momentum, the collective mode
In Fig. 2 we plot the imaginary part of the noninter- approaches and then enters the region of single-particle
acting density and longitudinal spin response functions excitations, where it becomes rapidly suppressed due to
in p-doped GaAs for different wave vectors. For a small Landau damping.
7
0.5 0.05
lh hh
250
0.16
0.4 0.04
V] 200
me 0.12
energy [eV]00..23 00..0023me broadening [ -1-1τ [ps]xx110500 0.08[eV]
eti
lif
0.04
0.1 0.01 50
0 0
0 0 0 0.2 0.4 0.6 0.8 1
0 0.02 0.04 0.06 0.08 0.1 0.12
q [Å-1] ω [eV]
FIG.4. (Coloronline)Dispersion(dashedblack)andlifetime FIG. 5. (Color online) Total (charge and spin) carrier re-
broadening(solidred)ofthevalencebandplasmoncalculated laxation rate for Ga0.948Mn0.052As with hole concentration
for the p-doped GaAs with the hole concentration of p = p = 3×1020 cm−3. Dashed line: static screening model.
3.5×1020 cm−3. Dotted lines correspond to the onset of the Solid line: evaluation of Eq. (10) with full dynamic TDDFT
intrabandsingle-particleexcitationswithinthelightholeand treatment of electron interaction. See discussion in text.
heavy hole bands.

300
The situation is different for the valence band. In
Fig.3weplotaschematicdiagramoftheexcitationspec- 250
trum. Theexcitationregionforsingle-particletransitions
200
withintheheavyholebandisqualitativelysimilartothat -1Ω]
othfethsiengcloen-pdaurcttiicolneebxacnitda.tioInnstpheectvraulemnciseebxatnendd,ehdowdueveetro, -1m 150
c
the intraband transitions within the light hole band and σ [
100
interbandtransitionsbetweenheavyandlightholebands
and between split-off and heavy/light hole bands (red
50
and blue arrows in Fig. 1). In Fig. 3 the corresponding
regions of single-particleexcitations are shadedwith dif-
0
ferentpatterns. Itcanbeseenthatthecollectivemodein 0 0.2 0.4 0.6 0.8 1
ω [eV]
thevalencebandfallsentirelywithintheregionofsingle-
particle excitations and, therefore, becomes suppressed
FIG.6. (Coloronline)Infraredconductivityofferromagnetic
even at the long-wavelength side of the spectrum. Error Ga0.948Mn0.052Aswithholeconcentrationp=3×1020 cm−3.
barsinFig.3indicatetheplasmonresonancebroadening
Calculations are performed according to Eq. (12), and using
due to Landau damping. a relaxation rate obtained through Eq. (10) (solid line) or a
To illustrate the effect we have performed numerical fixed τ−1 = 230ps−1 (dashed line). Symbols are the experi-
calculations of the plasmon dispersion and the lifetime mental data of Ref. 34.
broadening of the collective excitations in the valence
band of the p-doped GaAs. The plasmon frequencies
40 meV range. An additional sharp rise in the damping
weredetermined asthe zerosofthe realpartofthe RPA
takesplacewhenthe collectivemodeentersthe regionof
dielectric function and the lifetime broadening is asso-
heavy hole intraband excitations.
ciated with the imaginary part of the frequency poles.
In Fig. 4 the black and red lines correspond to the dis-
persion and the lifetime of the plasmon excitations, re-
spectively. The dotted lines indicate the regions of the B. Magnetically doped GaMnAs
intrabandsingle-particleexcitationswithinthelighthole
and heavy hole bands, compare with Fig. 3. At small The introduction of magnetic impurities in GaAs has
wavevectorsthe plasmon mode falls within the region of two consequences. First, charge and spin disorder are
inter-valence bandsingle-particleexcitations resulting in brough into the system and, second, the mean field part
alifetimebroadeningofthecollectiveresonanceofabout of the p-d exchange interaction between localized spins
5 meV.Once the plasmondispersionentersthe regionof and itinerant holes causes modifications of the valence
single-particleexcitationswithinthe lightholeband,the band structure once the system enters the magnetically
life-time broadening substantially increases into the 30- ordered phase.
8
0.1 k[]-1Åz0.2 1,000
0.1
0 0 800 70 K
65 K
-0.1 45 K
-0.1 -0.2 -1Ω] 600 5 K
-0.2-0.1 0 0.1 0.2 -1m
eV]-0.2 kx[Å-1] σ [c 400
y [
g
er-0.3 EF 200
n
e
-0.4 0
0 0.2 0.4 0.6 0.8
ω [eV]
-0.5
FIG. 8. (Color online) Temperature dependence of infrared
-0.6 conductivityofGa0.948Mn0.052Aswithholeconcentrationp=
3×1020 cm−3 and Tc =70 K calculated with weak disorder,
-0.2 0 0.2 0.4 lifetime broadening of Γ=5 meV.
-kx [Å-1] kz [Å-1]
FIG. 7. (Color online) Band structure of ferromagnetic
Ga0.95Mn0.05Aswithholeconcentrationp=3.5×1020 cm−3. were performed according to Eq. (12). Solid line corre-
Alignment of localized spins results in strongly anisotropic sponds to a relaxation rate obtained through Eq. (10),
valence band spin splitting. Inset shows a cut of the Fermi dashed line describes calculations with the fixed τ−1 =
surface bythe plane ky =0. 230ps−1. The theory shows qualitative agreement with
the experiment. The insensitivity of the calculations to
the frequency dependence of relaxation rate (minor dif-
Let us consider the effect of disorder first. In calcu- ference between solid and dashed lines in Fig. 6) sug-
lating carrier relaxation rates, most theoretical models geststhateffectsofthebandstructureplaythedominant
for GaMnAs use a static screening approach, where all role in determining the shape of the infrared conductiv-
many-body effects are reduced to the static screening of ity and overshadow the strong frequency dependence of
theCoulombdisorderpotential. Withinourmodel,how- τ obtained within our model and presented in Fig. 5.
ever,themomentumandfrequencydependentrelaxation An alternative possible experimental probe that could
rate(10)isexpressedthroughthesetofdensityandspin- reveal the details of the frequency and momentum de-
density response functions that allows us to use the full pendence of the carrier relaxation rate in more explicit
dynamic treatment of electron-electron interaction, thus ways are measurements of the position and lineshape of
accountingforthevarietyofmany-bodyeffectsincluding the plasmon resonance itself. It was shown in Ref. 25
correlations and collective modes. that these quantities are sensitive to the carrier relax-
In Fig. 5 we plot the frequency dependence of the ation time, with both real and imaginary part of τ and
total (charge and spin) relaxation rate calculated for its dynamic nature being essential. Our approach seems
Ga Mn As within the static screening model and to fit well to describe such experiments.
0.948 0.052
using the full dynamic treatment of electron-electronin- As was mentioned before, the magnetic impurities
teraction according to Eq. (10). The difference between bringlocalizedspinsintothesystem,whichinteractwith
the two curves in the static limit is due to the xc partof the itinerant carriers through the p-d exchange interac-
the electron-electroninteractionthat affects both charge tion. The fluctuating part of this interaction constitutes
and spin scattering. The most striking difference, how- the spindisorder. Themeanfieldpartofexchangeinter-
ever, is the pronounced feature appearing between 0.2 action,whichweabsorbintothecleansystembandstruc-
eV and 0.5 eV associated with the collective modes. Al- tureHamiltonianHˆ , isresponsibleforthe spinsplitting
e
thoughwehaveseenabovethatthecollectiveexcitations of the valence bands once the system enters the magnet-
are significantly damped in the valence band, they still ically ordered state. Due to the spin-orbit interaction
play an important role in the transportproperties of the within the valence band, this spin splitting strongly de-
system providing an effective channel for momentum re- pends both on the magnitude and direction of the wave
laxation. Their contributions give up to 50% increase vector k.
to the total carrier relaxation rate. Note that, due to In Fig. 7 we plot the band structure of ferromagnetic
their longitudinal character, the plasmon modes do not Ga Mn As. Strong anisotropy of the valence band
0.95 0.05
directly affect the optical response and enter only indi- spin-splitting is seen between directions along and per-
rectly through the tensor of frequency and momentum pendicular to the magnetization of localized spins (z-
dependent relaxation rates (10). direction). The inset shows a cut of the Fermi surface
In Fig. 6 we compare our calculations of the infrared by the plane k =0. One caneasily see the distortionof
y
conductivityofferromagneticGa Mn Aswiththe the Fermi surface from the spherical shape of the para-
0.948 0.052
experimental data of Singley et al.34 The calculations magnetic system (for clarity we have neglected here the
9
“spin-flip” transitions gain the intensities at the account
250 of “spin-conserving” heavy hole-light hole transitions.
-1-1cm Ω]112050000 52 5K K IdnuTcFthiigve.itr9yeawolfeGGcaoaMm0.n9p4Aa8rsMesneax0m.p0p5e2rleAimssaefnrretoammlduRacehtfa.mo3on4rwienidftrihasorceraddlcecuroelnad--.
σ [ 50 4750 KK tions using our model of Eqs. (12) and (10). The large
disorder induced life-time broadening blankets most of
0
0 0.2 0.4 0.6 0.8 1 thefeaturesdiscussedabove. Thesuppressionofthehigh
ω [ eV]
250 energyshoulderofsplitofftolightholetransitionsinthe
-1Ω]200 ferromagneticstate is seen, however,both onthe experi-
-1m 150 5 K mental and theoretical plots. Overall,for energies above
c100 25 K the main peak position around 0.2 eV, the calculations
σ [ 50 4750 KK are in good agreement with the experimental results.
0 Note also that unlike in Ref. 14, our calculations do
0 0.2 0.4 0.6 0.8 1
ω [eV] notrequireincorporationofanimpuritybandwithinthe
energy gap to avoid a drop in conductivity around 0.8-1
eV. At energies below the main peak position the agree-
FIG. 9. (Color online) Temperature dependence of the in-
ment with the experiment is worse. We should mention,
frared conductivityofGa0.948Mn0.052Aswithholeconcentra-
tion p=3×1020 cm−3 and Tc =70 K. Upperpanel: experi- however, that this is the region of ωτ ≤ 1 where our
mental data of Ref. 34. Lower panel: results from Eq. (12). calculations are less reliable due to approximate nature
of the expression (12). The self-consistent evaluation of
Eq.(8)shouldbeusedthereinstead. Oncethefrequency
goes to zero, the static conductivity should more appro-
valence band warping, but it is included in our calcula-
priately be calculated using an expression derived from
tions). The modification of the Fermi surface together
the semiclassical Boltzmann equation.13 We have inves-
withthesuppressionoflocalizedspinfluctuationsarere-
tigatedthisregimebefore22 todescribethedropinstatic
sponsible for the significant drop in static resistivity of
resistivity in the ferromagnetic phase.
GaMnAsduringthetransitionfromparamagnetictofer-
romagnetic state. This effect was considered before.22,35
Here we pointoutthat the modificationofthe valence
band structure during the transition from paramagnetic IV. CONCLUSIONS
to ferromagnetic state also modifies energies and oscil-
lator strengths of intervalence band optical transitions We have developed a comprehensive theory of trans-
affecting thus the infrared conductivity as well. To bet- port in spin and charge disordered media. The theory is
ter show the underlying physics of temperature induced basedontheequationofmotionoftheparamagneticcur-
changes, we plot in Fig. 8 the infrared conductivity for rent response function of the disordered system, treats
the sample parameters of Ref. 34, but with a small life- disorder and many-body effects on equal footings, and
timebroadeningofΓ=5meV.Intheparamagneticstate combines a k p based description of the semiconductor
·
(solidline)threefeaturescanbeidentified: astrongpeak valence band structure with a full dynamic treatment of
around0.2eVcorrespondingtotheheavyhole-lighthole electron-electron interaction by means of TDDFT. We
transitions,asmallerpeakwithabroadshoulderaround have applied our theory to the specific case of GaMnAs.
0.4 eV associated with the split off to light hole transi- We have shown that the multiband nature and spin-
tions and a wide background of split off to heavy hole orbit interaction within the valence band bring new ef-
transitions. fects for p-doped GaAs as compared to the conventional
With the temperature going below T = 70 K, two n-type systems. The density and spin-density responses
c
main phenomena occur. The first is the suppression of ofnoninteractingcarrierswithinthevalencebandarenot
the high energy shoulder of the split off to light hole the same anymore. Moreover, the long wavelength side
transitions. The second is the appearance of the tran- of the single-particle excitation spectrum is now com-
sitions between the spin-split heavy hole and light hole pletely dominated by the inter-valence band spin excita-
bands and the redistribution of the oscillator strength tions. Due to the extended region of single-particle ex-
amongthem. Thelowestenergypeakscorrespondtothe citationswithin the valence band,the collectiveplasmon
transitions between spin-split bands. Calculations were mode entirely falls within the region of these excitations
performed for light linearly polarized in the plane per- and,therefore,iseffectively dampedforallwavevectors.
pendicular to the magnetization. Due to the spin-orbit Forthemagneticallydopedsystemthemean-fieldpart
interaction within the valence band, the transitions be- of the p-d exchange interaction between itinerant holes
tween the spin-split states are optically allowed. The and localized spins substantially modifies the semicon-
additional peak at higher energy corresponds to heavy ductor band structure once the system enters a magnet-
hole-light hole “spin-flip” transitions. As the temper- ically ordered phase. This modification substantially af-
ature goes down, the spin splitting increases and the fects energies and oscillatorstrengths of the intervalence
10
band optical transitions. Our calculations are in good (A2) can be represented as
agreement with experimental data for the temperature
dependence of the infrared conductivity in GaMnAs. (H0+H1+H2)A=εA, (A3)
A full dynamical treatment of electron-electron inter-
actions is essentialto capture the influence of the collec- where A is the vector of coefficients An(k), H0 is the di-
tive excitations on the carrier relaxation rate. Our cal- agonal part of Hamiltonian, and H1 and H2 correspond
culations show that, by providing an effective channel of to the block-diagonaland off-block-diagonalparts of the
momentum relaxation, the collective excitations within k π matrix with respect to the included and remote
·
the valence band significantly (up to 50%) increase the bands. Next, we apply the canonical transformation
transport relaxation rate.
However, it turns out that the actual infrared absorp- A=eSB=eS1+S2B, (A4)
tionspectraarenotverysensitivetothedetailsofthefre-
quencydependenceoftherelaxationrate,butaremostly withS1 andS2 beingantihermitianoperatorsoffirstand
determinedby the featuresofthe bandstructure. Direct second order in the perturbation, respectively. The ma-
measurements of the position and lineshape of the plas- trix equation (A3) then has the form
mon resonance itself are likely to be more sensitive to
the details of the frequency and momentum dependence e−S1−S2(H0+H1+H2)eS1+S2 B=H¯B=εB.
(A5)
of the carrier relaxation rate.
(cid:8) (cid:9)
By choosing
The theory presented here, treating disorder and
many-bodyeffectsonequalfootings,providesaverygen-
H +[H ,S ]=0, [H ,S ]+[H ,S ]=0, (A6)
eral framework for describing electron dynamics in ma- 2 0 1 0 2 1 1
terials. It can, in principle, be made self-consistent and
where[...]denotesthecommutator,wewriteuptoterms
thus be applied beyond the weak-disorder limit; it can
of second order in the perturbations H and H
accommodatemanydifferenttypesofdisorder,aswellas 1 2
band structure models. This should make it well suited
1
forfurtherexplorationoftheopticalandtransportprop- H¯ H0+H1+ [H2,S1]. (A7)
≈ 2
erties of DMSs and other systems of practical interest.
The matrix elements between the Luttinger-Kohn peri-
odic amplitudes u n are
n,0
ACKNOWLEDGMENTS | i≡| i
¯h2k2
nH n = ε + δ , (A8)
This work was supported by DOE under Grant No. h | 0| ′i n,0 2m n,n′
(cid:18) 0(cid:19)
DE-FG02-05ER46213. ¯hk πα
α s,s′
sH1 s′ = , (A9)
h | | i m
0
α
Appendix A: Generalized k·p approach X¯hk πα
sH r = α s,r, (A10)
2
h | | i m
0
The derivation of the generalized k p perturbation Xα
· sH r
approach presented here is based on Ref. 23. First, the 2
sS r = h | | i
1
electronic wave function is expanded in the Luttinger- h | | i − sH0 s rH0 r
h | | i−h | | i
Kohn basis31 ¯hkαπsα,r 1
= . (A11)
Ψ= An(k)χn,k = 1 An(k)eikr un,0 , (A1) Xα m0 εr,0−εs,0
√V | i
n,k n,k For the last term in (A7) we can then write
X X
where u areperiodic partsofBlochfunctions atk=
0, and|Ann,(0ki) are the expansioncoefficients. This results hs|[H2,S1]|s′i= hs|H2|rihr|S1|s′i
inthefollowingmatrixformoftheSchr¨odingerequation: Xr n
sS r rH s (A12)
1 2 ′
¯h2k2 ¯h −h | | ih | | i
Xn,kAn(k)(cid:20)(cid:18)εn,0+ 2m0 −ε(cid:19)δn′,n+ m0k·πn′,n(cid:21)=0, = ¯h2mkα2kβ επsα,rπorβε,s′ + επsβ,rπrαε,s′ .
(A2) α,β 0 s′,0− r,0 s,0− r,0!
where ε are the band edge energies at k=0. Xr
n,0
The last term in Eq. (A2) mixes states with different HereweusedthefactthattheH andS operatorshave
2 1
n for k=0. Now we separatethe whole setof the bands only off-block-diagonal matrix elements between the s
6
n into those whose contributionwe aregoing to calcu- andrbands. Eqs.(A8)-(A12)definethematrixoftheef-
{ }
late exactly s , and the remote bands r that we will fectiveHamiltonian(A7). Nonvanishingmatrixelements
{ } { }
treat up to the second order in momentum. Equation are determined by the symmetry of the crystal.