(KS=0.12 for the MCMC and KS=0.16 for the MC DNN).

The disparity between the observed systemic redshift and the predictions given
by zELDA might come from several facts. One scenario could be that the
systemic redshift given by the $\mathrm{Ly}\alpha$ independent redshift
estimator is not properly constrained. However, usually, these estimators give
a good redshift accuracy. Other possibility is that the thin shell model might
be incomplete and not able to produce these kind of line with large shifts
between the $\mathrm{Ly}\alpha$ wavelength and the bulk of the line profile.
Although, we find that the outflow model reproduces quite well the shape of
the line (KS$\sim 0.06$). Another possible explanation is that the
$\mathrm{Ly}\alpha$ radiation might come from a different galaxy component
than the radiation used for measuring the systemic redshift.

Overall, the reason behind the tension between the systemic redshift given by
the $\mathrm{Ly}\alpha$ independent redshift estimator and zELDA remains
unknown. Therefore, we decided to exclude this particular line profile from
the analysis performed in §6.

##  Appendix E Best fitting parameters of the observed line profiles.

In Tab.3 and 4 we list the systemic redshift obtained by studying different
features than the $\mathrm{Ly}\alpha$ line $z^{\rm Sys}$, the name of the
sources, $L_{\rm Ly\alpha}$ and $EW_{\rm Ly\alpha}$. These last two were also
obtained from the Lyman alpha Spectral Database (LASD Runnholm et al., 2021).
In addition, we display the name and the ’Label’ of each source. This ’Label’
matches with the numbers shown in the bottom right of Fig. 17 and in the top
left of Fig. 14 and 15.

In Tab.5, 6 and 7 we list all the best fitting parameters for the 97 observed
line profiles analyzed in §6 for both, the MCMC and the MC DNN approaches. The
model parameters are the systemic redshift $z^{\rm Zelda}$, the outflow
expansion velocity $V_{\rm exp}$, the neural hydrogen column density $N_{\rm
H}$, the dust optical depth $\tau_{a}$, the rest-frame injected equivalent
width of the $\mathrm{Ly}\alpha$ line $EW_{\rm in}$ and its rest frame
intrinsic width $W_{\rm in}$.

Table 3:  Name, systemic redshift, $\mathrm{Ly}\alpha$ luminosity and observed
equivalent width for the observed galaxies. The column Label indicates the
number that appears with the spectra in Fig.13, 14 and 15.

Label | Name | $z^{\rm Sys}$ |  $\rm\log$Ly$\alpha$ |  $\rm\log$$EW_{\rm Ly\alpha}$
---|---|---|---|---
|  |  | [$\rm erg/s$] | [Å]
1 | SDSSJ0213+1259 | 0.219 | 41.7 | 0.84
2 | SDSSJ1025+3622 | 0.1265 | 42.31 | 1.39
3 | SDSSJ0150+1308 | 0.1467 | 41.37 | 0.59
4 | SDSSJ0055-0021 | 0.1674 | 41.45 | 0.67
5 | SDSSJ1112+5503 | 0.1316 | 41.88 | 1.02
6 | SDSSJ1144+4012 | 0.127 | 40.71 | 0.32
7 | SDSSJ1414+0540 | 0.0819 | 40.5 | 0.5
8 | SDSSJ0808+3948 | 0.0912 | 42.01 | 1.17
9 | SDSSJ1429+0643 | 0.1735 | 42.73 | 1.56
10 | SDSSJ1416+1223 | 0.1232 | 41.2 | 0.24
11 | SDSSJ1521+0759 | 0.0943 | 41.63 | 0.73
12 | SDSSJ1428+1653 | 0.1817 | 42.46 | 1.29
13 | SDSSJ1612+0817 | 0.1491 | 42.3 | 1.26
14 | SDSSJ0926+4427 | 0.1807 | 42.78 | 1.57
15 | GP0303–0759 | 0.1649 | 41.94 | 0.95
16 | J0159+0751 | 0.0611 | 41.59 | 2.17
17 | SDSSJ0938+5428 | 0.1021 | 41.35 | 0.55
18 | SDSSJ0021+0052 | 0.0984 | 42.59 | 1.41
19 | SDSSJ2103-0728 | 0.1369 | 42.21 | 1.25
20 | HARO11 | 0.0206 | 41.26 | 0.93
21 | GP0911+1831 | 0.2622 | 42.8 | 1.68
22 | GP1219+1526 | 0.1956 | 43.19 | 2.12
23 | GP1133+6514 | 0.2414 | 42.58 | 1.51
24 | GP1054+5238 | 0.2526 | 42.54 | 1.15
25 | GP1137+3524 | 0.1944 | 42.62 | 1.53
26 | GP1244+0216 | 0.2394 | 42.56 | 1.69
27 | GP1249+1234 | 0.2634 | 43.07 | 1.96
28 | GP1424+4217 | 0.1848 | 42.93 | 1.88
29 | J0925+1403 | 0.3012 | 42.83 | 1.82
30 | J0820+5431 | 0.0386 | 40.47 | 1.81
31 | J1205+4551 | 0.0654 | 41.65 | 2.41
32 | J1355+4651 | 0.0278 | 40.95 | 2.15
33 | J1242+4851 | 0.0623 | 41.44 | 2.28
34 | J1152+3400 | 0.3419 | 43.0 | 1.82
35 | J1503+3644 | 0.3557 | 42.86 | 1.75
36 | J1333+6246 | 0.3181 | 42.75 | 1.88
37 | J1442-0209 | 0.2937 | 43.16 | 1.94
38 | J0901+2119 | 0.2993 | 42.48 | 2.16
39 | J1154+2443 | 0.3689 | 42.87 | 2.01
40 | J1011+1947 | 0.3322 | 42.64 | 2.17
41 | J1243+4646 | 0.4317 | 43.08 | 1.89
42 | J1256+4509 | 0.353 | 42.54 | 2.01
43 | J1248+4259 | 0.3629 | 42.8 | 2.41
44 | SDSSJ1457+2232 | 0.1486 | 41.24 | 1.04
45 | SDSSJ0815+2156 | 0.141 | 42.34 | 1.79
46 | GALEX1417+5228 | 0.2083 | 42.06 | 2.03
47 | J0213+0056 | 0.0399 | 41.33 | 1.77
48 | GALEX1001+0233 | 0.3824 | 42.36 | 2.09
49 | GALEX1417+5305 | 0.2671 | 41.39 | 1.79

Table 4:  Name, systemic redshift, $\mathrm{Ly}\alpha$ luminosity and observed
equivalent width for the observed galaxies. The column Label indicates the
number that appears with the spectra in Fig.13, 14 and 15.

Label | Name | $z^{\rm Sys}$ |  $\rm\log$Ly$\alpha$ |  $\rm\log$$EW_{\rm Ly\alpha}$
---|---|---|---|---
|  |  | [$\rm erg/s$] | [Å]
50 | J0240-0828 | 0.0822 | 42.32 | 2.27
51 | GALEX1423+5246 | 0.3431 | 41.68 | 1.76
52 | J0808+1728 | 0.0442 | 41.4 | 1.44
53 | GALEX1418+5217 | 0.2398 | 40.81 | 1.25
54 | GALEX1419+5315 | 0.2637 | 41.25 | 1.54
55 | GALEX1418+5307 | 0.2034 | 41.63 | 1.58
56 | GALEX1418+5218 | 0.2388 | 41.21 | 1.82
57 | GALEX1420+5243 | 0.247 | 41.07 | 1.07
58 | GALEX1434+3532 | 0.1946 | 41.27 | 1.51
59 | J0851+5840 | 0.0919 | 41.64 | 1.7
60 | GALEX1436+3456 | 0.2684 | 41.72 | 1.65
61 | GALEX1437+3445 | 0.3237 | 41.88 | 1.44
62 | J1200+2719 | 0.0819 | 42.43 | 1.98
63 | KISSR1084 | 0.0321 | 39.46 | 0.52
64 | J1226+0415 | 0.0942 | 42.02 | 1.88
65 | KISSR1578 | 0.028 | 41.16 | 1.0
66 | KISSR1567 | 0.0426 | 39.05 | 0.93
67 | J1311-0038 | 0.0811 | 42.03 | 1.86
68 | J1509+3731 | 0.0325 | 41.0 | 1.57
69 | J1608+3528 | 0.0327 | 41.17 | 2.39
70 | J1735+5703 | 0.0472 | 42.17 | 1.88
71 | J2302+0049 | 0.0331 | 41.3 | 1.85
72 | GP0822+2241 | 0.2162 | 42.36 | 1.66
73 | GP0751+1638 | 0.2647 | 41.59 | 1.2
74 | GP0917+3152 | 0.3004 | 42.59 | 1.26
75 | GP1009+2916 | 0.2219 | 42.34 | 1.79
76 | GP0927+1740 | 0.2883 | 41.62 | 1.07
77 | GP1018+4106 | 0.237 | 41.89 | 1.48
78 | GP1122+6154 | 0.2046 | 42.26 | 1.73
79 | GP1339+1516 | 0.192 | 41.83 | 1.42
80 | GP1440+4619 | 0.3008 | 42.83 | 1.48
81 | GP1514+3852 | 0.3326 | 42.74 | 1.49
82 | GP1454+4528 | 0.2685 | 42.25 | 1.45
83 | GP1559+0841 | 0.297 | 42.57 | 1.77
84 | GP2237+1336 | 0.2935 | 42.26 | 1.13
85 | KISSR242 | 0.0378 | 41.37 | 1.34
86 | LARS04 | 0.0325 | 39.96 | 0.66
87 | LARS02 | 0.0298 | 41.03 | 1.61
88 | LARS03 | 0.0307 | 39.9 | 0.77
89 | LARS08 | 0.0382 | 40.14 | 0.53
90 | LARS11 | 0.0844 | 40.7 | 0.76
91 | GALEX0330-2816 | 0.2813 | 41.65 | 1.2
92 | LARS05 | 0.0338 | 41.28 | 1.19
93 | GALEX0333-2821 | 0.2471 | 41.51 | 1.34
94 | GALEX0332-2801 | 0.2155 | 41.5 | 1.52
95 | GALEX0331-2814 | 0.2803 | 42.02 | 2.03
96 | GALEX0332-2811 | 0.2043 | 42.07 | 1.56
97 | GALEX1000+0157 | 0.2647 | 42.11 | 1.38

Table 5:  Parameters associated with the line profiles. The column Label
indicates the number that appears with the spectra in Fig.13, 14 and 15.

| MCMC | DNN
---|---|---
Label | $z^{\rm Zelda}$ |  $\log$$V_{\rm exp}$ |  $\log$$N_{\rm H}$ |  $\log$$\tau_{a}$ |  $\log$$EW_{\rm in}$ |  $\rm\log$$W_{\rm in}$ | $z^{\rm Zelda}$ |  $\log$$V_{\rm exp}$ |  $\log$$N_{\rm H}$ |  $\log$$\tau_{a}$ |  $\log$$EW_{\rm in}$ |  $\rm\log$$W_{\rm in}$
|  | [$\rm km/s$] | [$\rm cm^{-2}$] |  | [Å] | [Å] |  | [$\rm km/s$] | [$\rm cm^{-2}$] |  | [Å] | [Å]
1 | $0.21867^{+1.6e-04}_{-4.9e-07}$ | $1.34^{+4.4e-01}_{-1.4e-03}$ | $19.21^{+1.5e-03}_{-7.1e-01}$ | $-3.86^{+1.2e+00}_{-1.4e-01}$ | $1.13^{+1.3e-03}_{-9.3e-03}$ | $0.04^{+6.8e-04}_{-3.8e-04}$ | $0.21872^{+5.0e-05}_{-3.1e-05}$ | $1.52^{+1.0e-01}_{-9.8e-02}$ | $18.68^{+2.0e-01}_{-2.1e-01}$ | $-3.56^{+3.1e-01}_{-1.6e-01}$ | $1.16^{+5.0e-02}_{-6.5e-02}$ | $-0.05^{+5.0e-02}_{-4.1e-02}$
2 | $0.12659^{+2.6e-06}_{-6.5e-08}$ | $2.1^{+3.3e-04}_{-8.2e-05}$ | $19.35^{+1.5e-04}_{-6.7e-03}$ | $-0.54^{+1.7e-03}_{-6.9e-03}$ | $1.58^{+4.5e-05}_{-5.8e-03}$ | $-0.19^{+2.9e-04}_{-1.0e-03}$ | $0.12671^{+7.0e-05}_{-7.6e-05}$ | $2.2^{+2.7e-02}_{-4.8e-02}$ | $18.85^{+1.5e-01}_{-1.5e-01}$ | $-3.02^{+2.6e-01}_{-2.8e-01}$ | $1.36^{+1.9e-02}_{-1.9e-02}$ | $-0.27^{+2.8e-02}_{-2.9e-02}$
3 | $0.147^{+6.4e-07}_{-3.5e-08}$ | $2.18^{+7.4e-08}_{-5.4e-05}$ | $19.66^{+5.4e-05}_{-1.0e-03}$ | $-0.06^{+4.6e-05}_{-2.4e-03}$ | $1.15^{+3.9e-05}_{-1.3e-03}$ | $-0.7^{+1.9e-07}_{-4.8e-05}$ | $0.14714^{+1.6e-04}_{-4.0e-04}$ | $2.2^{+9.9e-02}_{-1.7e-01}$ | $19.65^{+3.6e-01}_{-1.9e-01}$ | $-0.5^{+3.7e-01}_{-4.3e-01}$ | $1.01^{+2.9e-01}_{-2.6e-01}$ | $-0.72^{+2.1e-01}_{-1.6e-01}$
4 | $0.16825^{+3.2e-04}_{-5.3e-04}$ | $2.25^{+1.3e-01}_{-1.8e-01}$ | $19.54^{+4.7e-01}_{-2.6e-01}$ | $-2.5^{+1.5e+00}_{-5.8e-03}$ | $0.8^{+2.5e-01}_{-3.3e-03}$ | $-0.4^{+7.1e-01}_{-9.3e-02}$ | $0.16819^{+4.1e-04}_{-4.8e-04}$ | $2.4^{+1.4e-01}_{-2.3e-01}$ | $19.64^{+2.8e-01}_{-3.5e-01}$ | $-2.62^{+1.1e+00}_{-8.7e-01}$ | $1.08^{+1.6e-01}_{-4.0e-01}$ | $-0.08^{+2.6e-01}_{-2.4e-01}$
5 | $0.13193^{+4.8e-06}_{-1.1e-06}$ | $2.34^{+2.0e-03}_{-9.0e-04}$ | $19.56^{+3.4e-04}_{-7.7e-03}$ | $-0.2^{+1.5e-03}_{-5.9e-02}$ | $1.36^{+3.4e-04}_{-7.0e-03}$ | $-0.69^{+3.4e-02}_{-2.2e-04}$ | $0.1317^{+1.6e-04}_{-1.7e-04}$ | $2.23^{+1.0e-01}_{-9.1e-02}$ | $19.67^{+1.6e-01}_{-1.3e-01}$ | $-0.89^{+3.0e-01}_{-3.5e-01}$ | $1.09^{+1.1e-01}_{-7.5e-02}$ | $-0.53^{+7.7e-02}_{-1.3e-01}$
6 | $0.12701^{+5.7e-05}_{-1.7e-04}$ | $1.75^{+9.7e-02}_{-1.0e-01}$ | $20.3^{+2.4e-01}_{-5.6e-02}$ | $-0.09^{+4.5e-02}_{-3.2e-01}$ | $1.74^{+2.8e-01}_{-2.0e-01}$ | $-0.65^{+1.4e-01}_{-9.7e-01}$ | $0.12642^{+4.8e-04}_{-4.1e-04}$ | $1.77^{+2.0e-01}_{-2.3e-01}$ | $20.58^{+2.3e-01}_{-5.3e-01}$ | $-0.36^{+2.9e-01}_{-3.0e-01}$ | $1.45^{+3.3e-01}_{-3.9e-01}$ | $-0.57^{+3.8e-01}_{-1.6e-01}$
7 | $0.08187^{+6.0e-04}_{-7.2e-03}$ | $1.68^{+5.1e-01}_{-4.0e-01}$ | $19.16^{+7.8e-01}_{-4.1e-01}$ | $-1.41^{+1.1e+00}_{-1.1e+00}$ | $0.83^{+1.8e-01}_{-2.9e-01}$ | $0.15^{+1.2e-01}_{-1.2e+00}$ | $0.08204^{+3.3e-04}_{-9.0e-04}$ | $1.79^{+3.7e-01}_{-4.3e-01}$ | $19.6^{+7.7e-01}_{-6.5e-01}$ | $-1.3^{+9.3e-01}_{-1.6e+00}$ | $0.94^{+4.9e-01}_{-3.6e-01}$ | $0.24^{+2.4e-01}_{-4.2e-01}$
8 | $0.09138^{+4.8e-06}_{-2.0e-07}$ | $2.56^{+5.3e-02}_{-2.0e-04}$ | $18.45^{+1.6e-03}_{-4.7e-01}$ | $-2.0^{+1.6e+00}_{-3.7e-04}$ | $1.28^{+5.2e-05}_{-7.2e-03}$ | $-0.79^{+3.3e-04}_{-2.2e-02}$ | $0.09137^{+9.5e-06}_{-1.8e-05}$ | $2.59^{+6.9e-02}_{-4.4e-02}$ | $18.1^{+9.6e-02}_{-8.7e-02}$ | $-1.9^{+1.7e-01}_{-1.8e-01}$ | $1.25^{+1.6e-02}_{-1.8e-02}$ | $-0.79^{+1.7e-02}_{-1.4e-02}$
9 | $0.17389^{+5.0e-08}_{-3.6e-10}$ | $1.95^{+1.2e-04}_{-1.9e-06}$ | $19.25^{+2.2e-06}_{-1.1e-09}$ | $-0.69^{+6.8e-04}_{-5.9e-06}$ | $1.73^{+1.8e-04}_{-3.7e-06}$ | $0.3^{+2.1e-06}_{-4.7e-05}$ | $0.17398^{+3.6e-05}_{-5.6e-05}$ | $2.06^{+4.8e-02}_{-1.6e-02}$ | $19.08^{+6.9e-02}_{-8.9e-02}$ | $-0.81^{+4.3e-01}_{-2.9e-01}$ | $1.78^{+5.0e-02}_{-3.2e-02}$ | $0.3^{+4.2e-02}_{-1.6e-02}$
10 | $0.12327^{+1.2e-04}_{-1.3e-05}$ | $0.99^{+5.8e-01}_{-1.1e-01}$ | $19.9^{+6.5e-02}_{-4.4e-01}$ | $-0.67^{+3.5e-01}_{-1.2e-01}$ | $1.01^{+3.8e-02}_{-1.2e-01}$ | $0.07^{+7.3e-02}_{-2.1e-02}$ | $0.12325^{+7.6e-05}_{-1.6e-04}$ | $1.21^{+7.9e-02}_{-6.9e-02}$ | $19.51^{+2.0e-01}_{-2.4e-01}$ | $-0.38^{+1.4e-01}_{-3.1e-01}$ | $1.02^{+2.3e-01}_{-2.3e-01}$ | $0.06^{+1.3e-01}_{-1.6e-01}$
11 | $0.09423^{+1.4e-04}_{-8.4e-08}$ | $1.99^{+5.3e-02}_{-9.9e-04}$ | $19.56^{+1.4e-03}_{-1.6e-01}$ | $-0.2^{+2.5e-04}_{-2.6e-02}$ | $1.21^{+5.7e-05}_{-8.4e-02}$ | $-0.48^{+3.4e-04}_{-2.6e-01}$ | $0.09425^{+9.9e-05}_{-8.8e-05}$ | $2.17^{+5.0e-02}_{-9.6e-02}$ | $19.46^{+1.5e-01}_{-1.4e-01}$ | $-0.14^{+6.2e-02}_{-2.9e-01}$ | $1.27^{+1.1e-01}_{-1.6e-01}$ | $-0.74^{+9.0e-02}_{-8.3e-02}$
12 | $0.18209^{+5.5e-05}_{-2.9e-04}$ | $2.25^{+7.2e-02}_{-1.5e-01}$ | $18.25^{+6.4e-01}_{-1.6e-03}$ | $-2.5^{+1.7e+00}_{-2.7e-03}$ | $1.26^{+1.1e-01}_{-2.9e-04}$ | $-0.81^{+6.1e-02}_{-1.9e-03}$ | $0.18149^{+4.3e-04}_{-1.3e-04}$ | $2.12^{+1.7e-01}_{-2.4e-01}$ | $19.49^{+1.7e-01}_{-6.4e-01}$ | $-1.51^{+3.4e-01}_{-1.5e+00}$ | $1.41^{+1.4e-01}_{-1.4e-01}$ | $-0.63^{+6.1e-02}_{-1.5e-01}$
13 | $0.14936^{+2.4e-08}_{-1.2e-09}$ | $2.24^{+7.3e-06}_{-2.5e-06}$ | $19.43^{+6.6e-07}_{-7.7e-05}$ | $-0.0^{+2.4e-09}_{-1.4e-06}$ | $1.68^{+1.6e-06}_{-6.2e-05}$ | $-1.05^{+6.5e-05}_{-2.6e-06}$ | $0.14922^{+1.1e-04}_{-4.4e-05}$ | $2.27^{+4.9e-02}_{-3.4e-02}$ | $19.76^{+7.6e-02}_{-1.9e-01}$ | $-0.03^{+1.8e-02}_{-1.9e-02}$ | $1.69^{+8.3e-02}_{-1.2e-01}$ | $-0.83^{+2.6e-02}_{-6.7e-02}$
14 | $0.18093^{+7.8e-09}_{-4.1e-07}$ | $2.16^{+2.2e-05}_{-4.9e-04}$ | $18.94^{+4.1e-04}_{-6.9e-05}$ | $-1.5^{+1.5e-07}_{-4.9e-04}$ | $1.62^{+6.5e-06}_{-7.3e-04}$ | $0.06^{+2.0e-05}_{-1.9e-04}$ | $0.18117^{+3.4e-05}_{-1.4e-04}$ | $2.36^{+3.9e-02}_{-8.3e-02}$ | $18.53^{+2.9e-01}_{-9.6e-02}$ | $-3.14^{+2.0e+00}_{-2.0e-01}$ | $1.6^{+4.7e-02}_{-1.2e-02}$ | $-0.26^{+4.0e-02}_{-2.7e-02}$
15 | $0.1648^{+1.4e-07}_{-1.3e-09}$ | $1.81^{+1.7e-04}_{-2.6e-06}$ | $19.59^{+1.6e-06}_{-1.8e-04}$ | $-2.5^{+6.0e-06}_{-9.7e-10}$ | $0.95^{+6.2e-05}_{-6.8e-07}$ | $-0.32^{+5.2e-06}_{-1.0e-04}$ | $0.16462^{+1.5e-04}_{-1.6e-04}$ | $1.97^{+1.3e-01}_{-1.3e-01}$ | $19.71^{+2.1e-01}_{-2.6e-01}$ | $-2.86^{+5.7e-01}_{-3.9e-01}$ | $0.94^{+3.7e-02}_{-3.2e-02}$ | $-0.41^{+8.1e-02}_{-1.0e-01}$
16 | $0.06097^{+1.6e-06}_{-2.3e-07}$ | $1.78^{+9.9e-05}_{-1.3e-03}$ | $19.25^{+4.5e-04}_{-4.5e-03}$ | $-0.0^{+1.0e-04}_{-2.0e-03}$ | $2.45^{+7.6e-03}_{-1.5e-03}$ | $-0.83^{+5.7e-04}_{-3.1e-03}$ | $0.06102^{+1.7e-05}_{-3.2e-06}$ | $1.62^{+1.0e-01}_{-8.3e-03}$ | $19.09^{+2.0e-02}_{-8.4e-02}$ | $-0.84^{+5.1e-01}_{-7.2e-02}$ | $2.09^{+7.7e-02}_{-5.7e-03}$ | $-0.91^{+2.2e-02}_{-8.0e-03}$
17 | $0.10287^{+2.6e-05}_{-5.2e-07}$ | $2.07^{+2.3e-02}_{-2.4e-04}$ | $18.98^{+3.1e-03}_{-5.3e-02}$ | $-1.0^{+2.9e-04}_{-2.5e+00}$ | $0.96^{+4.1e-03}_{-2.8e-02}$ | $0.52^{+8.2e-02}_{-3.7e-03}$ | $0.10265^{+3.0e-04}_{-2.3e-04}$ | $1.7^{+3.5e-01}_{-4.3e-01}$ | $19.47^{+5.4e-01}_{-6.2e-01}$ | $-1.84^{+1.0e+00}_{-1.2e+00}$ | $0.63^{+1.7e-01}_{-2.1e-01}$ | $0.27^{+1.5e-01}_{-1.4e-01}$
18 | $0.09889^{+2.6e-07}_{-2.4e-06}$ | $2.32^{+6.0e-05}_{-5.1e-04}$ | $18.89^{+2.6e-03}_{-1.4e-04}$ | $-2.5^{+3.8e-04}_{-1.5e+00}$ | $1.42^{+5.2e-05}_{-2.2e-03}$ | $-0.8^{+1.3e-03}_{-5.2e-04}$ | $0.09865^{+6.9e-05}_{-7.9e-05}$ | $2.34^{+5.5e-02}_{-7.2e-02}$ | $19.04^{+1.0e-01}_{-1.3e-01}$ | $-1.84^{+5.4e-01}_{-2.5e-01}$ | $1.48^{+4.8e-02}_{-1.8e-02}$ | $-0.51^{+2.6e-02}_{-2.4e-02}$
19 | $0.1372^{+7.9e-09}_{-1.2e-07}$ | $2.4^{+1.2e-05}_{-1.3e-04}$ | $19.0^{+2.8e-08}_{-1.1e-05}$ | $-1.15^{+7.9e-04}_{-2.9e-05}$ | $1.46^{+1.3e-04}_{-5.2e-06}$ | $-0.66^{+1.5e-04}_{-1.0e-05}$ | $0.13701^{+1.0e-04}_{-1.9e-04}$ | $2.36^{+4.3e-02}_{-7.9e-02}$ | $19.08^{+1.9e-01}_{-8.9e-02}$ | $-0.25^{+1.1e-01}_{-1.1e+00}$ | $1.57^{+2.9e-02}_{-5.0e-02}$ | $-0.21^{+1.6e-01}_{-1.8e-01}$
20 | $0.02089^{+1.2e-08}_{-2.0e-07}$ | $2.34^{+1.1e-04}_{-1.4e-05}$ | $18.98^{+7.2e-04}_{-1.9e-05}$ | $-0.5^{+3.1e-05}_{-5.1e-08}$ | $1.13^{+1.2e-04}_{-6.3e-06}$ | $-0.85^{+5.6e-06}_{-3.1e-04}$ | $0.02088^{+2.2e-05}_{-2.7e-05}$ | $2.44^{+4.2e-02}_{-3.1e-02}$ | $19.29^{+8.5e-02}_{-9.5e-02}$ | $-1.17^{+5.7e-01}_{-4.6e-01}$ | $1.08^{+6.0e-02}_{-2.9e-02}$ | $-0.83^{+5.4e-02}_{-2.1e-02}$
21 | $0.26218^{+4.1e-07}_{-2.2e-06}$ | $1.9^{+2.1e-04}_{-2.7e-02}$ | $18.79^{+1.7e-03}_{-4.3e-02}$ | $-0.5^{+2.2e-03}_{-2.5e+00}$ | $1.87^{+1.1e-03}_{-1.1e-01}$ | $-0.2^{+1.1e-01}_{-3.2e-04}$ | $0.26216^{+2.1e-05}_{-2.1e-05}$ | $1.96^{+1.8e-02}_{-1.5e-02}$ | $18.77^{+3.8e-02}_{-3.9e-02}$ | $-1.62^{+9.0e-02}_{-9.0e-02}$ | $1.73^{+1.3e-02}_{-1.2e-02}$ | $-0.52^{+1.6e-02}_{-1.4e-02}$
22 | $0.19599^{+3.7e-09}_{-1.1e-08}$ | $1.89^{+9.5e-06}_{-8.6e-06}$ | $18.02^{+1.2e-04}_{-2.2e-05}$ | $-1.5^{+1.0e-07}_{-6.4e-05}$ | $2.17^{+3.9e-05}_{-8.5e-04}$ | $0.05^{+1.5e-05}_{-8.5e-06}$ | $0.19592^{+4.8e-05}_{-4.9e-05}$ | $1.81^{+9.2e-03}_{-2.2e-02}$ | $17.94^{+1.1e-01}_{-6.2e-02}$ | $-1.01^{+1.1e-01}_{-9.2e-02}$ | $2.14^{+6.2e-03}_{-6.2e-03}$ | $-0.0^{+2.5e-02}_{-5.8e-02}$
23 | $0.24198^{+1.4e-08}_{-4.9e-08}$ | $1.73^{+3.8e-04}_{-3.2e-06}$ | $18.44^{+3.0e-04}_{-5.4e-06}$ | $-0.07^{+6.9e-04}_{-9.7e-06}$ | $1.65^{+2.6e-05}_{-4.5e-05}$ | $-0.01^{+2.5e-06}_{-2.2e-04}$ | $0.242^{+2.6e-05}_{-2.9e-05}$ | $1.82^{+9.4e-02}_{-7.6e-02}$ | $18.06^{+7.3e-02}_{-7.2e-02}$ | $-0.08^{+8.3e-02}_{-1.1e-01}$ | $1.65^{+2.1e-02}_{-2.3e-02}$ | $-0.08^{+1.7e-02}_{-1.8e-02}$
24 | $0.25296^{+2.7e-07}_{-5.1e-08}$ | $2.0^{+2.7e-06}_{-1.3e-04}$ | $19.25^{+3.4e-04}_{-1.0e-05}$ | $-0.0^{+2.4e-05}_{-7.9e-04}$ | $1.53^{+4.8e-05}_{-5.2e-04}$ | $-0.27^{+2.5e-04}_{-1.4e-03}$ | $0.25293^{+6.6e-05}_{-8.9e-05}$ | $2.07^{+1.0e-01}_{-8.4e-02}$ | $18.98^{+1.6e-01}_{-1.2e-01}$ | $-1.06^{+4.3e-01}_{-4.5e-01}$ | $1.2^{+6.0e-02}_{-5.8e-02}$ | $-0.4^{+4.8e-02}_{-5.8e-02}$
25 | $0.19454^{+6.5e-07}_{-1.4e-08}$ | $2.16^{+2.9e-04}_{-9.6e-05}$ | $19.25^{+1.0e-06}_{-8.6e-04}$ | $-0.17^{+1.2e-03}_{-7.8e-03}$ | $1.8^{+6.0e-05}_{-5.5e-03}$ | $-0.07^{+1.2e-05}_{-5.5e-04}$ | $0.19432^{+3.9e-04}_{-1.2e-04}$ | $2.23^{+6.9e-02}_{-1.0e-01}$ | $19.45^{+1.8e-01}_{-5.9e-01}$ | $-1.67^{+3.0e-01}_{-7.4e-01}$ | $1.62^{+4.6e-02}_{-9.3e-02}$ | $-0.35^{+5.6e-02}_{-7.5e-02}$
26 | $0.23979^{+1.8e-08}_{-8.8e-10}$ | $2.02^{+5.6e-05}_{-6.0e-06}$ | $19.25^{+2.3e-08}_{-9.7e-06}$ | $-0.12^{+7.3e-05}_{-9.4e-06}$ | $1.91^{+1.6e-05}_{-1.6e-06}$ | $0.19^{+1.5e-06}_{-8.2e-06}$ | $0.2399^{+6.0e-05}_{-9.1e-05}$ | $2.1^{+3.0e-02}_{-4.6e-02}$ | $19.01^{+1.0e-01}_{-8.7e-02}$ | $0.09^{+1.6e-01}_{-3.8e-01}$ | $1.87^{+4.3e-02}_{-3.1e-02}$ | $0.28^{+3.0e-02}_{-2.7e-02}$
27 | $0.26279^{+5.7e-04}_{-1.7e-06}$ | $2.13^{+2.2e-01}_{-2.7e-03}$ | $19.67^{+2.2e-03}_{-5.6e-01}$ | $-3.48^{+1.1e+00}_{-2.2e-02}$ | $1.97^{+1.4e-02}_{-1.4e-02}$ | $-0.34^{+6.5e-02}_{-1.1e-01}$ | $0.26349^{+1.0e-04}_{-6.3e-04}$ | $2.41^{+3.5e-02}_{-1.5e-01}$ | $18.54^{+9.7e-01}_{-1.4e-01}$ | $-3.11^{+2.3e-01}_{-2.8e-01}$ | $2.03^{+2.4e-02}_{-3.2e-02}$ | $-0.34^{+1.5e-01}_{-4.6e-02}$
28 | $0.18518^{+6.0e-06}_{-1.1e-03}$ | $1.4^{+2.0e-02}_{-1.1e-01}$ | $19.2^{+1.3e-03}_{-1.4e+00}$ | $-1.14^{+5.3e-03}_{-1.4e+00}$ | $2.0^{+4.4e-04}_{-3.0e-01}$ | $0.2^{+2.1e-03}_{-1.7e-01}$ | $0.18513^{+1.2e-04}_{-1.8e-04}$ | $1.41^{+4.5e-01}_{-1.8e-01}$ | $19.25^{+3.7e-01}_{-6.4e-01}$ | $-2.09^{+1.2e+00}_{-1.1e+00}$ | $2.09^{+1.7e-01}_{-5.8e-01}$ | $0.06^{+1.8e-01}_{-3.2e-01}$
29 | $0.30134^{+8.7e-10}_{-1.9e-08}$ | $1.48^{+1.7e-08}_{-2.0e-06}$ | $19.25^{+8.0e-09}_{-6.5e-07}$ | $-0.79^{+1.7e-04}_{-1.5e-05}$ | $2.05^{+2.6e-04}_{-8.8e-06}$ | $-0.18^{+3.4e-06}_{-8.7e-05}$ | $0.30137^{+2.3e-05}_{-3.9e-05}$ | $1.49^{+7.4e-02}_{-4.2e-02}$ | $19.19^{+1.0e-01}_{-1.2e-01}$ | $-1.01^{+6.9e-01}_{-6.3e-01}$ | $2.09^{+4.6e-02}_{-6.7e-02}$ | $-0.15^{+7.6e-02}_{-1.2e-01}$
30 | $0.03847^{+7.6e-08}_{-1.2e-07}$ | $0.91^{+6.1e-05}_{-5.5e-05}$ | $19.36^{+1.4e-04}_{-5.0e-05}$ | $-3.0^{+6.6e-07}_{-2.5e-05}$ | $1.75^{+6.9e-04}_{-2.0e-05}$ | $-0.4^{+4.2e-05}_{-1.0e-04}$ | $0.03855^{+2.5e-05}_{-2.9e-05}$ | $1.14^{+3.6e-02}_{-2.7e-02}$ | $18.72^{+1.5e-01}_{-1.6e-01}$ | $-3.47^{+3.5e-01}_{-2.5e-01}$ | $1.96^{+2.5e-02}_{-2.7e-02}$ | $-0.47^{+2.9e-02}_{-2.9e-02}$
31 | $0.06533^{+3.7e-09}_{-1.5e-07}$ | $1.11^{+6.5e-05}_{-1.5e-04}$ | $19.0^{+9.2e-05}_{-4.3e-07}$ | $-3.5^{+3.3e-03}_{-5.4e-06}$ | $2.1^{+2.1e-04}_{-1.8e-04}$ | $-0.19^{+3.0e-05}_{-4.8e-04}$ | $0.06531^{+9.0e-06}_{-4.3e-06}$ | $1.06^{+6.7e-03}_{-6.8e-03}$ | $19.14^{+2.0e-02}_{-1.3e-02}$ | $-3.33^{+9.3e-02}_{-7.6e-01}$ | $1.75^{+9.2e-03}_{-6.7e-02}$ | $-0.42^{+1.2e-02}_{-1.7e-02}$
32 | $0.02813^{+4.2e-09}_{-3.6e-07}$ | $1.69^{+7.7e-05}_{-1.9e-03}$ | $18.75^{+1.7e-07}_{-1.8e-05}$ | $-0.0^{+8.5e-07}_{-1.6e-04}$ | $2.52^{+3.5e-04}_{-2.1e-04}$ | $-0.57^{+3.0e-05}_{-8.2e-04}$ | $0.02803^{+3.8e-06}_{-4.0e-06}$ | $1.45^{+8.5e-03}_{-9.0e-03}$ | $19.22^{+1.1e-02}_{-1.1e-02}$ | $-0.5^{+2.9e-02}_{-6.0e-02}$ | $2.18^{+1.8e-03}_{-2.3e-03}$ | $-0.54^{+4.0e-03}_{-4.2e-03}$
33 | $0.0622^{+1.3e-06}_{-2.8e-05}$ | $1.25^{+8.7e-04}_{-1.5e-01}$ | $19.0^{+2.1e-01}_{-1.4e-04}$ | $-0.39^{+2.1e-03}_{-1.1e-01}$ | $2.47^{+6.4e-02}_{-2.3e-03}$ | $-0.65^{+1.2e-03}_{-8.0e-02}$ | $0.06216^{+3.8e-06}_{-4.9e-05}$ | $1.05^{+1.4e-02}_{-3.0e-02}$ | $19.16^{+7.4e-02}_{-1.3e-02}$ | $-1.87^{+4.0e-02}_{-4.4e-02}$ | $2.11^{+1.1e-02}_{-5.9e-03}$ | $-0.54^{+2.8e-02}_{-9.1e-02}$

Table 6:  Parameters associated with the line profiles. The column Label
indicates the number that appears with the spectra in Fig.13, 14 and 15.

| MCMC | DNN
---|---|---
Label | $z^{\rm Zelda}$ |  $\log$$V_{\rm exp}$ |  $\log$$N_{\rm H}$ |  $\log$$\tau_{a}$ |  $\log$$EW_{\rm in}$ |  $\rm\log$$W_{\rm in}$ | $z^{\rm Zelda}$ |  $\log$$V_{\rm exp}$ |  $\log$$N_{\rm H}$ |  $\log$$\tau_{a}$ |  $\log$$EW_{\rm in}$ |  $\rm\log$$W_{\rm in}$
|  | [$\rm km/s$] | [$\rm cm^{-2}$] |  | [Å] | [Å] |  | [$\rm km/s$] | [$\rm cm^{-2}$] |  | [Å] | [Å]
34 | $0.34225^{+3.4e-10}_{-3.8e-11}$ | $1.71^{+5.8e-07}_{-1.6e-05}$ | $18.46^{+4.2e-07}_{-4.7e-05}$ | $-0.25^{+2.2e-05}_{-2.2e-07}$ | $1.93^{+2.6e-06}_{-2.4e-06}$ | $0.15^{+1.4e-07}_{-1.2e-05}$ | $0.34229^{+1.6e-05}_{-1.2e-05}$ | $1.77^{+5.0e-02}_{-4.2e-02}$ | $18.01^{+6.1e-02}_{-7.3e-02}$ | $-0.03^{+5.5e-02}_{-6.1e-02}$ | $2.04^{+3.0e-02}_{-3.5e-02}$ | $0.07^{+3.0e-02}_{-2.7e-02}$
35 | $0.35558^{+2.4e-08}_{-3.9e-07}$ | $2.0^{+5.1e-06}_{-2.9e-04}$ | $19.25^{+1.9e-05}_{-1.8e-07}$ | $-0.0^{+3.0e-06}_{-1.4e-04}$ | $2.26^{+2.2e-04}_{-2.5e-03}$ | $-0.33^{+7.5e-05}_{-2.3e-04}$ | $0.35546^{+4.3e-05}_{-7.3e-05}$ | $1.89^{+3.4e-02}_{-3.6e-02}$ | $19.24^{+1.0e-01}_{-6.4e-02}$ | $-0.47^{+1.5e-01}_{-2.3e-01}$ | $2.05^{+5.6e-02}_{-6.0e-02}$ | $-0.25^{+2.2e-02}_{-2.5e-02}$
36 | $0.31828^{+1.1e-06}_{-3.3e-06}$ | $2.32^{+1.1e-02}_{-4.2e-03}$ | $18.8^{+5.6e-02}_{-1.4e-02}$ | $-3.97^{+2.5e+00}_{-2.9e-02}$ | $1.84^{+2.3e-02}_{-6.1e-03}$ | $-0.87^{+1.5e-03}_{-1.2e-03}$ | $0.31817^{+8.5e-05}_{-1.1e-04}$ | $2.36^{+4.1e-02}_{-5.2e-02}$ | $18.81^{+1.7e-01}_{-1.6e-01}$ | $-1.95^{+3.9e-01}_{-3.8e-01}$ | $1.82^{+2.5e-02}_{-2.6e-02}$ | $-0.66^{+2.8e-02}_{-2.2e-02}$
37 | $0.29356^{+4.7e-08}_{-2.4e-10}$ | $2.12^{+1.8e-07}_{-2.0e-05}$ | $19.11^{+8.3e-07}_{-2.0e-04}$ | $-0.72^{+1.0e-06}_{-4.0e-04}$ | $2.03^{+8.6e-07}_{-7.7e-05}$ | $-0.18^{+3.8e-07}_{-2.8e-06}$ | $0.29353^{+5.2e-05}_{-4.4e-05}$ | $2.18^{+3.6e-02}_{-3.1e-02}$ | $18.9^{+8.6e-02}_{-9.1e-02}$ | $-2.04^{+3.1e-01}_{-2.1e-01}$ | $2.01^{+1.3e-02}_{-1.4e-02}$ | $-0.28^{+1.7e-02}_{-1.9e-02}$
38 | $0.29948^{+4.9e-08}_{-1.7e-08}$ | $1.87^{+2.1e-05}_{-2.0e-04}$ | $18.75^{+8.7e-08}_{-1.6e-05}$ | $-0.5^{+1.5e-05}_{-2.8e-03}$ | $2.33^{+2.3e-04}_{-8.1e-05}$ | $-0.18^{+1.7e-05}_{-8.1e-05}$ | $0.29953^{+9.2e-06}_{-1.1e-05}$ | $2.0^{+2.1e-02}_{-2.2e-02}$ | $18.54^{+6.6e-02}_{-5.6e-02}$ | $-1.22^{+1.4e-01}_{-1.4e-01}$ | $2.12^{+2.0e-02}_{-2.0e-02}$ | $-0.03^{+2.0e-02}_{-2.1e-02}$
39 | $0.36933^{+4.7e-11}_{-5.2e-09}$ | $1.74^{+1.1e-05}_{-3.0e-07}$ | $17.32^{+7.2e-05}_{-2.1e-06}$ | $-1.0^{+6.2e-06}_{-3.7e-09}$ | $2.1^{+2.6e-06}_{-2.4e-05}$ | $-0.24^{+2.8e-07}_{-2.0e-05}$ | $0.36922^{+6.5e-05}_{-6.9e-05}$ | $1.63^{+3.3e-02}_{-4.9e-02}$ | $17.53^{+1.4e-01}_{-9.5e-02}$ | $-1.01^{+2.7e-01}_{-2.9e-01}$ | $2.08^{+2.1e-02}_{-3.5e-02}$ | $-0.28^{+4.7e-02}_{-1.6e-02}$
40 | $0.33214^{+3.7e-08}_{-1.1e-09}$ | $1.27^{+1.4e-04}_{-4.2e-06}$ | $19.25^{+5.2e-10}_{-7.5e-07}$ | $-0.85^{+3.9e-06}_{-6.9e-04}$ | $2.29^{+7.6e-06}_{-4.2e-04}$ | $-0.44^{+1.6e-04}_{-1.4e-06}$ | $0.33214^{+1.6e-05}_{-1.5e-05}$ | $1.22^{+4.0e-02}_{-4.1e-02}$ | $19.26^{+4.8e-02}_{-5.8e-02}$ | $-1.34^{+2.6e-01}_{-2.5e-01}$ | $2.13^{+3.1e-02}_{-3.0e-02}$ | $-0.32^{+2.0e-02}_{-1.9e-02}$
41 | $0.43204^{+1.4e-07}_{-6.2e-05}$ | $1.52^{+2.9e-04}_{-3.7e-01}$ | $17.0^{+2.3e-02}_{-2.3e-04}$ | $-0.05^{+4.4e-02}_{-1.4e-02}$ | $1.96^{+1.1e-02}_{-1.5e-03}$ | $-0.18^{+1.2e-02}_{-1.1e-03}$ | $0.432^{+4.2e-05}_{-7.9e-05}$ | $1.35^{+1.5e-01}_{-1.4e-01}$ | $17.25^{+2.1e-01}_{-1.6e-01}$ | $-1.35^{+3.9e-01}_{-4.6e-01}$ | $2.01^{+3.6e-02}_{-3.4e-02}$ | $-0.2^{+6.2e-02}_{-4.3e-02}$
42 | $0.3533^{+1.8e-08}_{-3.4e-09}$ | $1.68^{+3.6e-06}_{-2.2e-04}$ | $18.75^{+1.3e-07}_{-2.3e-05}$ | $-0.5^{+9.0e-07}_{-5.3e-07}$ | $2.1^{+4.3e-05}_{-1.1e-03}$ | $-0.47^{+1.9e-05}_{-3.2e-04}$ | $0.35324^{+4.1e-05}_{-5.4e-05}$ | $1.57^{+4.5e-02}_{-4.5e-02}$ | $18.85^{+1.7e-01}_{-1.7e-01}$ | $-0.88^{+5.0e-01}_{-3.5e-01}$ | $2.08^{+3.2e-02}_{-4.9e-02}$ | $-0.52^{+3.9e-02}_{-3.1e-02}$
43 | $0.36314^{+1.1e-07}_{-4.7e-09}$ | $1.72^{+2.7e-04}_{-1.8e-05}$ | $18.57^{+2.6e-05}_{-1.1e-04}$ | $-0.0^{+3.2e-07}_{-2.4e-04}$ | $2.59^{+9.3e-05}_{-1.4e-03}$ | $-0.1^{+1.2e-04}_{-2.6e-06}$ | $0.36317^{+6.4e-05}_{-2.6e-05}$ | $1.88^{+5.0e-02}_{-6.3e-02}$ | $18.14^{+1.3e-01}_{-1.1e-01}$ | $0.04^{+7.9e-02}_{-8.7e-02}$ | $2.17^{+1.4e-02}_{-1.4e-02}$ | $-0.07^{+3.1e-02}_{-4.8e-02}$
44 | $0.1488^{+3.6e-05}_{-9.0e-05}$ | $1.08^{+1.1e-01}_{-3.2e-02}$ | $20.34^{+2.3e-01}_{-1.4e-01}$ | $-0.83^{+1.0e-01}_{-2.4e+00}$ | $1.6^{+6.5e-02}_{-6.5e-01}$ | $-0.03^{+2.6e-02}_{-8.9e-02}$ | $0.14875^{+1.3e-04}_{-8.8e-05}$ | $1.11^{+1.4e-01}_{-5.8e-02}$ | $20.27^{+1.3e-01}_{-2.3e-01}$ | $-1.79^{+6.2e-01}_{-9.8e-01}$ | $1.28^{+1.8e-01}_{-2.0e-01}$ | $-0.09^{+1.7e-01}_{-2.4e-01}$
45 | $0.14112^{+4.2e-06}_{-6.6e-05}$ | $1.75^{+6.5e-03}_{-1.0e-01}$ | $18.54^{+3.3e-01}_{-4.2e-04}$ | $-0.05^{+6.3e-04}_{-2.3e+00}$ | $2.06^{+2.0e-04}_{-1.8e-01}$ | $-0.25^{+1.5e-01}_{-1.1e-04}$ | $0.14103^{+7.2e-05}_{-6.7e-05}$ | $1.68^{+1.2e-01}_{-1.6e-01}$ | $18.52^{+4.7e-01}_{-2.3e-01}$ | $-1.35^{+9.7e-01}_{-1.2e+00}$ | $2.17^{+8.8e-02}_{-3.6e-01}$ | $-0.15^{+8.2e-02}_{-1.1e-01}$
46 | $0.20811^{+4.2e-08}_{-6.0e-10}$ | $1.7^{+1.3e-07}_{-8.9e-11}$ | $18.8^{+6.7e-06}_{-2.4e-04}$ | $-0.06^{+3.1e-06}_{-1.3e-04}$ | $2.4^{+3.0e-05}_{-2.8e-05}$ | $-0.85^{+1.7e-05}_{-1.4e-06}$ | $0.20804^{+7.7e-06}_{-8.0e-06}$ | $1.53^{+1.3e-02}_{-1.2e-02}$ | $19.06^{+5.4e-02}_{-5.4e-02}$ | $-1.13^{+2.1e-01}_{-1.8e-01}$ | $2.12^{+1.2e-02}_{-1.2e-02}$ | $-0.88^{+1.1e-02}_{-1.1e-02}$
47 | $0.03997^{+3.5e-07}_{-3.9e-08}$ | $1.86^{+3.1e-04}_{-4.1e-05}$ | $19.3^{+3.7e-04}_{-2.3e-04}$ | $-0.0^{+1.5e-06}_{-4.8e-04}$ | $2.27^{+5.5e-04}_{-1.6e-04}$ | $-0.46^{+1.3e-04}_{-1.0e-04}$ | $0.04001^{+9.5e-06}_{-1.7e-05}$ | $1.84^{+3.6e-02}_{-2.0e-02}$ | $19.21^{+5.6e-02}_{-3.0e-02}$ | $-0.26^{+1.2e-01}_{-7.0e-02}$ | $2.14^{+6.0e-02}_{-3.8e-02}$ | $-0.55^{+9.0e-03}_{-9.0e-03}$
48 | $0.38269^{+5.8e-10}_{-2.0e-08}$ | $2.01^{+4.0e-06}_{-5.2e-05}$ | $19.09^{+5.7e-05}_{-2.5e-06}$ | $-3.0^{+1.3e-03}_{-2.5e-06}$ | $1.69^{+1.5e-04}_{-1.7e-05}$ | $-0.26^{+7.2e-05}_{-7.7e-06}$ | $0.38277^{+3.2e-04}_{-3.3e-04}$ | $2.17^{+1.9e-01}_{-1.9e-01}$ | $19.06^{+4.7e-01}_{-6.4e-01}$ | $-1.85^{+9.5e-01}_{-1.4e+00}$ | $1.82^{+1.5e-01}_{-1.3e-01}$ | $-0.34^{+1.8e-01}_{-2.0e-01}$
49 | $0.26721^{+1.3e-04}_{-2.0e-04}$ | $1.84^{+8.8e-01}_{-1.4e-01}$ | $18.67^{+1.6e-01}_{-3.1e-01}$ | $-1.49^{+1.1e+00}_{-4.0e-01}$ | $2.02^{+9.3e-02}_{-4.8e-02}$ | $-1.3^{+3.1e-01}_{-1.8e-01}$ | $0.26731^{+1.3e-04}_{-2.6e-04}$ | $2.29^{+4.0e-01}_{-3.8e-01}$ | $18.43^{+8.3e-01}_{-9.1e-01}$ | $-1.33^{+1.1e+00}_{-9.5e-01}$ | $1.99^{+1.4e-01}_{-1.8e-01}$ | $-0.85^{+1.2e-01}_{-1.1e-01}$
50 | $0.08227^{+8.2e-08}_{-2.3e-09}$ | $1.51^{+2.7e-06}_{-4.9e-05}$ | $19.0^{+1.0e-07}_{-1.0e-09}$ | $-1.47^{+2.4e-03}_{-8.6e-06}$ | $2.34^{+2.7e-04}_{-7.6e-06}$ | $-0.11^{+2.4e-06}_{-3.8e-04}$ | $0.08224^{+3.7e-06}_{-3.0e-06}$ | $1.4^{+1.4e-02}_{-1.1e-02}$ | $19.22^{+1.2e-02}_{-1.6e-02}$ | $-1.1^{+1.2e-01}_{-9.3e-02}$ | $2.21^{+5.0e-03}_{-4.6e-03}$ | $-0.27^{+9.0e-03}_{-7.4e-03}$
51 | $0.3438^{+8.6e-08}_{-2.7e-07}$ | $1.3^{+1.1e-03}_{-1.2e-02}$ | $17.5^{+9.3e-05}_{-2.8e-03}$ | $-3.5^{+1.6e-02}_{-3.2e-04}$ | $1.62^{+6.5e-02}_{-2.0e-03}$ | $-0.07^{+3.2e-02}_{-8.3e-04}$ | $0.3435^{+5.8e-05}_{-6.6e-05}$ | $1.17^{+8.1e-02}_{-4.8e-02}$ | $19.16^{+2.5e-01}_{-1.7e-01}$ | $-3.77^{+1.4e+00}_{-4.6e-01}$ | $1.61^{+1.5e-01}_{-1.3e-01}$ | $0.22^{+1.3e-01}_{-1.4e-01}$
52 | $0.04436^{+2.4e-10}_{-1.3e-07}$ | $2.08^{+1.1e-06}_{-1.1e-04}$ | $18.56^{+1.9e-04}_{-2.0e-06}$ | $-1.0^{+4.7e-08}_{-1.0e-04}$ | $1.49^{+2.3e-04}_{-3.7e-06}$ | $-1.21^{+2.7e-04}_{-6.7e-06}$ | $0.04429^{+7.9e-06}_{-6.0e-06}$ | $2.03^{+1.0e-02}_{-9.5e-03}$ | $18.24^{+3.8e-02}_{-4.7e-02}$ | $-0.23^{+4.5e-02}_{-3.4e-02}$ | $1.64^{+1.2e-02}_{-1.2e-02}$ | $-0.82^{+5.6e-03}_{-5.9e-03}$
53 | $0.24031^{+4.7e-06}_{-1.3e-04}$ | $1.78^{+6.8e-02}_{-1.3e-02}$ | $18.44^{+7.3e-01}_{-1.1e-01}$ | $-2.0^{+1.5e+00}_{-1.5e-02}$ | $1.4^{+1.6e-01}_{-8.5e-02}$ | $-0.59^{+9.3e-02}_{-5.4e-01}$ | $0.24045^{+1.5e-02}_{-6.0e-04}$ | $1.71^{+4.0e-01}_{-4.6e-01}$ | $18.65^{+8.8e-01}_{-1.4e+00}$ | $-1.45^{+1.0e+00}_{-1.4e+00}$ | $1.05^{+3.8e-01}_{-6.1e-01}$ | $-0.42^{+5.7e-01}_{-3.5e-01}$
54 | $0.26394^{+2.3e-04}_{-2.4e-05}$ | $1.55^{+1.2e+00}_{-2.8e-02}$ | $17.0^{+1.5e+00}_{-1.2e-03}$ | $-0.73^{+4.5e-02}_{-9.0e-01}$ | $1.62^{+1.8e-02}_{-1.3e-02}$ | $-0.46^{+4.4e-02}_{-4.6e-01}$ | $0.26396^{+1.3e-04}_{-2.1e-04}$ | $2.51^{+2.5e-01}_{-5.2e-01}$ | $18.2^{+8.9e-01}_{-8.8e-01}$ | $-3.0^{+1.6e+00}_{-7.6e-01}$ | $1.74^{+1.5e-01}_{-1.5e-01}$ | $-0.59^{+1.3e-01}_{-1.3e-01}$
55 | $0.2035^{+4.2e-08}_{-1.9e-08}$ | $1.89^{+4.7e-06}_{-7.5e-06}$ | $18.73^{+5.6e-05}_{-1.9e-04}$ | $-2.5^{+4.8e-09}_{-1.5e-07}$ | $1.63^{+4.5e-06}_{-6.9e-06}$ | $-0.17^{+3.3e-06}_{-1.5e-05}$ | $0.20358^{+6.8e-05}_{-1.1e-04}$ | $2.0^{+4.7e-02}_{-6.2e-02}$ | $18.37^{+3.2e-01}_{-2.9e-01}$ | $-2.72^{+7.3e-01}_{-5.6e-01}$ | $1.72^{+5.5e-02}_{-5.2e-02}$ | $-0.19^{+4.6e-02}_{-4.8e-02}$
56 | $0.2394^{+9.8e-07}_{-8.9e-05}$ | $1.18^{+5.2e-01}_{-4.0e-03}$ | $17.0^{+1.5e+00}_{-2.2e-03}$ | $-2.97^{+1.2e+00}_{-2.6e-02}$ | $1.55^{+5.1e-02}_{-5.4e-03}$ | $-0.5^{+2.7e-03}_{-1.0e-01}$ | $0.23929^{+2.0e-04}_{-4.7e-04}$ | $2.05^{+2.9e-01}_{-2.4e-01}$ | $18.49^{+7.9e-01}_{-7.6e-01}$ | $-1.89^{+1.2e+00}_{-1.3e+00}$ | $1.62^{+1.4e-01}_{-1.2e-01}$ | $-0.55^{+1.6e-01}_{-1.2e-01}$
57 | $0.24814^{+1.8e-02}_{-4.4e-06}$ | $0.92^{+5.2e-01}_{-1.2e-02}$ | $17.1^{+7.5e-01}_{-1.0e-01}$ | $-2.04^{+2.3e-01}_{-6.4e-01}$ | $0.9^{+1.6e-02}_{-1.4e-01}$ | $-0.4^{+8.0e-03}_{-4.7e-01}$ | $0.24729^{+8.4e-04}_{-9.4e-04}$ | $2.07^{+3.5e-01}_{-3.6e-01}$ | $19.33^{+8.8e-01}_{-1.3e+00}$ | $-1.5^{+8.9e-01}_{-1.2e+00}$ | $1.09^{+2.9e-01}_{-2.5e-01}$ | $-0.39^{+3.1e-01}_{-2.4e-01}$
58 | $0.19494^{+9.5e-04}_{-5.1e-05}$ | $2.18^{+2.5e-04}_{-1.5e+00}$ | $19.25^{+4.5e-02}_{-1.1e+00}$ | $-1.1^{+2.6e-01}_{-3.3e-01}$ | $1.48^{+1.8e-02}_{-1.3e-01}$ | $-0.01^{+9.7e-03}_{-2.7e-01}$ | $0.19534^{+3.3e-04}_{-3.7e-04}$ | $2.27^{+2.4e-01}_{-5.3e-01}$ | $18.5^{+4.9e-01}_{-5.4e-01}$ | $-0.89^{+6.7e-01}_{-1.4e+00}$ | $1.38^{+1.5e-01}_{-1.2e-01}$ | $-0.18^{+1.7e-01}_{-1.5e-01}$
59 | $0.09199^{+1.3e-08}_{-9.6e-08}$ | $1.76^{+3.9e-05}_{-2.9e-04}$ | $19.25^{+2.0e-07}_{-7.1e-06}$ | $-0.0^{+5.2e-07}_{-3.1e-05}$ | $2.13^{+9.4e-05}_{-3.7e-05}$ | $-0.37^{+2.2e-05}_{-2.6e-04}$ | $0.09199^{+8.3e-06}_{-9.2e-06}$ | $1.7^{+7.6e-03}_{-8.0e-03}$ | $19.23^{+2.9e-02}_{-2.6e-02}$ | $-0.29^{+1.0e-01}_{-1.0e-01}$ | $2.12^{+4.5e-02}_{-4.7e-02}$ | $-0.29^{+7.4e-03}_{-7.5e-03}$
60 | $0.26901^{+1.4e-06}_{-1.6e-04}$ | $2.06^{+1.6e-03}_{-5.8e-02}$ | $18.99^{+1.0e-01}_{-4.4e-02}$ | $-2.5^{+1.6e+00}_{-6.7e-02}$ | $1.45^{+1.7e-01}_{-1.3e-02}$ | $-1.66^{+9.8e-01}_{-1.7e-01}$ | $0.26931^{+1.9e-04}_{-4.4e-04}$ | $2.43^{+1.5e-01}_{-2.0e-01}$ | $18.85^{+5.2e-01}_{-5.2e-01}$ | $-1.99^{+1.4e+00}_{-1.4e+00}$ | $1.65^{+1.1e-01}_{-1.2e-01}$ | $-0.58^{+1.4e-01}_{-8.8e-02}$
61 | $0.32333^{+3.0e-04}_{-1.7e-06}$ | $2.18^{+7.2e-04}_{-1.6e-01}$ | $19.75^{+1.2e-03}_{-6.1e-01}$ | $-0.01^{+8.8e-03}_{-1.9e+00}$ | $2.02^{+4.0e-03}_{-5.3e-01}$ | $-0.52^{+1.9e-02}_{-1.3e-01}$ | $0.32345^{+4.3e-04}_{-6.0e-04}$ | $2.2^{+2.7e-01}_{-3.1e-01}$ | $19.36^{+7.2e-01}_{-5.2e-01}$ | $-2.3^{+1.4e+00}_{-1.2e+00}$ | $1.44^{+1.6e-01}_{-1.3e-01}$ | $-0.4^{+2.5e-01}_{-1.9e-01}$
62 | $0.08194^{+2.5e-10}_{-1.1e-07}$ | $2.11^{+8.1e-07}_{-1.4e-05}$ | $18.85^{+2.9e-04}_{-2.4e-06}$ | $-0.37^{+3.0e-04}_{-7.9e-06}$ | $2.2^{+2.3e-04}_{-5.7e-07}$ | $-1.04^{+1.1e-05}_{-1.6e-06}$ | $0.08181^{+4.4e-05}_{-1.6e-05}$ | $2.09^{+3.1e-02}_{-7.3e-03}$ | $19.23^{+5.9e-02}_{-1.7e-01}$ | $-0.28^{+1.6e-01}_{-7.4e-02}$ | $2.14^{+3.4e-02}_{-2.9e-02}$ | $-0.8^{+1.0e-02}_{-1.5e-02}$
63 | $0.03215^{+1.8e-05}_{-1.0e-05}$ | $1.92^{+7.7e-02}_{-3.9e-02}$ | $19.57^{+3.1e-02}_{-6.8e-02}$ | $-0.05^{+3.6e-02}_{-4.7e-01}$ | $1.29^{+7.2e-02}_{-1.3e-01}$ | $-0.87^{+7.0e-02}_{-1.3e-01}$ | $0.03185^{+4.2e-04}_{-2.8e-04}$ | $1.83^{+3.6e-01}_{-2.4e-01}$ | $20.0^{+3.0e-01}_{-5.6e-01}$ | $-1.12^{+6.3e-01}_{-9.0e-01}$ | $0.93^{+4.1e-01}_{-3.5e-01}$ | $-0.62^{+2.1e-01}_{-2.3e-01}$
64 | $0.09439^{+2.8e-07}_{-2.3e-05}$ | $2.02^{+1.9e-04}_{-3.0e-02}$ | $18.57^{+1.0e-01}_{-7.7e-04}$ | $-3.0^{+2.2e+00}_{-8.9e-04}$ | $1.94^{+4.7e-02}_{-3.5e-03}$ | $-0.15^{+3.6e-04}_{-2.6e-02}$ | $0.09424^{+3.7e-05}_{-3.0e-05}$ | $1.97^{+2.0e-02}_{-2.1e-02}$ | $18.97^{+6.5e-02}_{-5.9e-02}$ | $-2.99^{+1.1e-01}_{-8.7e-02}$ | $2.06^{+1.1e-02}_{-9.6e-03}$ | $-0.18^{+1.3e-02}_{-1.0e-02}$
65 | $0.02813^{+1.7e-06}_{-3.5e-04}$ | $2.2^{+5.7e-04}_{-2.0e-01}$ | $19.16^{+4.0e-01}_{-8.0e-03}$ | $-0.41^{+2.4e-01}_{-9.2e-03}$ | $1.26^{+3.1e-01}_{-4.1e-03}$ | $-0.93^{+5.2e-01}_{-6.1e-03}$ | $0.02817^{+8.1e-05}_{-1.4e-04}$ | $2.29^{+3.7e-02}_{-4.6e-02}$ | $18.98^{+1.4e-01}_{-6.1e-02}$ | $-1.18^{+3.3e-01}_{-2.5e-01}$ | $1.08^{+4.3e-02}_{-3.4e-02}$ | $-0.71^{+4.9e-02}_{-4.0e-02}$
66 | $0.04313^{+2.2e-07}_{-3.6e-07}$ | $1.57^{+6.9e-03}_{-5.7e-04}$ | $18.39^{+5.8e-04}_{-5.5e-03}$ | $-3.5^{+7.7e-05}_{-4.5e-02}$ | $0.69^{+1.6e-02}_{-2.2e-04}$ | $-1.3^{+1.1e-05}_{-6.0e-04}$ | $0.04301^{+3.0e-04}_{-2.2e-04}$ | $1.85^{+2.7e-01}_{-3.4e-01}$ | $18.85^{+6.0e-01}_{-6.7e-01}$ | $-0.95^{+7.0e-01}_{-9.3e-01}$ | $1.14^{+2.4e-01}_{-2.3e-01}$ | $-0.5^{+2.7e-01}_{-2.0e-01}$

Table 7:  Parameters associated with the line profiles. The column Label
indicates the number that appears with the spectra in Fig.13, 14 and 15.

| MCMC | DNN
---|---|---
Label | $z^{\rm Zelda}$ |  $\log$$V_{\rm exp}$ |  $\log$$N_{\rm H}$ |  $\log$$\tau_{a}$ |  $\log$$EW_{\rm in}$ |  $\rm\log$$W_{\rm in}$ | $z^{\rm Zelda}$ |  $\log$$V_{\rm exp}$ |  $\log$$N_{\rm H}$ |  $\log$$\tau_{a}$ |  $\log$$EW_{\rm in}$ |  $\rm\log$$W_{\rm in}$
|  | [$\rm km/s$] | [$\rm cm^{-2}$] |  | [Å] | [Å] |  | [$\rm km/s$] | [$\rm cm^{-2}$] |  | [Å] | [Å]
67 | $0.08126^{+5.1e-08}_{-2.8e-10}$ | $1.91^{+1.9e-05}_{-1.4e-06}$ | $18.51^{+4.0e-06}_{-1.5e-04}$ | $-2.5^{+2.3e-06}_{-4.7e-04}$ | $1.91^{+2.6e-05}_{-1.5e-05}$ | $-0.33^{+4.0e-06}_{-9.8e-05}$ | $0.0812^{+9.4e-05}_{-1.7e-05}$ | $1.9^{+7.5e-02}_{-2.1e-02}$ | $18.59^{+8.6e-02}_{-3.2e-01}$ | $-2.2^{+2.7e-01}_{-1.8e-01}$ | $2.0^{+1.4e-02}_{-1.3e-02}$ | $-0.43^{+9.8e-03}_{-9.4e-03}$
68 | $0.0324^{+1.8e-07}_{-1.3e-06}$ | $1.25^{+1.9e-02}_{-1.4e-03}$ | $20.12^{+2.7e-03}_{-1.0e-02}$ | $-0.73^{+1.1e-03}_{-1.2e-02}$ | $2.41^{+5.7e-03}_{-5.9e-02}$ | $-1.96^{+9.0e-01}_{-4.3e-02}$ | $0.0325^{+1.5e-05}_{-3.2e-05}$ | $1.6^{+5.6e-02}_{-1.1e-01}$ | $19.5^{+1.1e-01}_{-4.9e-02}$ | $-1.5^{+4.2e-01}_{-1.8e-01}$ | $1.66^{+7.2e-02}_{-5.5e-02}$ | $-0.48^{+1.1e-02}_{-1.1e-02}$
69 | $0.03277^{+1.4e-08}_{-2.6e-07}$ | $1.54^{+5.3e-05}_{-4.4e-04}$ | $18.5^{+1.9e-06}_{-1.9e-04}$ | $-0.0^{+1.3e-05}_{-1.4e-03}$ | $2.7^{+4.6e-04}_{-5.9e-03}$ | $-0.86^{+3.8e-05}_{-2.5e-04}$ | $0.03267^{+1.1e-06}_{-1.1e-06}$ | $1.39^{+2.6e-03}_{-2.4e-03}$ | $18.82^{+6.2e-03}_{-6.2e-03}$ | $-1.65^{+1.5e-02}_{-4.0e-02}$ | $2.13^{+7.5e-04}_{-6.1e-04}$ | $-0.86^{+2.0e-03}_{-2.2e-03}$
70 | $0.04731^{+2.1e-07}_{-3.1e-09}$ | $1.92^{+2.3e-04}_{-3.8e-06}$ | $19.25^{+2.0e-05}_{-1.6e-07}$ | $-0.43^{+9.4e-04}_{-6.4e-05}$ | $2.14^{+2.5e-04}_{-2.3e-05}$ | $-0.24^{+7.4e-06}_{-3.3e-04}$ | $0.04739^{+2.7e-05}_{-6.2e-06}$ | $1.95^{+1.7e-02}_{-8.3e-03}$ | $18.92^{+2.2e-02}_{-1.2e-01}$ | $-0.55^{+7.1e-02}_{-2.2e-01}$ | $1.99^{+8.5e-03}_{-1.8e-02}$ | $-0.36^{+6.4e-03}_{-7.0e-03}$
71 | $0.03321^{+4.6e-05}_{-3.3e-06}$ | $1.6^{+2.0e-01}_{-5.0e-02}$ | $18.85^{+9.3e-03}_{-3.2e-01}$ | $-0.88^{+1.6e-01}_{-1.8e+00}$ | $1.92^{+2.4e-02}_{-9.8e-02}$ | $-0.14^{+5.3e-02}_{-3.8e-02}$ | $0.03318^{+8.3e-06}_{-8.3e-05}$ | $1.66^{+2.5e-02}_{-2.2e-02}$ | $18.75^{+1.9e-01}_{-5.7e-02}$ | $-2.65^{+2.0e-01}_{-1.6e-01}$ | $2.06^{+1.1e-02}_{-1.3e-02}$ | $-0.11^{+8.7e-03}_{-4.4e-02}$
72 | $0.21646^{+4.9e-09}_{-6.7e-08}$ | $1.6^{+7.9e-05}_{-6.8e-07}$ | $19.5^{+3.3e-06}_{-1.3e-08}$ | $-1.0^{+7.3e-08}_{-1.7e-05}$ | $1.85^{+3.0e-05}_{-3.7e-04}$ | $0.27^{+5.1e-05}_{-4.2e-04}$ | $0.21647^{+1.7e-05}_{-2.0e-05}$ | $1.74^{+6.2e-02}_{-5.5e-02}$ | $19.25^{+7.6e-02}_{-5.2e-02}$ | $-0.47^{+6.7e-02}_{-1.2e-01}$ | $1.93^{+4.1e-02}_{-3.8e-02}$ | $0.2^{+3.1e-02}_{-4.0e-02}$
73 | $0.26491^{+6.8e-05}_{-5.0e-05}$ | $1.86^{+9.7e-02}_{-1.8e-01}$ | $18.75^{+3.5e-02}_{-9.7e-03}$ | $-2.0^{+1.8e-02}_{-2.4e-02}$ | $1.37^{+4.8e-02}_{-9.6e-03}$ | $0.03^{+4.5e-02}_{-5.3e-02}$ | $0.26514^{+4.3e-04}_{-7.0e-04}$ | $1.88^{+4.5e-01}_{-4.3e-01}$ | $19.21^{+8.4e-01}_{-9.4e-01}$ | $-1.92^{+9.4e-01}_{-1.0e+00}$ | $2.04^{+3.1e-01}_{-8.0e-01}$ | $-0.29^{+3.5e-01}_{-2.5e-01}$
74 | $0.30045^{+3.3e-07}_{-2.8e-08}$ | $1.78^{+7.3e-06}_{-7.7e-03}$ | $18.75^{+1.6e-05}_{-1.3e-06}$ | $-2.5^{+5.0e-01}_{-6.7e-04}$ | $1.46^{+3.9e-03}_{-1.2e-04}$ | $0.03^{+8.3e-05}_{-4.3e-03}$ | $0.30046^{+1.4e-05}_{-2.1e-05}$ | $1.94^{+2.9e-02}_{-4.6e-02}$ | $18.5^{+8.5e-02}_{-7.0e-02}$ | $-1.4^{+3.8e-01}_{-6.7e-01}$ | $1.55^{+6.4e-02}_{-4.8e-02}$ | $0.0^{+2.9e-02}_{-3.9e-02}$
75 | $0.22224^{+3.5e-09}_{-1.6e-08}$ | $1.95^{+1.3e-06}_{-2.5e-09}$ | $18.74^{+1.4e-05}_{-1.3e-04}$ | $-0.03^{+4.9e-05}_{-1.8e-03}$ | $2.01^{+1.8e-05}_{-4.9e-04}$ | $-0.36^{+8.6e-06}_{-2.4e-05}$ | $0.22217^{+3.3e-05}_{-1.8e-05}$ | $2.01^{+2.8e-02}_{-3.2e-02}$ | $18.56^{+7.1e-02}_{-7.0e-02}$ | $-0.5^{+1.6e-01}_{-3.1e-01}$ | $1.92^{+3.1e-02}_{-3.5e-02}$ | $-0.23^{+2.7e-02}_{-4.4e-02}$
76 | $0.2899^{+4.3e-04}_{-8.8e-06}$ | $0.84^{+1.4e-01}_{-9.4e-01}$ | $17.25^{+1.3e-01}_{-2.3e-01}$ | $-2.63^{+6.2e-01}_{-3.7e-01}$ | $1.31^{+1.0e-01}_{-2.4e-01}$ | $-0.13^{+5.9e-02}_{-3.2e-01}$ | $0.28997^{+3.2e-04}_{-1.1e-03}$ | $1.43^{+4.3e-01}_{-2.2e-01}$ | $18.41^{+8.9e-01}_{-8.0e-01}$ | $-1.09^{+4.8e-01}_{-9.5e-01}$ | $0.78^{+2.6e-01}_{-2.2e-01}$ | $-0.4^{+2.6e-01}_{-2.2e-01}$
77 | $0.237^{+2.7e-05}_{-2.7e-06}$ | $1.6^{+6.5e-03}_{-2.8e-02}$ | $19.75^{+6.3e-05}_{-1.3e-03}$ | $-0.46^{+4.5e-02}_{-2.4e+00}$ | $1.97^{+3.5e-02}_{-4.5e-01}$ | $-0.24^{+1.6e-01}_{-1.2e-03}$ | $0.2368^{+7.3e-05}_{-7.4e-05}$ | $1.67^{+8.9e-02}_{-9.6e-02}$ | $19.74^{+1.4e-01}_{-1.6e-01}$ | $-2.98^{+9.2e-01}_{-6.5e-01}$ | $1.5^{+6.1e-02}_{-5.5e-02}$ | $0.04^{+4.8e-02}_{-6.2e-02}$
78 | $0.20493^{+4.1e-09}_{-2.4e-07}$ | $1.68^{+8.1e-05}_{-2.1e-05}$ | $18.36^{+9.8e-04}_{-4.1e-05}$ | $-0.0^{+1.2e-07}_{-4.7e-05}$ | $1.86^{+7.0e-04}_{-4.3e-05}$ | $-0.1^{+7.0e-05}_{-1.9e-05}$ | $0.20492^{+1.1e-05}_{-2.1e-05}$ | $1.48^{+3.4e-02}_{-4.0e-02}$ | $18.32^{+1.3e-01}_{-8.7e-02}$ | $0.05^{+4.5e-02}_{-8.7e-02}$ | $2.1^{+1.1e-02}_{-1.9e-02}$ | $-0.28^{+3.1e-02}_{-2.2e-02}$
79 | $0.19218^{+1.4e-08}_{-5.1e-10}$ | $2.15^{+4.5e-07}_{-4.7e-05}$ | $19.13^{+1.7e-06}_{-5.1e-05}$ | $-0.49^{+1.9e-04}_{-7.5e-06}$ | $1.7^{+4.3e-05}_{-2.0e-06}$ | $0.34^{+5.6e-09}_{-4.2e-06}$ | $0.19209^{+1.8e-04}_{-1.0e-04}$ | $2.14^{+1.1e-01}_{-1.4e-01}$ | $19.23^{+1.9e-01}_{-3.1e-01}$ | $-0.65^{+2.9e-01}_{-7.0e-01}$ | $1.6^{+5.9e-02}_{-6.2e-02}$ | $0.36^{+5.2e-02}_{-3.9e-02}$
80 | $0.30088^{+2.9e-05}_{-2.2e-07}$ | $2.35^{+1.6e-02}_{-4.9e-03}$ | $18.93^{+1.4e-03}_{-3.1e-02}$ | $-2.0^{+4.2e-04}_{-2.0e+00}$ | $1.47^{+5.1e-04}_{-1.3e-02}$ | $-0.85^{+9.9e-03}_{-1.3e-03}$ | $0.3008^{+7.0e-05}_{-6.6e-05}$ | $2.43^{+3.4e-02}_{-2.8e-02}$ | $19.28^{+4.9e-02}_{-1.5e-01}$ | $-1.15^{+4.5e-01}_{-9.7e-01}$ | $1.42^{+3.9e-02}_{-3.3e-02}$ | $-0.61^{+4.1e-02}_{-7.2e-02}$
81 | $0.33275^{+2.3e-06}_{-8.5e-08}$ | $1.85^{+8.8e-05}_{-3.7e-06}$ | $18.78^{+4.8e-04}_{-1.4e-02}$ | $-2.0^{+3.5e-04}_{-1.2e+00}$ | $1.72^{+2.1e-04}_{-3.5e-03}$ | $0.06^{+7.4e-03}_{-9.4e-04}$ | $0.33271^{+7.1e-05}_{-9.6e-05}$ | $2.06^{+5.2e-02}_{-7.3e-02}$ | $18.7^{+1.9e-01}_{-1.4e-01}$ | $-1.57^{+5.9e-01}_{-1.2e+00}$ | $1.71^{+4.9e-02}_{-4.4e-02}$ | $0.01^{+4.8e-02}_{-5.9e-02}$
82 | $0.26927^{+7.6e-08}_{-1.6e-08}$ | $1.08^{+1.0e-03}_{-3.8e-05}$ | $20.0^{+3.5e-05}_{-8.1e-08}$ | $-0.5^{+2.9e-07}_{-8.5e-05}$ | $2.17^{+3.1e-04}_{-4.6e-05}$ | $-0.08^{+9.3e-04}_{-2.4e-05}$ | $0.26919^{+6.8e-05}_{-7.5e-05}$ | $1.13^{+7.0e-02}_{-5.1e-02}$ | $19.9^{+1.3e-01}_{-1.5e-01}$ | $-1.34^{+5.7e-01}_{-3.9e-01}$ | $1.65^{+1.6e-01}_{-1.2e-01}$ | $0.06^{+6.3e-02}_{-6.7e-02}$
83 | $0.29686^{+6.1e-06}_{-1.2e-07}$ | $1.78^{+3.8e-05}_{-1.9e-03}$ | $19.74^{+4.1e-04}_{-1.4e-03}$ | $-0.0^{+1.4e-04}_{-1.4e-01}$ | $2.58^{+2.8e-04}_{-2.8e-01}$ | $-0.38^{+1.9e-04}_{-1.8e-03}$ | $0.2968^{+1.3e-04}_{-4.8e-05}$ | $1.88^{+5.9e-02}_{-5.1e-02}$ | $19.61^{+7.5e-02}_{-1.7e-01}$ | $-0.92^{+7.9e-02}_{-3.1e-01}$ | $2.01^{+7.1e-02}_{-8.3e-02}$ | $-0.22^{+8.0e-02}_{-3.9e-02}$
84 | $0.29329^{+7.5e-04}_{-4.9e-06}$ | $2.16^{+3.4e-01}_{-1.3e-02}$ | $19.75^{+2.5e-03}_{-4.5e-01}$ | $-0.0^{+3.1e-03}_{-9.6e-01}$ | $1.65^{+8.4e-03}_{-4.9e-01}$ | $-0.52^{+1.9e-04}_{-3.6e-01}$ | $0.29344^{+2.9e-04}_{-3.5e-04}$ | $2.39^{+9.6e-02}_{-1.8e-01}$ | $19.78^{+2.4e-01}_{-2.0e-01}$ | $-0.61^{+3.3e-01}_{-5.1e-01}$ | $1.2^{+1.1e-01}_{-1.2e-01}$ | $-0.42^{+1.3e-01}_{-1.8e-01}$
85 | $0.03804^{+9.3e-08}_{-1.2e-08}$ | $2.24^{+1.0e-05}_{-8.3e-05}$ | $19.47^{+3.2e-05}_{-3.1e-04}$ | $-0.0^{+2.0e-07}_{-1.7e-04}$ | $1.68^{+1.9e-05}_{-2.0e-04}$ | $-0.78^{+6.4e-05}_{-7.5e-06}$ | $0.03819^{+1.8e-05}_{-2.3e-05}$ | $2.37^{+2.7e-02}_{-1.0e-01}$ | $19.21^{+3.5e-02}_{-6.4e-02}$ | $-0.52^{+1.4e-01}_{-2.9e-01}$ | $1.43^{+2.4e-02}_{-2.8e-02}$ | $-0.81^{+1.6e-02}_{-3.3e-02}$
86 | $0.03203^{+5.1e-05}_{-8.3e-05}$ | $1.49^{+1.1e-01}_{-1.7e-01}$ | $21.12^{+1.9e-01}_{-2.5e-02}$ | $-0.27^{+1.6e-01}_{-3.0e-01}$ | $2.54^{+1.9e-01}_{-1.5e-01}$ | $-0.02^{+4.7e-02}_{-1.6e-01}$ | $0.03206^{+1.6e-04}_{-1.2e-04}$ | $1.48^{+1.6e-01}_{-1.4e-01}$ | $20.98^{+1.5e-01}_{-2.1e-01}$ | $-0.71^{+3.2e-01}_{-2.6e-01}$ | $1.76^{+3.8e-01}_{-2.7e-01}$ | $-0.3^{+2.8e-01}_{-3.7e-01}$
87 | $0.03018^{+5.6e-08}_{-7.0e-07}$ | $2.36^{+2.4e-05}_{-2.1e-04}$ | $18.06^{+2.9e-03}_{-2.0e-04}$ | $-2.1^{+1.6e-02}_{-1.0e-03}$ | $1.58^{+3.3e-04}_{-1.1e-05}$ | $-0.7^{+5.3e-05}_{-4.4e-08}$ | $0.03008^{+4.8e-05}_{-1.8e-04}$ | $2.31^{+3.1e-02}_{-1.0e-01}$ | $18.53^{+3.4e-01}_{-2.1e-01}$ | $-1.24^{+2.7e-01}_{-3.1e-01}$ | $1.66^{+4.5e-02}_{-3.4e-02}$ | $-0.71^{+2.2e-02}_{-2.4e-02}$
88 | $0.03045^{+3.5e-07}_{-1.4e-04}$ | $1.78^{+3.2e-04}_{-2.7e-01}$ | $20.57^{+2.4e-01}_{-3.4e-04}$ | $-0.17^{+2.6e-04}_{-3.9e-01}$ | $2.1^{+9.6e-04}_{-1.1e-01}$ | $-0.52^{+1.9e-02}_{-1.0e-01}$ | $0.03041^{+1.0e-04}_{-1.6e-04}$ | $1.64^{+1.6e-01}_{-2.1e-01}$ | $20.55^{+1.1e-01}_{-1.2e-01}$ | $-0.46^{+1.9e-01}_{-2.1e-01}$ | $1.77^{+2.5e-01}_{-1.7e-01}$ | $-0.63^{+1.5e-01}_{-1.4e-01}$
89 | $0.03751^{+4.6e-06}_{-8.9e-08}$ | $1.85^{+3.5e-05}_{-7.8e-04}$ | $20.25^{+9.4e-05}_{-6.5e-05}$ | $-0.04^{+2.0e-04}_{-2.6e-03}$ | $1.96^{+2.1e-03}_{-7.0e-04}$ | $-1.3^{+8.5e-04}_{-2.8e-02}$ | $0.03759^{+1.6e-04}_{-1.7e-04}$ | $1.96^{+1.3e-01}_{-1.5e-01}$ | $20.25^{+2.7e-01}_{-1.9e-01}$ | $-0.22^{+1.2e-01}_{-2.6e-01}$ | $1.68^{+1.9e-01}_{-2.1e-01}$ | $-0.63^{+1.1e-01}_{-1.2e-01}$
90 | $0.08488^{+8.7e-06}_{-1.2e-03}$ | $2.74^{+4.4e-03}_{-7.8e-01}$ | $19.68^{+7.9e-02}_{-5.7e-01}$ | $-0.09^{+8.4e-02}_{-2.6e+00}$ | $0.91^{+6.6e-01}_{-2.7e-02}$ | $-1.07^{+5.5e-01}_{-1.9e-02}$ | $0.08471^{+1.9e-04}_{-1.1e-03}$ | $2.61^{+1.3e-01}_{-2.2e-01}$ | $19.47^{+5.7e-01}_{-3.0e-01}$ | $-2.58^{+1.9e+00}_{-1.0e+00}$ | $1.35^{+5.5e-01}_{-3.1e-01}$ | $-0.59^{+1.8e-01}_{-1.4e-01}$
91 | $0.28123^{+1.2e-07}_{-4.3e-08}$ | $1.85^{+2.4e-06}_{-9.8e-05}$ | $19.25^{+7.7e-06}_{-4.1e-04}$ | $-0.0^{+5.0e-06}_{-2.6e-04}$ | $1.75^{+2.0e-04}_{-1.6e-04}$ | $-0.56^{+8.1e-05}_{-3.5e-04}$ | $0.28131^{+1.8e-04}_{-2.0e-04}$ | $2.01^{+1.4e-01}_{-1.7e-01}$ | $19.22^{+2.1e-01}_{-2.3e-01}$ | $-0.22^{+1.6e-01}_{-4.0e-01}$ | $1.54^{+2.0e-01}_{-1.4e-01}$ | $-0.56^{+1.9e-01}_{-1.6e-01}$
92 | $0.03307^{+7.6e-06}_{-3.6e-05}$ | $1.73^{+2.6e-02}_{-1.3e-02}$ | $20.64^{+3.2e-02}_{-3.0e-02}$ | $-0.12^{+2.0e-02}_{-4.2e-02}$ | $2.67^{+1.5e-01}_{-2.2e-02}$ | $-0.6^{+1.4e-01}_{-1.4e+00}$ | $0.03344^{+1.0e-04}_{-8.8e-05}$ | $2.11^{+2.1e-02}_{-2.6e-01}$ | $19.94^{+1.4e-01}_{-1.1e-01}$ | $-0.04^{+1.3e-02}_{-1.5e-01}$ | $1.8^{+9.1e-02}_{-6.1e-02}$ | $-0.59^{+1.5e-01}_{-8.4e-02}$
93 | $0.24833^{+3.3e-07}_{-7.9e-05}$ | $1.44^{+1.2e-01}_{-1.3e-03}$ | $17.01^{+1.0e+00}_{-6.0e-03}$ | $-3.0^{+1.3e-01}_{-2.0e-03}$ | $1.17^{+1.3e-02}_{-1.2e-03}$ | $-0.55^{+9.4e-04}_{-3.4e-02}$ | $0.24813^{+1.3e-04}_{-2.7e-04}$ | $1.8^{+2.2e-01}_{-1.8e-01}$ | $18.01^{+7.5e-01}_{-5.9e-01}$ | $-2.9^{+1.1e+00}_{-5.5e-01}$ | $1.1^{+8.5e-02}_{-7.3e-02}$ | $-0.44^{+1.8e-01}_{-1.2e-01}$
94 | $0.21511^{+7.0e-08}_{-2.1e-06}$ | $1.32^{+1.3e-02}_{-4.9e-05}$ | $20.55^{+1.1e-04}_{-8.8e-03}$ | $-1.03^{+3.9e-04}_{-8.5e-03}$ | $2.17^{+8.5e-05}_{-2.3e-02}$ | $-0.7^{+7.3e-02}_{-4.6e-04}$ | $0.21506^{+2.0e-04}_{-9.4e-05}$ | $1.55^{+3.0e-01}_{-1.5e-01}$ | $20.31^{+1.1e-01}_{-4.1e-01}$ | $-2.5^{+1.0e+00}_{-8.4e-01}$ | $1.53^{+1.5e-01}_{-1.1e-01}$ | $-0.42^{+2.1e-01}_{-2.6e-01}$
95 | $0.28026^{+1.6e-05}_{-8.1e-05}$ | $2.78^{+2.0e-01}_{-7.9e-02}$ | $17.75^{+2.5e-01}_{-2.6e-01}$ | $-1.48^{+4.8e-01}_{-1.5e+00}$ | $2.08^{+2.7e-02}_{-3.4e-02}$ | $-0.35^{+1.4e-02}_{-6.5e-03}$ | $0.28017^{+9.4e-05}_{-9.9e-05}$ | $2.75^{+6.7e-02}_{-1.0e-01}$ | $17.47^{+6.2e-01}_{-3.7e-01}$ | $-2.84^{+1.9e+00}_{-9.6e-01}$ | $2.07^{+7.1e-02}_{-8.4e-02}$ | $-0.17^{+6.7e-02}_{-7.6e-02}$
96 | $0.20497^{+2.8e-08}_{-1.6e-10}$ | $2.22^{+1.4e-05}_{-6.7e-08}$ | $19.33^{+7.5e-07}_{-7.1e-05}$ | $-0.55^{+3.4e-06}_{-2.6e-04}$ | $1.63^{+2.4e-07}_{-4.2e-05}$ | $-0.8^{+3.5e-05}_{-1.9e-07}$ | $0.20513^{+9.3e-05}_{-1.6e-04}$ | $2.42^{+4.6e-02}_{-1.3e-01}$ | $19.22^{+1.4e-01}_{-1.5e-01}$ | $-0.49^{+3.5e-01}_{-5.1e-01}$ | $1.61^{+6.6e-02}_{-8.9e-02}$ | $-0.63^{+6.9e-02}_{-3.6e-02}$
97 | $0.26491^{+3.1e-08}_{-4.3e-07}$ | $2.18^{+1.6e-05}_{-4.6e-07}$ | $19.26^{+8.1e-05}_{-1.2e-03}$ | $-0.53^{+2.3e-03}_{-5.7e-02}$ | $1.62^{+3.5e-04}_{-1.4e-02}$ | $0.08^{+2.0e-03}_{-1.2e-04}$ | $0.26484^{+2.4e-04}_{-2.1e-04}$ | $2.12^{+1.4e-01}_{-2.0e-01}$ | $19.33^{+3.2e-01}_{-4.7e-01}$ | $-1.99^{+8.0e-01}_{-1.2e+00}$ | $1.36^{+8.8e-02}_{-7.9e-02}$ | $0.2^{+9.1e-02}_{-1.6e-01}$


# Gate-defined Two-dimensional Hole and Electron Systems in an Undoped InSb
Quantum Well

Zijin Lei<EMAIL_ADDRESS>Solid State Physics Laboratory, ETH Zurich,
CH-8093 Zurich, Switzerland  Erik Cheah  Solid State Physics Laboratory, ETH
Zurich, CH-8093 Zurich, Switzerland  Filip Krizek  Solid State Physics
Laboratory, ETH Zurich, CH-8093 Zurich, Switzerland  IBM Research Europe -
Zurich, Säumerstrasse 4, 8803 Rüschlikon, Switzerland  Rüdiger Schott  Solid
State Physics Laboratory, ETH Zurich, CH-8093 Zurich, Switzerland  Thomas
Bähler  Solid State Physics Laboratory, ETH Zurich, CH-8093 Zurich,
Switzerland  Peter Märki  Solid State Physics Laboratory, ETH Zurich, CH-8093
Zurich, Switzerland  Werner Wegscheider  Solid State Physics Laboratory, ETH
Zurich, CH-8093 Zurich, Switzerland  Mansour Shayegan  Department of
Electrical and Computer Engineering, Princeton University, Princeton, New
Jersey 08544, USA  Thomas Ihn  Solid State Physics Laboratory, ETH Zurich,
CH-8093 Zurich, Switzerland  ETH Zurich Quantum Center, CH-8093 Zurich,
Switzerland  Klaus Ensslin  Solid State Physics Laboratory, ETH Zurich,
CH-8093 Zurich, Switzerland  ETH Zurich Quantum Center, CH-8093 Zurich,
Switzerland

###### Abstract

Quantum transport measurements are performed in gate-defined, high-quality,
two-dimensional hole and electron systems in an undoped InSb quantum well. For
both polarities, the carrier systems show tunable spin-orbit interaction as
extracted from weak anti-localization measurements. The effective mass of InSb
holes strongly increases with carrier density as determined from the
temperature dependence of Shubnikov–de Haas oscillations. Coincidence
measurements in a tilted magnetic field are performed to estimate the spin
susceptibility of the InSb two-dimensional hole system. The g-factor of the
two-dimensional hole system decreases rapidly with increasing carrier density.

##  I Introduction

Silicon, with both p-type and n-type transport characteristics, is the
building block of today’s complementary metal–oxide–semiconductor (CMOS)
circuits. However, not every semiconductor exhibits high quality for both
carrier types, and it is even more challenging to achieve ambipolar operation
in a single device. Nowadays, in the widely used silicon CMOS technology, the
integration of p- and n-type components is often achieved with sophisticated
fabrication techniques. For example, ion implantation is adopted to reverse
the polarity on the wafer locally [1, 2, 3]. In contrast to the established
processes for silicon, many challenges still remain in devices based on III-V
compounds, whose mobilities are in general much higher. The p- and n-type
components in III-V devices mostly rely on different doping methods and
materials, which have to be combined with specific metals for realizing ohmic
contacts [4, 5]. So far, most approaches rely on complex bilayer system
designs, where p- and n-type layers are grown subsequently along the growth
direction. There, the two-dimensional electron and hole gases (2DEG and 2DHG)
are produced either by different choice of dopants, different semiconductor
materials, or by electrostatic gating using both a top-gate and a back gate
[6, 7, 8, 9, 10, 11, 12]. Recently, incorporating both a high-quality 2DHG and
a 2DEG into a single device became possible in 2D materials, for example, in
graphene, MoS2 or black phosphorous [13, 14, 15, 16], since in these materials
one can contact carriers in both conduction bands and valence bands [17, 18,
19, 20, 21].

Narrow-gap III-V compounds, such as InAs and InSb, have also been proposed as
systems that can potentially host both p- and n-type polarities [22, 23]. InSb
is known for the small effective mass of electrons and light holes in the bulk
material, high carrier mobility at room temperature, strong spin-orbit
interaction (SOI), and a large g-factor in both the n- and p-type regimes [24,
25, 26, 27, 28, 29, 30]. These unique properties are interesting for potential
applications such as high-frequency electronics [31], optoelectronics [32],
and spintronics [33]. More recently, InSb, together with InAs, has attracted
attention as a potential platform for topological quantum information
processing [34, 35, 36]. Various low-dimensional n-type InSb systems have been
studied, including quantum wells [37, 38], nanowires [39, 40], and other as-
grown nanomaterials [41, 42, 43]. In previous work on n-type quantum wells we
have shown that the parabolic band approximation agrees with experimental data
[44, 45, 46]. On the contrary, for lower-dimensional, p-type InSb, similar to
other p-type III-V compounds, the band structures are predicted to be more
complicated than their n-type counterparts. In general, they strongly depend
on the dimensions, shape of the potential well, global or local strain fields,
SOI, and the carrier confinement [47, 48, 49]. Since the growth and processing
methods for p-type devices are not as mature as for n-type, there is a lack of
in-depth experimental studies of p-type InSb [50]. So far, quantum transport
measurements have been reported for p-type InSb nanowires [51, 52] and quantum
wells [50]. Also, room temperature experiments on p- and n-type InSb field-
effect transistors were published [53]. However, limited by both the material
quality and device design, the important properties of p-type InSb, such as
scattering mechanisms, effective mass, g-factor, and SOI, have not been
precisely characterized.

In this work, we present quantum transport measurements on a gate-defined 2DHG
and 2DEG in an ambipolar Hall bar device based on a single, undoped InSb
quantum well. Both the 2DHG and the 2DEG have relatively high mobility, even
though the quantum well is close to the surface to allow for electrostatic
top-gating. Shubnikov–-de Haas (SdH) oscillations and the integer quantum Hall
effect are observed for both carrier polarities. From the temperature
dependence of SdH oscillations, we extract the effective mass of InSb holes
and observe a large band non-parabolicity. Moreover, the strong and tunable
SOI of both the 2DHG and the 2DEG is characterized by weak antilocalization
(WAL) measurements. Finally, a coincidence measurement of the g-factor in the
2DHG is performed, showing a strong dependence of the spin susceptibility on
the carrier density.

##  II Sample preparation

Figure 1: (a) High-angle-annular-dark-field scanning transmission electron
microscopic (HAADF STEM) image of the InSb quantum well and neighboring
In0.9Al0.1Sb barriers. There is no extended defects observed in the quantum
well area. (b) Two-terminal conductance $G_{1,4}$ measured between contacts 1
and 4 vs. top-gate voltage $V_{\rm{TG}}$. A constant voltage of 0.5 mV is
applied at 1.7 K. The blue and red traces show the down- and up-sweep,
respectively. Inset: a schematic diagram of the Hall bar sample and the
definition of tilt angle $\theta$. A global top-gate covers both the 2DEG or
2DHG and the contacts.

The InSb quantum well sample we discuss here is grown on a GaAs (100)
substrate by molecular beam epitaxy (MBE). The metamorphic buffer
heterostructure, which is based on a previous publication by Lehner et al.
[37], is used to filter the crystallographical defects related to the lattice
mismatch. As shown in the high-angle-annular-dark-field (HAADF) scanning-
transmission-electron-microscope (STEM) image of the InSb quantum well sample
in Fig. 1 (a), a 21.5-nm-thick InSb quantum well is sandwiched in-between
$\rm{In_{0.9}Al_{0.1}Sb}$ confinement barriers, with an 8.5-nm-thick top
barrier and a 2-$\rm{\mu}$m-thick bottom barrier. We note that there is no
intentional doping during growth.

After defining a standard
$400\times$200\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}^{2}$$ Hall bar
using chemical etching, Ti/Au ohmic contacts are evaporated. This combination
of metals has been reported to provide ohmic contacts for both p- and n-type
InSb [51, 52]. In the next step, the sample is coated with a 40-nm-thick
aluminum oxide dielectric layer using atomic layer deposition (ALD) at
$150\text{\,}\mathrm{\SIUnitSymbolDegree}\mathrm{C}$. Finally, a Ti/Au top-
gate covering the Hall bar and overlapping the contacts is deposited by
electron-beam evaporation, as illustrated in the inset of Fig. 1(b).

Figure 2: (a) and (b) Density and mobility vs. $V_{\rm{TG}}$ measured in p and
n-type regimes, respectively. (c) The mobilities vs. densities for both p- and
n-type. The dashed lines are linear fits.  Figure 3:  (a) The SdH oscillations
and integer quantum Hall effects of 2DHG. The filling factors are labeled in
the figure. (b) and (c) The Landau fan diagrams of 2DHG. Both $\rho_{xx}$ and
$\rho_{xy}$ are presented. (d), (e), and (f) Same as (a), (b), and (c), but in
the n-type regime. The filling factors for 2DHG and 2DEG are labeled as
$\nu_{\rm{h}}$ and $\nu_{\rm{e}}$, respectively. The white dashed lines
correspond to the same carrier density $4\times 10^{11}\>\rm{cm^{-2}}$, for
both p- and n-types.

##  III SdH oscillations and integer quantum Hall effects

Transport measurements are performed using standard low-frequency lock-in
techniques in a cryostat with a base temperature of 1.7 K. Figure 1(b)
presents the two-terminal conductance $G_{1,4}$ between contacts 1 and 4 as a
function of top-gate voltage $V_{\rm{TG}}$. The conductance $G_{1,4}$
increases with increasing $V_{\rm{TG}}$ for positive voltages and with
decreasing $V_{\rm{TG}}<$-2\text{\,}\mathrm{V}$$. In-between, the sample is
insulating. This indicates that by sweeping $V_{\rm{TG}}$ from negative to
positive, the polarity of the charge carriers changes from p-type to n-type,
while in-between, the Fermi level traverses the bandgap. Comparing the up- and
down-sweep of $V_{\rm{TG}}$ we find hysteresis. Therefore, we keep the sweep-
direction fixed for the following measurements. In the n-type regime, down-
sweeps of $V_{\rm{TG}}$ are used, while up-sweeps are used in the p-type
regime.

Now we measure the Hall bar sample with the standard method where a constant
current is applied from contact 1 to contact 4. The longitudinal (between
contact 2 and 3) and the transverse (between contact 3 and 5) resistivities,
$\rho_{xx}$ and $\rho_{xy}$, are measured. Figures 2(a) and (b) show the
densities $n_{\rm{h}}$ of holes and $n_{\rm{e}}$ of electrons, determined from
the classical Hall resistance ($\rho_{xy}$) at small magnetic fields, and the
corresponding Drude mobilities $\mu_{\rm{h}}$ and $\mu_{\rm{e}}$ obtained from
$\rho_{xx}$ as a function of $V_{\rm{TG}}$. Figure 2(c) summarizes the data
into plots of density vs. mobility for electrons and holes. The hole density
$n_{\rm{h}}$ increases up to
$5.0\text{\times}{10}^{11}\text{\,}\mathrm{c}\mathrm{m}^{-2}$ with decreasing
$V_{\rm{TG}}$ and reaches the mobility $\mu_{\rm
h}=$21\,000\text{\,}\mathrm{c}\mathrm{m}^{2}\mathrm{/}\mathrm{V}\mathrm{s}$$.
For comparison, also $n_{\rm{e}}$ increases up to
$5\text{\times}{10}^{11}\text{\,}\mathrm{c}\mathrm{m}^{-2}$ with increasing
$V_{\rm{TG}}$, but the corresponding $\mu_{\rm e}$ reaches its maximum of
$1.1\text{\times}{10}^{5}\text{\,}\mathrm{c}\mathrm{m}^{2}\mathrm{/}\mathrm{V}\mathrm{s}$
at $n_{\rm{e}}=$4.5\text{\times}{10}^{11}\text{\,}\mathrm{c}\mathrm{m}^{-2}$$
and starts to decrease beyond. The decreasing electron mobility at increasing
density suggests that interface roughness scattering starts to dominate.

Figures 3(a) and (d) present $\rho_{xx}$ and $\rho_{xy}$ vs. magnetic field
$B$ in the hole and the electron regimes, where $V_{\rm{TG}}$ =
$-5.1\text{\,}\mathrm{V}$ and $1.5\text{\,}\mathrm{V}$, respectively. The
densities of both holes and electrons are
$4\text{\times}{10}^{11}\text{\,}\mathrm{c}\mathrm{m}^{-2}$. We find
pronounced SdH oscillations in $\rho_{xx}$ and quantum Hall plateaus close to
$\rho_{xy}=h/\nu e^{2}$, where $\nu$ is the integer filling factor, in both
the p-type and n-type regimes. The 2DEG and the 2DHG show a Zeeman splitting
in the SdH oscillations starting at and below very different filling factors
($\nu=3$ for holes and $\nu=9$ for electrons), indicating significant
differences in effective mass and g-factors between electrons and holes.
Related measurements are discussed further below. Landau fan diagrams of
$\rho_{xx}$ and $\rho_{xy}$ for both 2DHG and 2DEG are presented in Figures
3(b,c) and (e,f), respectively. The corresponding filling factors for 2DHG and
2DEG are labeled as $\nu_{\rm{h}}$ and $\nu_{\rm{e}}$, respectively. Both
regimes show the fan diagram of a single subband within the range of our
measurements. Furthermore, for both 2DHG and 2DEG, there is no low-field,
classical-positive magnetoresistance observed in $\rho_{xx}$ confirming that
only a single subband is occupied for both polarities [38]. These measurements
demonstrate the high quality of both the single-channel 2DEG and 2DHG
electrically induced in the same device. Both of them are single-band systems,
without the second electronic subband populated and without a noticeable
splitting of the heavy-hole band due to SOI [54]. We can quantitatively
estimate $\Delta n$, the carrier density at which the second electronic
subband starts to be populated. Using the infinite-high quantum well model,
$\Delta n$ is deduced to be
$\sim$10\text{\times}{10}^{11}\text{\,}\mathrm{c}\mathrm{m}^{-2}$$, which is
much larger than $n_{\rm{e}}$ and $n_{\rm{h}}$ in this work.

##  IV Effective mass and band non-parabolicity of 2D holes

Figure 4: Effective mass measurement for 2DHG. (a) Temperature dependence of
SdH oscillations with $n_{\rm{h}}=3.95\times 10^{11}$ cm-2. (b) Dingle factor
fitting at different $B$. The squares are data and the lines are fitted
curves. (c) The fitting of the quantum lifetime at 1.75 K. Inset: effective
mass of holes vs. $B$ when $n_{\rm{h}}=3.95\times 10^{11}$ cm-2. (d) Effective
mass as a function of carrier density. Error bars here are determined with the
deviation of the fitting results. Inset: Dependence of $\tau_{\rm{q}}$ on
$n_{\rm{h}}$ at 1.75 K.

In our previous work, we have measured the effective mass of electrons in InSb
quantum wells with the same thickness to be $\sim 0.016$ $m_{\rm{e}}$ [38, 44,
46]. In the following, we present measurements of the effective mass of holes
in the sample through the temperature dependence of the SdH oscillations at
small magnetic fields. With the increase of the measurement temperature, the
amplitude of the SdH oscillations will decrease until the oscillations vanish
completely. The lower the effective mass, the higher the temperature at which
the SdH oscillations can still be observed. The temperature dependence of the
SdH oscillation is usually described by the Ando formula valid in the low
magnetic field regime [55].

Figure 4(a) shows SdH oscillations for a hole density
$n_{\rm{h}}=$3.95\text{\times}{10}^{11}\text{\,}\mathrm{c}\mathrm{m}^{-2}$$,
measured at temperatures from $1.75\text{\,}\mathrm{K}$ to
$7\text{\,}\mathrm{K}$. The oscillating part of the resistivity,
$\Delta\rho_{xx}$, is obtained by subtracting the smooth background
($\bar{\rho}_{xx}$) from the measured $\rho_{xx}$. We select the local maxima
and minima in the traces of $\Delta\rho_{xx}$ vs. $B$ to extract the
temperature dependence of the SdH oscillations and plot the quantity
$\ln{(\Delta\rho_{xx}T_{0}/(\bar{\rho}_{xx}T))}$ as a function of temperature
in Figure 4(b), where $T_{0}$ is the lowest temperature at which we measured
the SdH oscillations. The effective mass $m^{\star}$ is then obtained by
fitting to the function $-\ln(\sinh(2\pi^{2}m^{\star}k_{\mathrm{B}}T/\hbar
eB))$ with the effective mass being the fitting parameter. The deduced
effective mass of holes is $m^{\star}=0.046\pm 0.003$ $m_{\rm{e}}$, roughly
constant within the magnetic field range of the measurement (inset of Fig.
4(c)).

The quantum lifetime was extracted by fitting the data in Fig. 4(c) with the
Dingle factor and using the effective mass determined above. At
$1.75\text{\,}\mathrm{K}$, we get a quantum lifetime of $\tau_{\rm{q}}=110\pm
10\>\rm{fs}$ from the slope of the linear fit
$\ln{\Delta\rho_{xx}/(\Delta\bar{\rho}_{xx})f(B,T)}$ vs. $1/B$, where
$f(B,T)=(2\pi^{2}k_{B}T/\hbar\omega_{c})/\sinh{(2\pi^{2}k_{B}T/\hbar\omega_{c})}$.
The quantum lifetime $\tau_{\rm{q}}$ is roughly a constant within the
temperature range investigated. Considering the scattering time
$\tau_{\rm{tr}}=0.435$ ps deduced from the Drude model at this carrier
density, the Dingle ratio is $\tau_{\rm{tr}}/\tau_{\rm{q}}\approx 4$. With the
same carrier density in the n-type regime, the Dingle ratio is about 20.8 in
this device. The large difference of the Dingle ratio may imply a large
difference in the scattering mechanism in 2DHG and 2DEG.

We repeat the measurement of the effective mass at different hole densities.
In Fig. 4(d), $m^{\star}$ is plotted as a function of $n_{\rm{h}}$. With the
increase of $n_{\rm{h}}$ from 3.1 to
$4.85\text{\times}{10}^{11}\text{\,}\mathrm{c}\mathrm{m}^{-2}$, the Fermi
level shifts deeper into the valence band. At the same time, $m^{\star}$
increases from 0.03 to 0.057 $m_{\rm{e}}$. This indicates a very large band
non-parabolicity. Compared to the effective mass of holes in InSb bulk
material, our results show a slightly larger value than for the light holes
(0.015 $m_{\rm{e}}$) but a much smaller value than the effective mass of heavy
holes ($\sim 0.43$ $m_{\rm{e}}$). This indicates that the band structure of
p-type InSb may have similarities to p-type GaAs with the light-hole heave-
hole degeneracy at the $\Gamma$ point being lifted due to the confinement in
the growth direction. Theory predicts that the in-plane effective mass of the
heavy-hole band is smaller than that of the light-hole band (mass inversion)
[54]. The effective mass depends strongly on density, which either implies
that bands are strongly altered by self-consistent effects, or strongly non-
parabolic bands, or both [48]. Compared with previous publications, our
$m^{\star}$ are within the range of the results obtained through theoretical
calculations and cyclotron resonance measurements [49, 50]. Nevertheless, the
value of $m^{\star}$ for holes depends on many factors, including the shape
and width of the quantum wells, the strain, the carrier densities, and
possibly the SOI. Using the measured $m^{\star}$, we also obtain
$\tau_{\rm{q}}$ as a function of $n_{\rm{h}}$ (inset of Fig. 4(d)), showing an
increase of $\tau_{\rm{q}}$ with increasing carrier density, similar to
mobility [Fig. 2(c)].

Using the values determined for $m^{\star}$, we qualitatively estimate the
background impurity density $n_{\rm{imp}}$ in the heterostructure. Since
$\mu_{\rm{h(e)}}$ is proportional to $n_{\rm{h(e)}}$ in a wide range, we may
assume that scattering by ionized background impurities is the dominant
scattering mechanism for both p- and n-type conduction [56]. Therefore, we
estimate $n_{\rm{imp}}$ with

$\begin{split}\frac{1}{\tau_{\rm{tr}}}&\approx\frac{n_{\rm{imp}}m^{\star}}{2\pi\hbar^{3}k_{\rm{F}}^{3}}(\frac{e^{2}}{2\epsilon_{0}\epsilon_{\rm{InSb}}})^{2}\\\
&\int_{0}^{2k_{F}}\frac{qdq}{(q+q_{\rm{TF}})^{2}\sqrt{1-(q/2k_{\rm{F}})^{2}}},\end{split}$
(1)

where $q_{\rm{TF}}$ is the Thomas–Fermi wave vector, $\epsilon_{\rm{InSb}}$ is
the dielectric constant of InSb, and $\tau_{\rm{tr}}$ is the mean scattering
time due to background impurity scattering [57, 58]. We calculate
$n_{\rm{imp}}$ in the range where mobilities are proportional to carrier
densities for 2DHG and 2DEG, respectively. $n_{\rm{imp}}$ is between
$1.7\text{\times}{10}^{17}\text{\,}\mathrm{c}\mathrm{m}^{-3}$ and
$3.8\text{\times}{10}^{17}\text{\,}\mathrm{c}\mathrm{m}^{-3}$ when $n_{h}$ is
in a range of
$3\sim$4.7\text{\times}{10}^{11}\text{\,}\mathrm{c}\mathrm{m}^{-2}$$. And when
$n_{h}$ is in a range of
$1.2\sim$3\text{\times}{10}^{11}\text{\,}\mathrm{c}\mathrm{m}^{-2}$$,
$n_{\rm{imp}}$ is between
$1.3\text{\times}{10}^{17}\text{\,}\mathrm{c}\mathrm{m}^{-3}$ and
$1.8\text{\times}{10}^{17}\text{\,}\mathrm{c}\mathrm{m}^{-3}$. This value is
slightly higher than previous publications on InSb quantum wells [59, 46] and
InAs quantum wells [60]. Then, it is worth comparing the mobilities of our
2DHG and 2DEG in the light of Eq. (1) considering the different effective
masses of electrons and holes at
$n_{\rm{e}}=n_{\rm{h}}=$3\text{\times}{10}^{11}\text{\,}\mathrm{c}\mathrm{m}^{-2}$$.
If $n_{\rm{imp}}$ is the same for both 2DHG and 2DEG, this equation predicts
$\mu_{\mathrm{e}}/\mu_{\mathrm{h}}\approx 2.4$. However, the measured mobility
ratio is $\sim 6.2$, showing that the mobility of 2DHG in our InSb quantum
well is lower than this prediction. This is similar to the high-mobility GaAs
2D systems [58, 61], and more investigation is needed to understand the
scattering mechanism in InSb 2D systems quantitatively.

##  V SOI of 2DHG and 2DEG

Figure 5: Density dependence of WAL for both 2DHG and 2DEG. (a) and (b)
present $\Delta\rho_{xx}$ and $\Delta\sigma_{xx}$ vs. $B$ for 2DHG,
respectively. Circles are the data while black curves are fits. (c) and (e).
Same as (a) and (b), but for the 2DEG. There are vertical offsets added in all
the figures here for a clear presentation.

Both the high-quality 2DHG and 2DEG in undoped InSb quantum wells display
tunable SOI, which we characterize by WAL measurements. Figure 5 shows the WAL
measurement at different carrier densities for the 2DHG and the 2DEG at
$1.7\text{\,}\mathrm{K}$, respectively. For the convenience of comparison, we
present $\Delta\rho_{xx}(B)=\rho_{xx}(B)-\rho_{xx}(0)$. To increase the
signal-to-noise ratio, each trace presented represents the average of more
than 5 consecutive measurements. In Fig. 5(a), with the increase of
$n_{\rm{h}}$ from 1.74 to
$3.86\text{\times}{10}^{11}\text{\,}\mathrm{c}\mathrm{m}^{-2}$, the local
maximum of $\Delta\rho_{xx}$ at zero magnetic field develops into a minimum,
which indicates the evolution from weak localization (WL) to WAL. Furthermore,
in the WAL regime, $\Delta\rho_{xx}$ first increases and then decreases with
the increase of the absolute value of $B$. At higher carrier densities, the
turn-over magnetic field of $\Delta\rho_{xx}$ vs. $B$ moves to higher values.
This implies an increase of the SOI with increasing $n_{\rm{h}}$. For the
2DEG, we find WAL throughout the whole range of carrier densities which is
from 1.85 to $4.92\text{\times}{10}^{11}\text{\,}\mathrm{c}\mathrm{m}^{-2}$
(Fig. 5(c)). Here we point out that whether or not the 2D system is in the WL
or WAL regimes mainly depends on the ratio between $l_{\rm{SO}}$ and the mean-
free-path $l_{\rm{e}}$ [55]. In our device, $l_{\rm{SO}}/l_{\rm{e}}$ can be
tuned over a larger range in the 2DHG than in the 2DEG. This is the reason why
the evolution from the WL regime to the WAL regime is not observed in the
2DEG.

Figure 6: (a) $l_{\rm{SO}}$ and $l_{\phi}$ as a function of carrier density,
are presented in (a) and (b) for 2DHG and 2DEG, respectively.

To understand this process in-depth, we extract the coherence length and the
spin-orbit length using the Hikami-Larkin-Nagaoka (HLN) expression [62, 63].
Similar methods were also adopted in publications where WAL effects were
measured in doped InSb quantum wells [35], InSb nanosheets [64], and in our
previous work [46]. As shown in Figs. 5(b) and (d), we convert the $\rho_{xx}$
and $\rho_{xy}$ to longitudinal conductivity ($\sigma_{xx}$) and subtract a
classical polynomial background from the measurement in both p- and n-type
regimes. The conductivity correction $\sigma_{xx}$ of WALs reads:

$\begin{split}\Delta\sigma_{xx}(B)&=\frac{e^{2}}{2\pi^{2}\hbar}\left\\{\Psi(\frac{1}{2}+\frac{H_{\phi}}{B}+\frac{H_{\rm{SO}}}{B})\right.\\\
&+\frac{1}{2}\Psi(\frac{1}{2}+\frac{H_{\phi}}{B}+\frac{2H_{\rm{SO}}}{B})\\\
&-\frac{1}{2}\Psi(\frac{1}{2}+\frac{H_{\phi}}{B})-\ln{(\frac{H_{\phi}+H_{\rm{SO}}}{B})}\\\
&\left.-\frac{1}{2}\ln{(\frac{H_{\phi}+2H_{\rm{SO}}}{B})}+\frac{1}{2}\ln{\frac{H_{\phi}}{B}}\right\\}.\end{split}$
(2)

Here, the two fitting parameters $H_{\phi}$ and $H_{\rm{SO}}$ are the phase-
coherence field and the spin-orbit field, respectively, and $\Psi$ is the
digamma function. The two fitting parameters can be converted to phase-
coherence length $l_{\phi}$ and spin-orbit length $l_{\rm{SO}}$ using
$l_{\phi}=\sqrt{\hbar/4eH_{\phi}}$ and
$l_{\rm{SO}}=\sqrt{\hbar/4eH_{\rm{SO}}}$. We first focus on the 2DHG. As shown
in Fig. 6(a), the phase coherence length $l_{\phi}$ increases from 0.31 to
0.59 $\rm{\mu}$m with the increase of $n_{\rm{h}}$. The reduction of
decoherence agrees with the increase of mobility. In contrast to $l_{\phi}$,
the spin-orbit length $l_{\rm{SO}}$ decreases from 0.37 to 0.18 $\rm{\mu}$m,
which indicates an increase of the SOI with higher $n_{\rm{h}}$.

Like for other systems with strong SOI, we estimate the energy scales of the
SOI with a parabolic band approximation [65, 66]. Since we have determined the
effective mass of the holes, $\Delta_{\rm{SO}}$, the energy splitting at the
Fermi level due to SOI can be calculated according to
$\Delta_{\rm{SO}}=\sqrt{2\hbar^{2}/\tau_{\rm{tr}}\tau_{\rm{SO}}}$. The spin-
orbit time $\tau_{\rm{SO}}$ is calculated from
$\tau_{\rm{SO}}=\hbar/(4eDH_{\rm{SO}})$. Here,
$D=v_{\rm{F}}^{2}\tau_{\rm{tr}}/2$ is the diffusion constant in 2D with
$v_{\rm{F}}$ being the Fermi velocity, which depends on $m^{\star}$. Then, we
normalize $\Delta_{\rm{SO}}$ to get the spin–orbit coefficient
$\alpha_{\rm{SO}}=\Delta_{\rm{SO}}/2k_{\rm{F}}$. The calculation shows that
$\Delta_{\rm{SO}}$ increases from 1.37 to 1.74 meV, while the spin–orbit
coefficient $\alpha_{\rm{SO}}$ is constant at
$\sim$53\text{\,}\mathrm{meV}\text{\,}\mathrm{\SIUnitSymbolAngstrom}$$. The
constant $\alpha_{\rm{SO}}$ is mainly due to the increasing mass. If we
roughly estimate the Fermi energy using
$E_{\rm{F}}=\hbar^{2}k_{\rm{F}}^{2}/2m^{\star}$, the ratio
$\Delta_{\rm{SO}}/E_{\rm{F}}$ increases from 5.2 $\%$ to 8.2 $\%$ in the range
of $n_{\rm{h}}$ measured.

The results for the 2DEG are qualitatively similar to those for the 2DHG.
Here, as shown in Fig. 6(b), $l_{\rm{SO}}$ decreases from 0.66 to 0.23
$\rm{\mu}$m, which implies an increasing SOI with increasing $n_{\rm{e}}$. For
electrons, $\Delta_{\rm{SO}}$ increases from 0.73 to 3.2 meV, and
$\alpha_{\rm{SO}}$ increases from 34 to
$91\text{\,}\mathrm{meV}\text{\,}\mathrm{\SIUnitSymbolAngstrom}$. This value
is approaching the results obtained recently in InAsSb quantum wells [67].
Interestingly, $l_{\phi}$ reaches a maximum of 16 $\rm{\mu}$m and decreases
with higher $n_{\rm{e}}$. This is likely because of the mobility in the 2DEG,
when the electron density $n_{\rm{e}}$ exceeds
$4\text{\times}{10}^{11}\text{\,}\mathrm{c}\mathrm{m}^{-2}$.

##  VI Coincidence measurements of 2DHG

Finally, we present measurements of the spin susceptibility of the 2DHG using
the coincidence method. This method has been introduced in previous
publications [68, 44]. The cyclotron energy is proportional to the
perpendicular magnetic field $B_{\perp}$, while the Zeeman energy is
proportional to the total magnetic field $B_{\rm tot}$, if an isotropic
$g$-factor is assumed. The ratio between these two energies can be
continuously changed by changing $\theta$, the angle between the directions of
sample normal and total magnetic field $B_{\rm tot}$ [inset of Fig. 1(b)].
Therefore, $B_{\perp}$, the projection of the magnetic field onto the sample
normal, is $B_{\perp}$ = $B_{\rm tot}\cos{\theta}$. Here, we introduce the
parameter:

$r=\frac{g^{\star}\mu_{\rm B}B_{\rm tot}}{\hbar\omega_{\rm
c}}=\frac{g^{\star}\mu_{\rm B}B_{\rm tot}}{e\hbar B_{\perp}/m^{\star}}$ (3)

to represent the ratio between Zeeman energy and cyclotron energy ($\mu_{\rm
B}$ is the Bohr magneton). Hence, we obtain
$r\cos{\theta}=g^{\star}m^{\star}/2m_{\rm{e}}$. Whenever $r$ takes integer
values, SdH-minima in $\rho_{xx}$ turn into maxima because the energy gap
between the Landau levels neighboring in energy vanishes. The value of $r$ can
therefore be obtained by following the behavior of the local maxima and minima
of $\rho_{xx}$ for changing tilt angles $\theta$. In general, for example in
InAs and InSb 2DEGs, the value of $\rho_{xx}$ at integer filling factors will
change periodically between local minima and maxima, with increasing $r$ [69,
68]. Knowing $r$ and $\theta$, one can estimate the spin susceptibility to be
$\chi=(g^{\star}m^{\star})/(2\pi\hbar^{2})$.

Figure 7: (a) and (b) The SdH oscillations measured for different tilt angles
$\theta$ when $n_{\rm{h}}=2.9$ and 4.9 $\times 10^{11}$ cm-2, respectively.
For a better presentation, there is a constant vertical offset of 0.05
k$\Omega$ between each trace in (b) but no offset in (a). The filling factors
are labeled with dashed lines.

Figure 7 shows $\rho_{xx}$ as a function of $B_{\perp}$ with $\theta$ varying
from 0 to $\sim 80^{\circ}$ for densities $n_{\rm{h}}=2.9\mbox{ and
}$4.9\text{\times}{10}^{11}\text{\,}\mathrm{c}\mathrm{m}^{-2}$$. The tilt
angle $\theta$ is calibrated precisely (within $0.1^{\circ}$) using the Hall
effect at small $B_{\perp}$. As shown in Fig. 7(a), despite being limited by
the range of the magnetic field, we observe local minima and maxima in
$\rho_{xx}(B)$ to change as $\theta$ is increased. For instance, with
increasing $\theta$, the local $\rho_{xx}$ maxima at $\nu_{\rm{h}}=5$ and 7
change to minima, while the pronounced minima at $\nu_{\rm{h}}$ = 6 and 4
change to maxima or vanish. This means that $r$ is approaching 1 when we
increase $\theta$ from 0 to $78^{\circ}$. As an estimation of the upper limit
of the g-factor, we assume $r\approx 1$ when $\theta\geq 78^{\circ}$. This
corresponds to a g-factor $g^{\star}\leq 14$ for the measured
$m^{\star}\approx 0.03m_{\rm{e}}$. This estimation of the g-factor is in
agreement with the value in bulk p-type InSb, which is $\sim 16$ [24, 54].
Here we also find that the local $\rho_{xx}$ minima for $\nu_{\rm{h}}=4$ and 3
appear in smaller $B_{\perp}$ when $\theta=0$. This could be because of the
effect of attractive scattering centers in the quantum well in the low density
regime [70].

However, coincidence measurements at higher densities show different results.
Figure 7(b) presents the coincidence measurement performed at
$n_{\rm{h}}=$4.9\text{\times}{10}^{11}\text{\,}\mathrm{c}\mathrm{m}^{-2}$$.
Even when $\theta$ is increased to $79.1^{\circ}$, within the range of the
magnetic field we can apply, there is no obvious dependence of $\rho_{xx}$ on
$\theta$. Since p-type InSb is regarded to be a material with strong SOI and
large g-factor, it is unlikely that the g-factor is so strongly reduced in the
higher density regime that there is no observable change of $\rho_{xx}$ up to
$\theta=79.1^{\circ}$. Since we have observed that the SOI of the 2DHG
increases significantly with increasing $n_{\rm{h}}$, we hypothesize that the
spin of the holes is locked with the orbits due to the larger SOI, implying a
strongly anisotropic $g$-factor. Similar results have been reported in other
p-type quantum wells and also in 2D materials, such as MoS2 [71, 72].

##  VII Conclusion

We have shown that both 2DHG and 2DEG can be induced by electrostatic gating
in a single, shallow InSb heterostructure without any intentional doping.
Thanks to the high quality of the sample, we are able to observe SdH
oscillations and integer quantum Hall effects in both the p- and n-type
regimes. In each regime, only a single electronic band is populated and
contributes to transport. From the density dependence of the effective mass,
we find that a large band non-parabolicity is present in the p-type regime.
Furthermore, both 2DHG and 2DEG show tunable SOI. Finally, using coincidence
measurements, we find that the spin susceptibility of InSb 2DHG strongly
depends on the carrier density. Our work provides a better understanding of
the detailed band structure and quantum properties of narrow-bandgap III-V
compounds, and paves the way for applications of InSb in high-mobility
electronics and quantum information processing devices.

###### Acknowledgements.

We thank Marilyne Sousa and the Cleanroom Operations Team of the Binnig and
Rohrer Nanotechnology Center (BRNC), IBM Research Zurich and Peng Zeng of the
ScopeM, ETH Zurich for assistance during TEM characterization. We thank
Shuichi Iwakiri, Lev Ginzburg, and Christoph Adam for fruitful discussions. We
acknowledge funding from QuantERA. Mansour Shayegan acknowledges support
through a QuantEmX travel grant from the Institute for Complex Adaptive Matter
(ICAM) and the Gordon and Betty Moore Foundation through Grant No. GBMF5305.
This work was supported by the Swiss National Science Foundation through the
National Center of Competence in Research Quantum Science and Technology.

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