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Data for the Publication "Two-carrier model-fitting of Hall effect in semiconductors with dual-band occupation: A case study in GaN two-dimensional hole gas"

Author: Dill, Joseph; Chang, Chuan; Jena, Debdeep; Xing, Huili
Publisher: Zenodo
DOI: 10.5281/zenodo.17673936
Source: https://zenodo.org/records/17673936/files/2carrier-equations.pdf
De i a ion o Two-Ca ie Hall effec
Equa ions
Joseph E. Dill, Chuan F. C. Chang, Debdeep Jena, Huili G ace Xing
Co nell Uni e si y
In oduc ion
This Ma hema ica no ebook p esen s a de i a ion o Eqs. 3-13 in he manusc ip “Two-ca ie model-
i ing o Hall effec in semiconduc o s wi h dual-band occupa ion: A case s udy in GaN wo-dimen-
sional hole gas” (doi: 10.1063/5.0248998 )
The inal esul s ha a e p in ed in he manusc ip a e ma ked wi h Blue ex .
De i a ion
In he p esence o m D ude-like pa allel-conduc ing channels, he XX and XY conduc i i ies a e gi en by
[Eq. (3a), (3b)]
σxx(B) = ∑i
mqiniμi
1+(μiB)2
σxy(B) = ∑i
mσxx,i(μiB)
He e, we conside he case o wo conduc ing channels
In[1]:= σxx1 =q1 n1 μ1
1+ (μ1 B)2;
σxx2 =q2 n2 μ2
1+ (μ2 B)2;
σxx =σxx1 +σxx2;
σxy1 = (μ1 B)σxx1;
σxy2 = (μ2 B)σxx2;
σxy =σxy1 +σxy2;
He e, and in he ex , we adop he con en ion ha q and
μ
always ca y he same sign, while
n
is
always posi i ely signed. This means ha σxx is always posi i e, while σxy,i will ake he same sign as qi
(by way o
μ
i
).
P in ed by Wol am Ma hema ica S uden Edi ion
F om he conduc i i y enso equa ion, [Eq. (4)]
J=σxx σxy
-σxy σyy
E⟺E=ρxx ρxy
-ρxy ρyy
J,
The XX ans XY esis i i ies a e gi en by
Eq. (5a)
In[7]:= ρxx =σxx
σxx2+σxy2// FullSimpli y
Ou [7]=
n2 q2 1+B2μ12μ2+n1 q1 μ11+B2μ22
n12q12μ12+2 n1 n2 q1 q2 μ1μ2+n22q22+B2(n1 q1 +n2 q2)2μ12μ22
Eq. (5b)
In[8]:= ρxy =σxy
σxx2+σxy2// FullSimpli y
Ou [8]=
B n1 q1 μ12+Bn2 q2 +B2(n1 q1 +n2 q2)μ12μ22
n12q12μ12+2 n1 n2 q1 q2 μ1μ2+n22q22+B2(n1 q1 +n2 q2)2μ12μ22
To exp ess ρxx and ρxy in e ms o σ1, σ2, μ1, and μ2. We will de ine a cas ing se ha eplaces ni wi h
σi
qiμi
.
In[9]:= Cas σ=n1 σ1
q1 μ1
, n2 σ2
q2 μ2;
Eq. (6a)
In[10]:= ρxx /. Cas σ// FullSimpli y
Ou [10]= σ1+B2μ22σ1+σ2+B2μ12σ2
1+B2μ22σ12+1+B2μ12σ22+2σ1σ2+B2μ1μ2σ2
Eq. (6b)
In[11]:= ρxy /. Cas σ// FullSimpli y
Ou [11]=
Bμ1σ1+B2μ22σ1+μ2σ2+B2μ12μ2σ2
1+B2μ22σ12+1+B2μ12σ22+2σ1σ2+B2μ1μ2σ2
I is also con enien o exp ess ρxx(B) and ρxy(B) in hei se ies-expanded o ms, as seen below:
Eq. (7a)
2 2ca ie -equa ions.nb
P in ed by Wol am Ma hema ica S uden Edi ion
In[12]:= Se ies[ρxx, {B, 0, 5}] // FullSimpli y
Ou [12]=
1
n1 q1 μ1+n2 q2 μ2
+n1 n2 q1 q2 μ1(μ1-μ2)2μ2 B2
(n1 q1 μ1+n2 q2 μ2)3
-
n1 n2 q1 q2 (n1 q1 +n2 q2)2μ13(μ1-μ2)2μ23B4
(n1 q1 μ1+n2 q2 μ2)5
+O[B]6
Eq. (7b)
In[13]:= Se ies[ρxy, {B, 0, 6}] // FullSimpli y
Ou [13]= n1 q1 μ12+n2 q2 μ22B
(n1 q1 μ1+n2 q2 μ2)2
-n1 n2 q1 q2 (n1 q1 +n2 q2)μ12(μ1-μ2)2μ22B3
(n1 q1 μ1+n2 q2 μ2)4
+
n1 n2 q1 q2 (n1 q1 +n2 q2)3μ14(μ1-μ2)2μ24B5
(n1 q1 μ1+n2 q2 μ2)6
+O[B]7
ρxx has s ic ly e en B-dependence, while ρxy is s ic ly odd, bo h wi h al e na ing sign. We deno e he
j
-
h se ies coefficien by cj as ollows:
ρxx(B)≈c0+c2B2-c4B4+c6B6- …
ρxy(B)≈c1B-c3B3-c5B5+ …
We can calcula e he alue o each se ies coefficien in e ms o q1, q
2
, n1, n2, μ1, μ
2
as ollows:
In[14]:= C0 =Rsh =Se iesCoe icien [ρxx, {B, 0, 0}] // FullSimpli y
Ou [14]=
1
n1 q1 μ1+n2 q2 μ2
In[15]:= C1 =RH =Se iesCoe icien [ρxy, {B, 0, 1}] // FullSimpli y
Ou [15]=
n1 q1 μ12+n2 q2 μ22
(n1 q1 μ1+n2 q2 μ2)2
In[16]:= C2 =Se iesCoe icien [ρxx, {B, 0, 2}] // FullSimpli y
Ou [16]=
n1 n2 q1 q2 μ1(μ1-μ2)2μ2
(n1 q1 μ1+n2 q2 μ2)3
In[17]:= C3 = -Se iesCoe icien [ρxy, {B, 0, 3}] // FullSimpli y
Ou [17]=
n1 n2 q1 q2 (n1 q1 +n2 q2)μ12(μ1-μ2)2μ22
(n1 q1 μ1+n2 q2 μ2)4
In[18]:= C4 = -Se iesCoe icien [ρxx, {B, 0, 4}] // FullSimpli y
Ou [18]=
n1 n2 q1 q2 (n1 q1 +n2 q2)2μ13(μ1-μ2)2μ23
(n1 q1 μ1+n2 q2 μ2)5
2ca ie -equa ions.nb 3
P in ed by Wol am Ma hema ica S uden Edi ion
In[19]:= C5 =Se iesCoe icien [ρxy, {B, 0, 5}] // FullSimpli y
Ou [19]=
n1 n2 q1 q2 (n1 q1 +n2 q2)3μ14(μ1-μ2)2μ24
(n1 q1 μ1+n2 q2 μ2)6
We can de ine a cas ing se o exp ess ρxx and ρxy in e ms o c0, c1, c2, c3.
In[20]:= Cas Cj =Sol e[{c0 C0, c1 C1, c2 C2, c3 C3},{n1, n2, μ1, μ2}]〚1〛// FullSimpli y
Ou [20]= n1 2 c23-c1 c2 c3 +c3 c0 c3 -c22c12+4 c0 c2-2 c0 c1 c2 c3 +c02c32
2c22-c1 c3c22c12+4 c0 c2-2 c0 c1 c2 c3 +c02c32q1
,
n2  -2 c23+c1 c2 c3 -c3 c0 c3 +c22c12+4 c0 c2-2 c0 c1 c2 c3 +c02c32
2c22-c1 c3c22c12+4 c0 c2-2 c0 c1 c2 c3 +c02c32q2,
μ1c1 c2 +c0 c3 +c22c12+4 c0 c2-2 c0 c1 c2 c3 +c02c32
2 c0 c2
,
μ2-
-c1 c2 -c0 c3 + (c1 c2 +c0 c3)2+4 c0 c2 c22-c1 c3
2 c0 c2 
Doing so, we ob ain he ollowing simpli ica ion o ρxx and ρxy, which is e y con enien o i ing
measu ed da a
Eq. (9a)
In[21]:= ρxx /. Cas Cj // FullSimpli y
Ou [21]= c0 +B2c23
c22+B2c32
Eq. (9b)
In[22]:= ρxy /. Cas Cj // FullSimpli y
Ou [22]= B c1 -B3c22c3
c22+B2c32
The se ies coefficien s ollow he ecu si e condi ion Eq. (8)
cj≥2=c3j-2
c2j-3
exempli ied below wi h c4 and c
5
In[23]:= C4 C34-2
C24-3
Ou [23]= T ue
In[24]:= C5 C35-2
C25-3
Ou [24]= T ue
4 2ca ie -equa ions.nb
P in ed by Wol am Ma hema ica S uden Edi ion
So, we can de ine a gene ic cj a iable as ollows:
In[25]:= Cj =C3j-2
C2j-3
Ou [25]=
n1 n2 q1 q2 (n1 q1 +n2 q2)μ12(μ1-μ2)2μ22
(n1 q1 μ1+n2 q2 μ2)4
-2+jn1 n2 q1 q2 μ1(μ1-μ2)2μ2
(n1 q1 μ1+n2 q2 μ2)3
3-j
We’ll con i m ha i wo ks o a ious j≥2.
In[26]:= (Cj /. j 2)C2
Ou [26]= T ue
In[27]:= (Cj /. j 3)C3
Ou [27]= T ue
In[28]:= (Cj /. j 4)C4
Ou [28]= T ue
In[29]:= (Cj /. j 5)C5
Ou [29]= T ue
In he manusc ip , we gi e an equa ion o cj≥2 in e ms o β=n2
n1+n2
and γ=μ2
μ1
, de i ed below.
In[30]:= Cas βγ =Sol e
 sh Rsh,
h RH,
β  n2
n1 +n2
,
γ  μ2
μ1
,{n1, n2, μ1, μ2}〚1〛// FullSimpli y
Ou [30]= n1 (-1+β)q1 (-1+β) - q2 βγ2
h (q1 -q1 β+q2 βγ)2, n2 βq1 -q1 β+q2 βγ2
h (q1 -q1 β+q2 βγ)2,
μ1 h (q1 -q1 β+q2 βγ)
sh q1 -q1 β+q2 βγ2,μ2 h γ(q1 -q1 β+q2 βγ)
sh q1 -q1 β+q2 βγ2
Case: q1 = q
2
In[31]:= Cj /. Cas βγ /.q1q2 // Expand // FullSimpli y
Ou [31]= -
sh (-1+β)β(-1+γ)2- h3(-1+β)β(-1+γ)2γ2
sh21+β-1+γ23j- h2(-1+β)β(-1+γ)2γ
sh 1+β-1+γ22-j
γ
2ca ie -equa ions.nb 5
P in ed by Wol am Ma hema ica S uden Edi ion

In[32]:= -
sh (-1+β)β(-1+γ)2
- h3(-1+β)β(-1+γ)2γ2
sh21+β-1+γ23
- h2(-1+β)β(-1+γ)2γ
sh 1+β-1+γ22
j
γ
Ou [32]= -
sh (-1+β)β(-1+γ)2 h γ
sh 1+β-1+γ2j
γ
Eq. (8) [q1=q2case]
In[33]:=
sh (1-β)β(-1+γ)2 hjγj
γ shj1+β-1+γ2j
Ou [33]= hj sh1-j(1-β)β(-1+γ)2γ-1+j1+β-1+γ2-j
Case: q1 = -q2
In[34]:= Cj /. Cas βγ /.q1-q2 // Expand // FullSimpli y
Ou [34]=
sh (-1+β)β(-1+γ)2 h3(-1+β)β(-1+2β) (-1+γ)2γ2
sh2-1+β+βγ23j h2(-1+β)β(-1+γ)2γ
sh -1+β+βγ22-j
(1-2β)2γ
In[35]:=
sh (-1+β)β(-1+γ)2
h3(-1+β)β(-1+2β) (-1+γ)2γ2
sh2-1+β+βγ23
h2(-1+β)β(-1+γ)2γ
sh -1+β+βγ22
j
(1-2β)2γ
Ou [35]=
sh (-1+β)β(-1+γ)2 h (-1+2β)γ
sh -1+β+βγ2j
(1-2β)2γ
In[36]:=
sh (-1+β)β(-1+γ)2 hjγj(2β-1)j
(1-2β)2γ shj-1+β+βγ2j
Ou [36]=
hj sh1-j(-1+β)β(-1+2β)j(-1+γ)2γ-1+j-1+β+βγ2-j
(1-2β)2
Eq. (8) [q1= -q2case]
In[37]:=
hj
shj-1(-1)j-1(1-β)β(-1+γ)2γj-1(1-2β)j-2
βγ2+1-1j
Ou [37]= (-1)-1+j hj sh1-j(1-2β)-2+j(1-β)β(-1+γ)2γ-1+j-1+β1+γ2-j
No e ha in he case o q1= -q2, he sign o all cj∈odd e ms is se by whe he ca ie 1 o ca ie 2 has
highe conduc i i y.
6 2ca ie -equa ions.nb
P in ed by Wol am Ma hema ica S uden Edi ion
In[38]:= nAP =1
q RH
// FullSimpli y
Ou [38]=
(n1 q1 μ1+n2 q2 μ2)2
qn1 q1 μ12+n2 q2 μ22
In[39]:= μAP =RH
Rsh
// FullSimpli y
Ou [39]=
n1 q1 μ12+n2 q2 μ22
n1 q1 μ1+n2 q2 μ2
In[40]:= nAP /.{q2 -q1, q q1} /.{μ2 γμ1} /.{n2  βn1} // FullSimpli y
Ou [40]= -n1 (-1+βγ)2
-1+βγ2
Eq. (11)
We de ine he ollowing subs i u ion o use in Eq. 10:
c*=c22c12+4 c0 c2-2 c0 c1 c2 c3 +c02c32
In[41]:= cs =Sq c22c12+4 c0 c2-2 c0 c1 c2 c3 +c02c32;
Eq. (10)
In[42]:= Cas nμToCj =n1 q1
q2
2 c23-c1 c2 c3 +c3 (c0 c3 +Sign[q2]cs)
2 q c1 c3 -c22cs
,
n2 -2 c23-c1 c2 c3 +c3 (c0 c3 -Sign[q2]cs)
2 q c1 c3 -c22cs
,
μ1c1 c2 +c0 c3 -Sign[q2]cs
2 c0 c2
,
μ2c1 c2 +c0 c3 +Sign[q2]cs
2 c0 c2 ;
We will alida e Eqs. 10a-c by subs i u ing in alues o n1, n2, μ1, and μ
2
and con i ming ha we ge he
same ou pu
In[43]:= ({n1, n2, μ1, μ2} /. Cas nμToCj) /.{c0 C0, c1 C1, c2 C2, c3 C3} /.
{n1 4, n2 1, μ1200, μ2800, q1 +1, q2 +1} /. q 1
Ou [43]= {4, 1, 200, 800}
In[44]:= ({n1, n2, μ1, μ2} /. Cas nμToCj) /.{c0 C0, c1 C1, c2 C2, c3 C3} /.
{n1 4, n2 1, μ1-200, μ2800, q1 -1, q2 +1} /. q 1
Ou [44]= {4, 1, -200, 800}
2ca ie -equa ions.nb 7
P in ed by Wol am Ma hema ica S uden Edi ion
In[45]:= ({n1, n2, μ1, μ2} /. Cas nμToCj) /.{c0 C0, c1 C1, c2 C2, c3 C3} /.
{n1 4, n2 1, μ1+200, μ2-800, q1 +q, q2 -q} /. q 1
Ou [45]= {4, 1, 200, -800}
In[46]:= ({n1, n2, μ1, μ2} /. Cas nμToCj) /.{c0 C0, c1 C1, c2 C2, c3 C3} /.
{n1 4, n2 1, μ1-200, μ2-800, q1 -q, q2 -q} /. q 1
Ou [46]= {4, 1, -200, -800}
Fo equa ions 12 and 13, we will de ine a new cas ing se wi h β, γ, n, and μ as he ee pa ame e s.
These equa ions a e explici ly only in he case o q1=q2.
In[47]:= Cas βγnμ=Sol e
μ  n1 μ1+n2 μ2
n1 +n2
,
nn1 +n2,
β  n2
n1 +n2
,
γ  μ2
μ1
,{n1, n2, μ1, μ2}〚1〛// FullSimpli y
Ou [47]= n1 n-nβ, n2 nβ,μ1μ
1+β(-1+γ),μ2γμ
1+β(-1+γ)
Eq. (12)
In[48]:= nAppa en =1
q2 C1
/. Cas βγnμ/.{q1 q2} // FullSimpli y
%n(1+β(γ-1))2
1+βγ2-1// FullSimpli y
Ou [48]=
n(1+β(-1+γ))2
1+β-1+γ2
Ou [49]= T ue
Eq. (13)
In[50]:= μAppa en =RH
Rsh
/. Cas βγnμ/.{q1 q2} // FullSimpli y
% μ 1+βγ2-1
(1+β(γ-1))2// FullSimpli y
Ou [50]= μ+β-1+γ2μ
(1+β(-1+γ))2
Ou [51]= T ue
8 2ca ie -equa ions.nb
P in ed by Wol am Ma hema ica S uden Edi ion